INFORMATION TO USERS

The most advanced technology has been used to photograph and

reproduce this manuscript from the microfilm master. UMI films the

text directly from the original or copysubmitted. Thus, some thesis and

dissertation copies are in typewriter face, while others may be from any

type of computer printer.

The quality of this reproduction is dependent upon the quality of the

copy submitted. Broken or indistinct print, colored or poor qualityillustrations and photographs, print bleedthrough, substandard margins,

and improper alignment can adversely affect reproduction.

In the unlikely event that the author did not send UMI a complete

manuscript and there are missing pages, these will be noted. Also, if

unauthorized copyright material had to be removed, a note will indicate

the deletion.

Oversize materials (e.g., maps, drawings, charts) are reproduced by

sectioning the original, beginning at the upper left-hand corner and

continuing from left to right in equal sections with small overlaps. Each

original is also photographed in one exposure and is included in

reduced form at the back of the book.

Photographs included in the original manuscript have been reproduced

xerographically in this copy. Higher quality 6" x 9" black and white

photographic prints are available for any photographs or illustrations

appearing in this copy for an additional charge. Contact UMI directly

to order.

U-M-IUniversity Microfilms International

A Bell & Howell Information Company300 North Zeeb Road, Ann Arbor, MI 48106-1346 USA

3131761-4700 800/521-0600

Order Number 9107056

Maximum principles and Liouville theorems for elliptic partialdifferential equations

Zhou, Chiping, Ph.D.

University of Hawaii, 1990

V·M·I300 N. Zeeb RdAnn Arbor, MI48106

MAXIMUM PRINCIPLES AND LIOUVILLE THIDREMS

FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF

THE UNIVERSITY OF HAWAI I IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN MATHEMATICS

AUGUST 1990

By

Chiping Zhou

Dissertation Committee:

Gerald N. Hile, Chairman

Thomas B. Hoover

Marvin E. Ortel

Xerxes R. Tata

Rui Zong Yeh

ACKNOWLEDGEMENTS

I wish to express my sincere thanks to my advisor, Professor

Gerald N. Hile, for his helpful suggestions and for his encouragement

in writing this dissertation. It has been a pleasure working under

the guidance of a leading researcher, who with his tremendous

expertise has kept my research running smoothly.

In addition, I would like to acknowledge all those who, directly

or indirectly, helped with my mathematical education, especially,

Chuanzhong Chen, Zongyi Hou, Mingzhong Li, Robert P. Gilbert, William

Lampe, Wayne Smith, and the members of my dissertation committee.

Finally, I wish to thank my wife Xiaoyu for her understanding and

support during my mathematical endeavours.

iii

iv

ABSTRACT

Part I : Maximum Principles for Elliptic Systems 0

Maximum principles and bounds of solutions for weakly coupled

second order elliptic systems,

Lu + Cu = 0 or f in D C IRn,

are established by using the idea of Liapunov's Second Method. Here

n a2L:= L a. o (x)ax ax +

10 JO -1 1 J 10 JO, -n aL bo(X)ax

i=l 1 i

is a second order real elliptic operator, and the mx1 vectors f, u,

and the mxm matrix C all have entries which are complex-valued

functions. It is shown that:

*If there exists a complex constant matrix B > ° such that C (x)B

+ BC(x) :;;; ° cor « -E < .0, E is a constant matrix) in D, then for all

C2(D)nC(D) solutions u of Lu + Cu = ° (or = f),

Ilullo,D:;;; K1l1ullo,aD (or Ilullo,D:;;; K11lullo,aD + K2I1fllo,D).

(P 1/2 2K

1Here K = __!!!..J and K = -- , where PI and f3 are the smallest1 PI ) 2 J.1.1 m

and biggest eigenvalues of B, respectively, and J.1.1

is the smallest

eigenvalue of EB-1.

Part II Schauder Estimates and Existence Theory for Entire Solutions

of Linear Fourth Order Elliptic Equations 0

Solutions in IRn (n ~ 2) of the linear, fourth order, variable

coefficient, elliptic equation,

and the

n 84p}: aiJ'kl ax ax ax ax +

i,j,k,l=l i j k I

n 82p n f!!!..-+ }: c . . ax ax + }: d . ax + e,p = f,

i,j=l 1J i j i=l 1 i

related homogeneous equation are investigated. An entire

v

solution is a solution of the equation defined in all 'IRn ,

Schauder-type a priori estimates are developed for entire solutions

with prescribed behavior at infinity. These estimates, of perhaps

independent interest themselves, lead to an existence and uniqueness

theory for entire solutions, with certain behavior required at

infinity, when the operator L can be separated as L = L2L 1

" Here L1

and L2

are two second order elliptic operators.

Under some suitable conditions that guarantee the operator L

approaches the biharmonic operator a2 near infinity at a certain

rate, it is shown that there exists a unique entire solution in IRn

(n ~ 2) of Lr/J = f (f(x) = O(lx\-4-E) near infinity), such that

4-n~(x) - L 1 .Dsr(x) ~ 0 as x ~ 00,ss=o

for some constant number 10

, constant vector ~l and constant symmtric

matrix 12

, Here r is a fundamental solution of the biharmonic

equation, Moreover, a one-to-one correspondence between the entire

4-nsolutions of Lr/J = f, with ,p(x) - }: 1 .Dsr(x) = O(lxl m) nearss=o

infinity, and the biharmonic polynormials of degree no greater than m

is established. When f =0, the result is an extension of a

Liouville-type theorem.

TABLE OF CONTENTS

Acknowledgements

Abstract

Part I. Maximum Principles for Elliptic Systems

1. Introduction

2. Notation and a Liapunov Stablity Theorem

3. Maximum Principles .....

4. Uniqueness Theorems for Some Boundary Value Problems

5. Estimate of K

6. Bounds for Solutions of the Nonhomogeneous Systems

7. Applications to Higher Order Equations ..

References for Part I . . .

iii

iv

1

2

4

6

13

17

24

28

32

vi

Part II. Schauder Estimates and Existence Theory for Entire

Solutions of Linear Fourth Order Elliptic Equations 35

1. Introduction. 36

2. Norms and a Schauder Estimate for Entire Solutions 38

3. Bounds for Entire Solutions of the Nonhomogeneous

Biharmonic Equation . . . . . . . . . . . 47

4. An A Priori Bound for the Fourth Order Elliptic Equation 63

5. Existence of Solutions

References for Part II

83

94

PART I

MAXIMUM PRINCIPLES FOR ElLIPTIC SYSTEMS

1

1. INTRODUCTION

*Positive definite solutions B of the matrix equation C B + BC =

-E (E > 0) have been successfully used to construct Liapunov

functions, and then to prove the stability of some ordinary

2

differential systems dudx = Cu (cf: [11], [24]). This method usually

is called Liapunov's Second Method. In 1974, Chow and Dunninger [2]

applied this method to the study of n-metaharmonic functions, and

obtained a generalized maximum principle for some classes of

n-metahamonic functions.

In this paper, we transfer the idea of Liapunov's second method

to the study of weakly coupled second-order elliptic systems

Here

Lu + Cu = 0 or f in D C IRn .

a2 8L == a .. (x)ax ax + a. (x)-a '

IJ ., 1 X.1 J 1

a .. = a ..IJ J 1

Tis a second-order real elliptic operator, and f = (f1

, .,. ,fm) ,T'

u = (u1

' .. , ,urn) ,and the mxm matrix C all have entries which are

complex-valued functions.

We establish the following generalized maximum principle for

certain classes of the homogeneous system (Theorem 3.1);

If there exists a complex crnstant matrix B > 0 such that

C*(x)B + BC(x) ~ 0 in D, then for all C2 (D) nC(D) solutions u of

Lu + CU = 0 ,

lI ull o,D ~ K lIullo,8D .

Here K ((3(3m ) 1/2 , where (3 and (3 are the smallest and biggest

1 m1

eigenvalues of B, respectively.

We also find a simple sufficient condition for the classical maximum

principle (K = 1 in the above inequality) holding; this condition is

*C (x) + C(x) ~ 0 in D (Theorem 3.3). These results extend the result

of Winter and Wong [23] for C being negative semidefinite to a more

general form of C. Generalized maximum principles for weakly coupled

3

in D C IRn

second-order elliptic systems have also been obtained by Dow [3], Hile

and Protter [8], Szeptycki [21] and Wasowski [22] under different

conditions on the coefficients.

We further show how our maximum principles may be used to prove

the uniqueness of various boundary value problems of some classes of

elliptic systems over bounded or unbounded domain D C ~n. By using a

recent result of Hile and Yeh [10], we even obtain uniqueness for a

boundary value problem with an exceptional boundary set r c an with

the Hausdorff dimension of r less than n - 1.

An estimate of the best possible K in our maximum principle

inequality is given when C is a 2 by 2 real matrix. The condition for

the classical maximum principle (K = 1) holding can be written as

Re{a} ~ 0, Re{d} ~ 0, Ib + cl 2~ 4Re{a}Re{d}

when C = [:~~~ ~~~~] is any 2 by 2 complex-valued matrix function.

We also study the nonhomogenuous system. Miranda [13) has

studied the weakly coupled real elliptic system

n auLU+L~ak+Cu=f

k=l

giving a rather complicated condition which implies the bound for

solutions u,

4

When all the matrices ~ = 0, k=1, ••• ,n, the condition required by

Miranda reduces to eTce ~ -colel2 for anye E Rm. We will extend

this result to C being a complex matrix function such that

*C + C ~ -2coI (Corollary 6.3). Moreover, we prove the following

(Theorem 6.1):

*If there exist B > 0 and E > 0 such that C (x)B + BC(x) ~ -E in

D, then for all C2(D) nC(D) solutions u of Lu + Cu = f ,

Here K] =

eigenvalue

(:~ f/2-1

of EB .

and 2 (/3 )1/2K2 =~~ , where ~1 is the smallest

1 1

Results for systems are later used to yield maximum principles

and bounds for some higher order elliptic homogeneous and

nonhomogeneous equations. Our maximum principles include those of

Chow and Dunninger [2], [6] for real metaharmonic functions as a

special case. Various maximum principles for higher order elliptic

equations were also studied in the papers of Agmon [1], Duffin [4],

[5], Fichera [7], Payne [14], Scheafer and Walter [16], [17] and the

books of Miranda [12] and Sperb [19].

2. NOTATION AND A LIAPUNOV STABILITY THEOREM

Unless otherwise stated, all matrices considered in this paper

will be over the complex field. Let X be any mxn matrix. Its

transpose, complex conjugate and adjoint will be denoted by XT, Xand

* *-;;Tx (X =x ), respectively. For the sake of brevity, both Hermitian

positive definite and real symmetric positive definite matrices will

be named positive. Similar abbreviations hold for semipositive,

negative and seminegative definite matrices. The notations B > 0,

B ~ 0, B < 0 and B ~ 0 mean that the square matrix B is positive,

semipositive, negative and seminegative, respectively. The norm

11-110 D means the sup norm over D; thus for complex-valued vector,

I 12 I 12 1/2lIullo D ;= sup [uf x) I = sup ( u1(x ) + --- + um(x) .

'xeD xeD

The following well~known result in Liapunov stability theory will

be applied several times in this paper. We state this result as a

Lemma.

LIAPUNOV LEMMA. Let C be an mxm complex or real matrix.

(a) Assume that no eigenvalue of C has positive real part, and

moreover that the elementary divisors of C corresponding to

eigenvalues with vanishing real part are linear. Then there exist

*matrices B > 0 and E ~ 0 such that C B + BC =-E;

(b) If each eigenvalue of C has negative real part, then for any

*E > 0, there exists a unique B > 0 such that C B + BC = -E.

The proof of this Lemma, both for real and complex versions, can

be found in many papers, such as [11], [18] and [24].

5

3. MAXIMUM PRINCIPLES

Consider a second-order operator,

6

Here the summation convention is

(3.1)

in a bounded

a2

L=a .. (x)axalJ . X.

1 J

domain D in IRn .

a+ a.(x).:l.. ,1 VA.

1

a .. = a .. ,lJ J 1

employed. We assume that L is elliptic in D; i.e., for all XED and

(3.2)

holds.

a .. (x)y.y. > 0lJ 1 J

We also suppose that the coefficients a .. and a. are boundedlJ 1

and real-valued functions in D.

Now consider the following weakly coupled second-order elliptic

system,

Lu (x) + c 1 (xru, (x) = 0sSt K

or, in more brief matrix form,

s=1,2, ... ,m, in D ,

(3.3) Lu(x) + C(x)u(x) = 0 in D .

Here C(x) = (csk(x)) is an mxm complex matrix function and u is a C2

mx1 complex vector function. Associated with (3.3) is the following

characteristic equation of C,

(3.4) h(A) - IAI-ci = 0 .

THEOREM 3.1. Assume that there exists a constant matrix B > 0 such

that

(3.5) *C (x)B + BC(x) ~ 0 , XED .

Then for all C2 (D) nC(D) solutions u of (3.3), there exists a

positive constant K such that

(3.6) Ilullo D s K Ijullo an, ,

--------------------- ._----

7

f3 1/2Here K = (pm) ,where PI and Pm are the smallest and biggest

1

eigenvalues of B, respectively.

Proof: Define

where "."

*v = u Bu =ueBu = Bu.u =bksukus '

is the dot product in ~n defined by x.y = y*x

Then v is a nonnegative function and,

etc. ;

2_ a uk= ax.ax.

1 J~,ij

v . =~. = bk uk·u + bk uku .,1 VAl S ,1 S S S,1

_ a2vv .. = ax ax = bk Uk ., u + bk uku .. + 2Re{bk uk .u . }

,IJ ., S, IJ S S S, IJ S ,1 S, J1 J

aukuk . =ax.,1 1

where

and,

Lv = a .. v .. + a. v .IJ ,IJ 1 ,1

= bk a. 'Uk ..u + bk uka ..u .. + 2bk a ..uk.u .SIJ ,IJS S IJS,IJ SIJ ,IS,J

+ bk a,uk .n + bk uka.u .S 1 ,1 S S 1 S,1

= bk (Luk)u + bk uk(Lu ) + 2a ..u .•Bu .S S S S IJ ,1 ,J

* * 1/2 1/2= (Lu) Bu + u B(Lu) + 2a ..B u .•B u.IJ ,1 ,J

Thus,

(3.7)

since

vn

* * 1/2 1/2Lv = -u (C B+BC)u + 2a ..B u .•B u. ~ 0 ,1 J ,1 ,J

*C B + BC ~ 0 and a .. v .•v. ~ 0 for any vectors vI' v 2, .. , ,1 J 1 J

Therefore, by the maximum principle for the elliptic operator L,

we have

(3.8) vex) ~ max v(y)yean

for all xeD.

Suppose that {Pi}~=1 are the eigenvalues of B with PI ~ P2~•••~ Pm .

Since B > 0, we know that PI > ° and

P1Iu(x)12 ~ v(x) = u(x)*Bu(x) ~ Pmlu(x)12 .

Hence, from (3.8),

2 Pm 2[utx) I ~ -P- max [uf y) I ,

I ydJD

8

where K -_ ( ~ml J1/2and consequently, lIullo,D ~ K lIullo,aD ' fJ )

The Liapunov Lemma yields the following corollary.

I

COROLLARY 3.2. Let C(X) = r(x)I + C in (3.3), where r(x) ~ 0 in D

-and C is a constant matrix over the complex field. Assume that none

-of the eigenvalues of C has a positive real part, and moreover that

-the elementary divisors of C corresponding to eigenvalues with

vanishing real part are linear. Then there exists a positive constant

K such that for all C2(D) nC(n) solutions u of (3.3),

(3.6) Ilullo D ~ K Ilullo aD ., ,

Proof: By the Liapunov Lemma in Section 2, there exist matrices B > 0

-* -and E ~ ° such that C B + BC = -E ~ o. Since r ~ 0 in D, we get

* -* -C (x)B + BC(x) = 2r(x)B + C B + BC ~ ° .Now the result of this corollary follows from Theorem 3.1. I

Remark. By the Liapunov Lemma, we know that at least one positive

-* - -definite B, satisfying C B + BC = -E ~ 0, exists if the matrix C meets

-the assumption of Corollary 3.2. In fact, if C satisfies the condition

of Corollary 3.2 and has at least one eigenvalue with vanishing real

part, then there is an infinite number of positive definite B such

-* -that C B + BC = -E ~ 0 (see [11] and [18]).

Theorem 3.1 and Corollary 3.2 are generalized maximum principles

since the value of K in (3.6) may be larger than 1. The best

conceivable value of K in (3.6) , for any matrix C, is K = 1 , which

corresponds to the classical maximum principle.

THEOREM 3.3. (The Classical Maximum Principle)

(a) A sufficient condition that

9

(3.6)1

holds, for all

(3.9)

lIullo D s lIullo an, ,C2 (D) nC(D) solutions u of (3.3), is

*C (x) + C(x) ~ 0

(b) Assume that the variable matrix C = C(X) in (3.3) is normal

* *(i.e., C (x)C(x) = C(x)C (x), x E D), and all its eigenvalues have

nonpositive real parts for all XED. Then (3.6)1 holds for all

C2(D) nC(D) solutions u of (3.3).

Proof: (a) By choosing B = I in Theorem 3.1, (3.6) with K = 1

(i.e., (3.6)1) follows from the condition (3.9).

(b) Suppose

A1

(x) , A2

(x ) , ••• , Am(X)

are all the eigenvalues of C(X). Since C(X) is normal, there exists a

unitary matrix U(x) such that

and then (3.6)1 follows from (a).

2ReoX (x)m

*u (x)C(x)U(x) =

Therefore, by the assumption,

* *U (x)(C (x) + C(x»U(x) =

* *Hence C + C =UAU ~ 0

••

2Re\ (x)••••••

•oX (x)m

-. A(x) ~ o.

I

10

Remarks. 1. The condition (3.9) is also a "necessary" condition for

the proof of the classical maximum principle by the method imposed

here. In fact, if, in Theorem 3.1, (3.6) holds with K = 1, then P1

=

Pm' and so there exists a B > 0, with an m-multiple eigenvalue P > 0,

* *such that C B + BC ~ 0; hence B = PI, and then C + C ~ o.

2. Theorem 3.3 contains the result of Winter and Wong [23] for

real negative semidefinite C =C(x, u, Vu) as a special case; one

may view, for given u, C(x, u(x), Vu(x» as a matrix function C1(x).

EXAMPLE 3.4. For n = 2 , consider

Lu + [: ~] u = 0 , a, b, c, d E IR .

The associated characteristic equation,

oX2- (a+d)oX + (ad-bc) = 0 ,

has roots

Hence, by Corollary 3.2, the inequality (3.6) is valid provided one of

_ .. _.._--------- -_._---- ----

The inequality (3.6) is not valid for the general case, when

a + d > 0 or a + d = 0, ad - be ::;:; O. In fact, u = sinx Siny[_~J

solves the systems

and

~u + [_~ - ~] u = 0

in D = (O,~)x(O,~) and vanishes on an, but (3.6) does not hold.

Theorem 3.1 gives a sufficient condition for which (3.6) holds.

It raises some open questions as to whether Theorem 3.1 can be

extended to a more general system (3.3) with weaker restrictions on

11

the matrix C, and as to whether necessary conditions can be determined

so that (3.6) holds.

(3.7)

in the

Following from the inequality

* * * 1/2 1/2L(u Bu) ~ -u (C B + BC)u + 2a .. B u .•B u. ~ 0 ,1 J ,1 ,J

proof of Theorem 3.1, and from Protter and Weinberger's book

[15], are the following two maximum principles for system (3.3).

COROLLARY 3.5. If u E C2 (D) nC(D) is a solution of (3.3), and if

*u Bu attains a maximum in D for some positive definite matrix B such

*that C (x)B + BC(x) is negative semidefinite in D, then u is a complex

*constant vector in D. Moreover, if C (x)B + BC(x) is negative

12

which implies that

1/2and then, Bu. =,1

definite at some xED, or, if C(x) is invertible for some XED, then

u == 0 in D.

Proof: Under the assumption of this corollary, by the proof of

Theorem 3.1, inequality (3.7) holds. Thus, by the maximum principle

*for the second-order elliptic equation (see [15]), u Bu == constant.

Hence, from (3.7) again, we have

* * * 1/2 1/2o = L(u Bu) = -u (C B + BC)u + 2a ..B u .•B u. ~ 0 in D;1 J ,1 , J

* * 1/2 1/2u (C B + BC)u = 0 and a ..B u .•B u. = 0IJ ,1 , J

o and u . = 0 in D, for 1 ~ i ~ n Thus u is a,1

*complex constant vector in D. Moreover, if C (x)B + BC(x) < 0 at

* *some XED, then, from u (C (x)B + BC(x))u = 0 , we have u == 0 in D;

and if C(x) is invertible for some xED, then, from the system (3.3),

we still have u == 0 in D. I

Note that Corollary 3.5 actually holds even if D is unbounded.

COROLLARY 3.6.2 nC(l)) be a solution of (3.3). SupposeLet u E C CD)

* *that u Bu ~ M in D and that u Bu = M at a piont P E an for some

*positive definite B such that C B + BC is negative semidefinite.

Here M is a nonnega~ive constant. Assume that P lies on the boundary

of a ball in D, and that the outward directional derivative ~ exists

at P. Then

unless u is a complex constant vector such that

*a(u Bu)av = 2Re[u*B%;] = 2Re[(B1/ 2u)* a(~:/2u)J > 0

*u Bu == M

at P

equivalently,

o1B1/ 2ul

OV > 0 at P

unless u is constant and IB1/2ul =M1

/2

4. UNIQUENESS THEOREMS FOR SOME BOUNDARY VALUE PROBLEMS

As applications of the maximum principles in section 3, we can

prove uniqueness theorems for various boundary value problems.

As an example, consider the first boundary value problem for the

elliptic system (3.3). By Theorem 3.1, the problem

13

Lu(x) + C(x)u(x) = f(x)

ulan = g(x)

xED

where C satisfies the assumption of Theorem 3.1, has at most one

solution.

As a second example, we have uniqueness for the following mixed

boundary problem:

Lu(x) + C(x)u(x) = f(x)

. ufx) = g_ (x)

t ~(x) + :(x)u = g2(x)

in D C IRn

on r 1 '

on r 2 '

where v = vex) is a given outward direction on r2 ' and r2 = OD\r1

COROLLARY 4.1. Suppose u1 and u2 satisfy (3.3)N and (4.2) in a

bounded domain D C IRn and C satisfies the assumption of Theorem 3.1.

Assume that each point of r2

lIes on the boundary of a ball in D. If

1 2L is elliptic as defined by (3.1) and a(x) ~ 0 on r2

' then u =u ,

except when a = 0, r1

is vacuous and C(x) is singular for all xED,

. hi h 1 2. I tIn w IC case u - u IS a comp ex cons ant vector.

Proof: Define 1 2 Then u satisfiesu =u - u

(3.3) Lu + au = 0 in D

{:= 0on r

1,

(4.3)0 on r

2- + au =iJv

Choose positive definite B, as in the proof of Theorem 3.1, such that

* *C B + BC is negative semidefinite, and let v = u Bu ; then (3.7)

holds. If v is not a constant, by Corollary 3.5, a nonzero maximum of

14

v must occur at a point P on r2

;

Re[u*B~]

Hence,

and by Corollary 3.6,

> 0 at P.

Re [u*B(~ + au)] > 0 at PEr2 '

which contradicts the boundary condition (4.3). Thus *v = u Bu must

be a constant. From (3.7), by the proof of Corollary 3.5, u must be a

complex constant vector. Then, from (3.3) and (4.3), we know that

O. 1 2u= ,I.e., u =u in D, except when a = 0, r

1is vacuous and C(x)

is singular for all xED. I

Besides having applications to some boundary value problems with

bounded domain D, the maximum principle we obtained in section 3 can

also be used to establish uniqueness theorems for various boundary

value problems with unbounded domain D eRn In this case, the point

00 is called the exceptional boundary point, and an appropriate growth

restriction on the solution at 00 is required. Moreover, by using a

recent result of Hile and Yeh [10], we even obtain uniqueness for the

boundary value problem with an exceptional boundary set r such that

the Hausdorff dimension of r is less than n - 1.

As a third example of this section, we have uniqueness for the

following boundary value problem with unbounded domain D eRn

15

Lu(x) + C(x)u(x) = f(x) in D ,

{

u(x) = g(x) on an(4.4)

lu(x)1 is bounded in D .

COROLLARY 4.2. Let D be an unbounded domain contained inside a cone,

let L be given in D by (3.1), with

a.(x) = o(r-1) as x ~ 00 in D, i = 1, ... , n, (r = Ixl),

1

and with the uniform ellipticity condition

nxeD,yeR,

holding for some positive constants 8 and A. Suppose u1 and u2

satisfy (3.3)N ' (4.4) in D and matrix C satisfies the assumption of

Th 3 1 Then U1 =u 2 . Deorem . . In .

Proof: 1 2 *Let u = u - u and v = u Bu, where B, as in Theorem 3.1, is

*positive definite such that C (x)B + BC(x) is negative semidefinite

for all xeD. Then v is nonnegative, and the proof of Theorem 3.1

gives

* * 1/2 1/2-u (C B + BC)u + 2a ..B u.B u ~ 0IJ

= 0 on an,is bounded in D.

in D,

Hence, by the Phragmen-Lindelof Principle ([9], Corollary 2),

lim vex) = 0 , as x ~ 00 in D.

16

Therefore, the maximum principle (Theorem 3.1) implies that v =0 in

D · u1 =u

2; 1. e., in D. •

Remark. When L =~ (Laplace operator) in (3.3)N ' the growth

restriction of lui being bounded in Corollary 4.2 can be replaced by

the following weak restriction:

max [uf x) 12

1 · . f Jxl=r 0nn In =r~ log r

or

1~I~r lu(x)12

lim inf . . = 0n-2r~ r

(see [15]).

if n = 2 ,

if n;:: 3

As a last example, we prove uniqueness for the solution of the

following boundary problem without knowing the data on an exceptional

boundary set reaD (the domain D can be unbounded):

(3.3)N (~ + a i (x)~Ju(x) + C(x)u(x) = f'(x) in D,1

{

u(x) = g(x) on aD \ r ,(4.5)

\u(x)1 is bounded in D .

COROLLARY 4. 3 .1 2 2Suppose u and u are two C solutions of the problem

(3.3)N, (4.5); and assume that \ai(x)\, 1 ~ i ~ n, are bounded in D

and the matrix C(x) satisfies the assumption of Theorem 3.1. Let r be

a subset of aD such that, for each y on r, D is contained on one side

of an (n-1)-dimensional hyperplane passing through y. Assume also that

the Hausdorff dimension of r is less than n - 1. Then u1 =u

2in D.

Proof:

17

Let u = u 1- u 2 and v = u*Bu where B, as in Theorem 3.1, is

*positive definite such that C (x)B + BC(x) is negative semidefinite

over D. Then v is nonnegative, and the proof of Theorem 3.1 implies

that

{

(~ +

v(x)

v(x) ~ M in D, for some M> 0 .

Hence, by a result of Hile and Yeh ([10], Corollary 2),

in D,

Therefore,

v == 0

v == 0

on r.

on an.Thus, by the maximum principle (if D is bounded) or Phragmen-Lindelof

principle (if D is unbounded) for the second order elliptic equation,

we have v == 0 in D. This gives u1== u2 in D.

5. ESTIMATE OF K

I

P 1/2In Theorem 3.1, the constant K = ( pm ) in (3.6) depends on C;

1

in fact, (3 and (3 are the smallest and largest eigenvalues of B,1 m

where B > 0 is chosen so that

C (x)B + BC(x) ~ o.

The value of K in (3.6) is important in applications, such as in

numerical computation and estimation. The best conceivable value of

K, for any C, is K = 1, which corresponds to the classical maximum

principle; and Theorem 3.3 gives a sufficient condition (3.9) to

guarantee the classical maximum principle (K = 1) for solutions of

(3.3). However, for a general matrix C, the best possible value of K

can be larger than 1.

EXAMPLE 5.1. For the system

Luf x) + [:~~~ ~~~nu(x) = 0 ,

where a, b, c and d are complex-valued functions, by Theorem 3.3 the

classical maximum principle (3.6)1 holds if

18

[a b] * + [a b] = [2Re{~}cdc d (b+c)

which is equivalent to

b+c ] s 0 ,2Re{d}

(5.1) - 2Re{a} :;:; 0, Re{d} :;:; 0; and Ib + cl :;:; 4Re{a}Re{d}.

The following two examples can be used to compute the best choice

(f3 )1/2 [ b]of K = \ f3: when C = : d is a real constant matrix with

eigenvalues A :;:; A :;:; 0 (A , A not both zero).- + +-

EXAMPLE 5.2. Consider

(5.2) c E IR .

In order to determine the best K, we choose a positive definite

B := ~ ~J which minimizes

(5.3) K = ( o:+"y) +

(0:+')') -

A/ (a-y)2

I 2Iv· (a-,)

and satisfies

Without lost of generality, we can assume that

19

(5.4) a+1=1.

Then the problem can be reduced to the following equivalent problem:

Find (a, P) E R x ~ which

minimizes (2a-1)2 + 41PI 2

subject to: a > a2 + IpI 2

ECP+{i) s 2 ,

a2( E

2+4) + 41pI 2- 4a :;;; 0 .

Obviously, P = 0 is the best. Thus the problem reduces to

minimize (2a - 1)2

subject to: a > a2

a 2( E

2+4) - 4a :;;; 0 .

It is easy to conclude that

{' if lEI :;;; 2 ,('2,0)

(a,pl = [4 ~E 2 ,0) , if lEI > 2

K -- (~m)1/2Hence by (5.3) and (5.4), the best value of ~ for the1

system (5.2) is

{

1 ,K = ill

2 '

if lEI:;;; 2 ,

if lEI >2

EXAMPLE 5.3. Now we consider the generalized form of (5.2),

( 5 . 5) Lu + [- ~ _~] u = 0 , o ;:;: 0, E E IR

By going through the same process, we can reduce the problem of

minimizing K in (3.6) to finding B = ~ 1~a] > 0 , which

minimizes (201.-1)2 + 41PI 2

subject to: a > 0 ,

a - 01.2

- IP12

> 0 ,

(u-1) (p+p)aE + 4u(a-a2) - a2E - (1+u)2IPI 2 ~ U .

20

Using lagrange's method of multipiers and after going through a rather

complicated computation, we get

1(2,0)

2, E ::;40-,

(a,p)=

Hence we

is

«1+U)2 E2 + 2vCU (1+U)V/E2[E2+(U-l)2] (E2-4U)(U-1)E) E2>4u.

2{ E2(E2+(1_u)2) + 2VU (1+U)\tIE2[E2+(1-U)2] }

find that the best choice of K in (3.6) for the system (5.5)

1 2, E :;;;40-,

K =

[

E2[E2+(1-u) 2]

E2[E2+(1_u)2]

For a general real matrix C = [: ~] with eigenvalues A_ ~ A+ ~ 0,

A < 0, by matrix theory (see [20], Chapter 5), there exists an

orthogonal matrix P := [~ ~] and a real number EO such that

[A_ EO] = pTcp = [p~a+prc+prb+r2d Pia+rqc+sPb+S~d] .o A+ pqa+psc+rqb+rsd q a+qsc+qsb+s d

From 0 = pqa+psc+rqb+rsd, we have

Let

EO = pqa+rqc+spb+srd = (ps-rq)(b-c) = detP.(b-c) = ±(b-c).

TB = PBl. Then

CTB + Be = P { [> ::]\ + B1 [> ::]} pT

s 0

is equivalent to

21

or

EO _ b-cBy letting E = - -A--- = + ---A--- and

5.3 we have the following:

~ ° ,

~ f~ ]s 0 .

A+

U =-A---' from the Example

f3 1/2Conclusion: The best value of K = ( f3~ ) for C = [: ~] E IR2X2

in (3.3) with eigenvalues

(a+d)± A/ (a-d)2+4bcA± = 2 ~ 0, and (a+d) < 0,

is

1

if (b_c)2 ~ 4detC ;

Remarks. 1. By using

AA = deW ,- +

A + A = trC ,- +E~ = (b_c)2 = a2 + b2 + c2 +d2 _ (a+d)2 + 2(ad-bc)

= (trCTC)2 - (trC)2 + 2detC ,

(A - A )2 = (trC)2 - 4detC ,- +

we can express K in terms of the matrix C.

2. The condition for the classical maxim~m principle (K = 1) is

(b_c)2 :;; 4detC ,

and the equivalent condition for it is

(b+c)2 :;; 4ad (which is the same as (5.1)),

or

It is clear that not only the normal matrices, but also some other

matrices satisfy one of the above conditions.

22

In case C -- [ac

bd] E 1R2X2 h t 1 • t' 1as wo comp ex conJuga e elgenva ues

with nonpositive real parts,

(5.6) '± = (a+d) ± v/-2(a-d)2-4bC i1\ - : = J.L ± vi

( J.L,vEIR, )lp,:;;0, V>o

the problem of finding the best choice of K in (3.6) is very

complicated. The method we used above for real eigenvalues does not

succeed. Here we can only give a formula for an upper bound for the

best possible K.

From (5.6), we have

2J.L =a+d :;; 0,

2v = A/-(a-d)2-4bc > o.

Suppose that e .- a+ip E ~, (when 1>=[::]' p=~:] E JR') is an

eigenvector of C corresponding to the eigenvalue A =J.L+vi.+

Then

23

C(a-ip) = (~-vi)(a-ip),

which gives

or,

Ca = ua -vp,

Cp = J.Lp + va,

in matrix form C(a P) = (a P) [ J.L VJ .-v J.L

The general solution for (a P) in the above equation is

(5.7)

i. e.,

(5.8)

a-:j! b= .fJ +.:::.au ": V2'

_ .£. d-y- ~l + II 132 '

for any PI' P2 E ~\{o}.

Let

then

Since II > 0 and bc ~ 0, we see that a and p are independent over

~2. Hence the matrix (a P) is nonsingular and

C = (a P)[ J.L VJ(a p)-l-II J.L

B = (a p)-lTBl(a p)-l > 0;

CTB+BC = (a p)-lT[ J.L II]T(a p)-lTB (a p)-l-II J.L 1

+ (a p)-lTBl(a p)-l(a p)r J.L VJ(a p)-lI.-V J.L

Hence,

= (IV fJ)-lT( [ ~ IIJTB B [ J.L PJ ) ( fJ)-l.... ,... -II J.L 1 + 1 -II J.L a ,... .

CTB + BC :;:; 0 iff

(5.9)

The inequality (5.9) holds for Bl = I, and in this case,

B = (a p)-lT(a p)-l = [(a p)(a p)T]-l > o.

The characteristic equation for (a p)(a p)T is

--- - - - --- ------

2 2

[11 [ ,x-(al+Pl) -(ala2+PlP2)]

det ,xI-(a p)(a P)-J = det -(a a +P P) ,x_(a2+p2)1 2 1 2 2 2

= ,x2 _ tr(a p)(a p)T,x + det(a p)(a p)-l

\2 (2 2 2 2 \ 2= A + al+a2+Pl+P2)A + (a lP2-a2Pl)

Let .xl , ,x2 (0 < ,xl ~ ,x2) be the two eigenvalues of (a p)(a p)T.

-1 \-1Then the eigenvalues of Bare 0 <,xl ~ A 2 ' and

2 222 /22 222 2(a l+a2+Pl+P2) + V. (a l+a2+Pl+P2) - 4(alP2-a2Pl)

2222 /22222 2(a l+a2+Pl+P2) - V (a l+a2+Pl+P2) - 4(alP2-a2Pl)

By (5.8), we have

2 2 2 2 1 I II 2 21a l + a2 + PI + P2 =~ b-c -cPl+(a-d)p P2+bP ,v 1 2

1 2 2(a

lP

2- a

2P

l) = v(-cPl+(a-d)PlP2)+bP2)

Hence, by choosing Pl,P2E ~\{O} such that

-cp~ + (a-d)PlP 2 + bP~ ~ 0 ,

we obtain

1 [b-e] 1 / 2 2+- Iv. (b-c) -4v 1/(b+c) 2«(a-d) 22 v 2=~+(5.10) K2 v=

1 [b-e] 1 / 2 2 \b-cl - / 2 2-- Iv. (b-c) -4v Iv. (b+c) +(a-d)2 v 2V

Since the choice of Bl = I in (5.9) may not be the best choice,

the formula (5.10) for K2 is not the best one. From formula (5.10),

we obtain K = 1 when a = d and b = -c .

6. BOUNDS FOR SOLUTIONS OF THE NONHOMOOENEOUS ELLIPTIC SYSTEM

In section 3 we obtained maximum principles for some homogeneous

elliptic systems. Now we extend these results to the nonhomogeneous

24

25

elliptic system

Lu(x) + C(x)u(x) = f(x) , xED.

Under the restriction of C being a real matrix function such that

eTce ~ -colel2 in D, for some Co > 0 and for anye E Rm, Miranda [13]

obtained a bound for solutions of the elliptic system (3.3)N:

Ilullo D ~ Ilullo an + c~lllfilo D ., , ,

In this section, we obtain a similar result for more general complex

matrix functions C in the elliptic system (3.3)N

THEOREM 6.1. Suppose, for C =C(x), there exist two complex constant

*matrices B > 0 , E > 0 such that C (x)B + BC(x) ~ -E. Then for

all C2(D) nC(D) solutions u of (3.3)N' there exist positive constants

K1 and K2 such that

(6.1)

2 l/2m

1J1 12ILl'"1

, where PI and Pm are the

smallest and biggest eigenvalues of B, respectively, and ILl is the

-1smallest eigenvalue of EB .

Proof: Define 1/2v=B u; i.e. ,-112

U = B v .

First we will prove that (for v),

(6.2) Ilvllo D s IIvllo an + K IIfllo D, , , K > 0 .

It is sufficient to show that at an internal relative maximum of Ivl

(or of Ivl 2 = IBl / 2ul 2

= Bueu) ,

(6.3) [vtx) I s Klf(x) I

we have

= 0 ,

We assume v(x) ~ 0 at such a maximum; otherwise the inequality is

trivial.

2At a relative maximum of Ivl =Bueu,

a I 2ax vi = 2Re(Bu.u k)k '

and the matrix of second derivatives is negative semidefinite; i.e.,

[~ ] [ ]ax.ax = 2Re(v i·v k + v.v ik) ~ 01 k '" nxnnxn

Hence,

26

Since

Re(Bu.u k) = Re(v.v k) = 0 ,, ,

[Re(v .•v k + v-v 'k) ] ~ 0 0

,1, ,1 nxn

(a ik) > 0, we have that

a·k(v .•v k + Re(v.v ok)) ~ 0 °1 ,1 , ,1

Into this inequality we substitute the system (3.3)N and

Re(Bu.u k) = 0 to get,o ~ a·kv oeV k + Re(Bu.aoku 'k)

1 ,1 , 1 ,1

=a·kv .•v k + Re[Bu.(-a.u . - Cu + f)]1 ,1 , 1 ,1

1 *=a·kv .•v k - -2(0 B+BC)ueu + Re(Bu.f) .1 ,1 ,

Therefore,

Since

that

1 *~ -Re(Bu.f) ~ a·kv .•v k - -2(0 B+BC)u.u

1 ,1 ,

1 1 -1/2 -1/2~ 2 Eu.u =2(B EB )VeV

and EB- 1 have the same eigenvalues, it follows

f.J.p1 / 2 1vl If I ~ 1 Ivl 2

,m 2

and then (6.3) holds with

K =ILl

Hence we have proved (6.2).

27

By using (6.2) and substituting 1/2v = B u, we have

so (6.1) holds with

_ (,BmJ

l / 2

K1 - l ,01and

2 ,01/2m I

From the Liapunov Lemma, the following corollary is easily

obtained:

-COROLLARY 6.2. Assume that C(x) = r(x)I + C in (3.3)N ' where r ~ 0

-in D and each eigenvalue of the complex constant matrix C has negative

2 -real part. Then, for all C (D) nC(D) solutions u of (3.3)N' there

exist positive constants K1 and K2

such that (6.1) holds.

COROLLARY 6.3. (a) A sufficient condition that

(6.4) lIullo D s lIullo an + c;ll1 f llo D, , ,2 -holds, for all C (D) nC(D) solutions u of (3.3)N' is

*(6.5) C (x) + C(x) ~ -2coI < 0, for some Co E R

(b) Suppose that the variable matrix C = C(X) is normal, and all

its eigenvalues {A.(X)}~ 1 have uniformly negative real parts in D;1 1=

i.e., ReAi(x) ~ -co < 0 in D, for 1 ~ i ~ m. Then (6.4) holds for all

2 -C (D) nC(D) solutions u of (3.3)N .

Proof: (a) By choosing B = I in Theorem 6.1, the inequality (6.4)

follows from the conditon (6.5).

(b) Since C(x) is normal, there exists a unitary matrix U(x)

such that

28

[

2Re\ (x)

U* (x) [c* (x) + C(x)]U(x) = •••••

·2ReA (x)m

*Hence C (x) + C(X) ~ -2coI; and then (6.4) follows from (a).

7. APPLICATIONS TO HIGHER ORDER EQUATIONS

I

Note that the results we obtain are valid for complex systems

which may be considered generalizations of some higher order complex

equations. For example, consider a 2m-order homogeneous equation of

the form

(7.1)

and the nonhomogeneous equation of the same form,

Pu = F.

Here ao ' aI' 0 •• ,am_ 1and F are complex-valued functions, ~ := L +

rex) where r ~ 0 in D and L is the elliptic operator defined by (3.1);

Let m-l= tu , ••• , urn = I u;

then the equations e7.1) and e7.1)N reduce to the equivalent systems

(3.3) and e3.3)N ' respectively, where

0 -1 u1

0o -1 u

2 •e7.2) C = r I • • and f •+ • • u = = •• • • 0

0 -1 ••ao a

1•••••a um_ 1

Fm-l

Associated with each system is the characteristic equation

(7.3) h(A) =(_A)m + a 1(-A)m-1 + 00. + a (-A) + a = O.m- 1 0

29

Hence, under the same assumptions on C in (7.2), Theorem 3.1 and

(7.4)

Theorem 3.3 hold, but (3.6) changes to

(m-1. )1/2 (m-1. )1/2

sup 1 !tJu(x)1 2~ K sup 1 ItJu(x)1 2 ,

xED j =0 xeaD j =0

also, under the same assumptions on C in (7.2), Theorem 6.1

(K ~ 1);

holds, but

-1 ]a

1(x)

(6.1) changes to

(m- 1 . 2)1/2 (m-1. 2)1/2

(7.4)N sup .1 ItJu(x) I s K1sup .1 \tJu(x) I + K211Fllo an'xED J=O xeaD J=O '

In the case of m = 2 and r =0 in D , the matrix in (7.2) is

a ± v/a2-4awith eigenvalues A± = 1 ~ 1 0

Hence the requirement for Theorem 3.1 holding is that there exists

B := [; ~J > 0 such that

= [ 2Re(sao)

qaO+sa 1

- p

-1 ]

a1(x)

which is equivalent to the existence of p,q e ~, s e [ such that

(7.5){

p > 0

pq _ Is 12 > 0and

{

Re(sao) ~ 0, R~(qa1;S) ~ 0

4Re(sao)oRe(qa1-s) ~ !qao+sa1-pI2.

If we choose s = 0 then

B = [~ ~J 'K2 _ max(p,q)

- min(p,q) ,

and (7.5) becomes

{

p > 0

q > 0and

30

So for the 4th-order equations,

(7.6)

and

we have

in D ,

in D ,

COROLLARY 7.1. Assume that Re{a1(x)} ~ 0 in D and ao > 0 , then

(a) for all complex-valued C4 (D) nC2(D) solutions u of (7.6),

I 12 I 12 1/2 1 -1 I 12 I 12 1/2sup( u + Iu) ~ V max(ao,ao ) sup( u + Iu) .D ~

(b) if rex) ~ -ro < 0 in D, then for all complex-valued

C4 (D) nC2(D) solutions u of (7.6)N'

s~p(luI2+IIUI2)1/2s v/max(ao,a~1)[sag(luI2+IIUI2)1/2 + r~11IFllo,~].

m-1When {aj}j=O are complex constants, by applying Corollary 3.2

and Corollary 6.2 we have the following corollary for the 2m-order

equations (7.1) and (7.1)N'

COROLLARY 7.2.

the conditions,

Suppose the roots {A.}.m1 of (7.3) satisfy both of1 1=

(i) Re A. s 0 ;1

(Li ) if Re A. = 0, then the corresponding elementary divisor1

is linear.

Then,

(a) there exists a positive constant K such that (7.4) holds for

all complex-valued regular solutions of (7.1).

31

(b) if ReA. < 0 ,1

1 :::; i :::; m ; or rex) :::; -r < 0o in D, there

exist positive constants K1 and K2 such that (7.4)N holds for all

complex-valued regular solutions of (7.1)N'

Remark. Corollary 7.2 contains the result of Chow and Dunninger [2],

[6] for real metaharmonic functions as a special case. The first part

of the Corollary 7.2 was obtained in [2] for the case of L =~

(Laplace operator) and ao ' a1

' ••• ,a being real constants.m-1

REFERENCES FOR PART I

[1] S. AGMON, Maximum theorems for solutions of higher order

elliptic equations, Bull. Amer. Math. Soc., Vol.66, (1960),

77-80.

[2] S. N. CHOW and D. R. DUNNINGER, A maximum principle for

n-metaharmonic functions, Proc. Amer. Math. Soc., 43 (1974),

79-83.

[3] M. A. DOW, Maximum principles for weakly coupled systems of

quasilinear parabolic inequalities, J. Austral. Math. Soc., 19

(1975), 103-120.

32

[4] R. J. DUFFIN, On aquestion of Hadamard concerning

super-biharmonic function, J. Math and Phys., 27 (1948), 253-258.

[5] R. J. DUFFIN, The maximum principle and biharmonic functions, J.

Math. Anal. Appl., 3 (1961), 399-405.

[6] D. R. DUNNINGER, Maximum principles for solutions of some fourth

order elliptic equations, J. Math. Anal. Appl., 37 (1972),

665-658.

[7] G. FICHERA, EI teorema del massimo modulo per l'equazione

dell'elastostatica tridimensionale, Arch. for Rat. Mech. and

Anal., 7 (1961), 373-387.

[8] G. N. HlLE and M. H. PROTTER, Maximum principles for a class of

first-order elliptical systems, J. Diff. Equ., 24 (1977),

136-151.

[9] G. N. HlLE and R. Z. YEH, Phragmen-Lindelof principles for

solutions of elliptic differential inequalities, J. Math. Anal.

Appl., 107 (1985),478-497.

33

[10] G. N. HILE and R. Z. YEH, Exceptional boundary set for solutions

of elliptic partial differential inequalities, Indiana Univ.

Math. J., 35 (1986), 611-621.

[11] J. L. MASSERA, Contributions to stability theory, Ann. of Math.,

(2) 64 (1956), 182-206. MR18 , 42.

[12] C. MIRANDA, "Partial Differential Equations of Elliptic Type,

Second Revised Edition," Springer-Verlag, Berlin, 1970.

[13] C. MIRANDA, SuI teorema del massimo modulo per una classe di

sistemi ellitici di equazioni de secondo ordine e per ie

equazioni a coefficienti complessi, 1st. Lombardo Accad. Sci.

Lett. Rend., A 104 (1970), 736-745.

[14] L. E. PAYNE, Some remarks on maximum principles, J. Analyse

Math., 30 (1976), 421-433.

[15] M. H. PROTTER and H. F. WEINBERGER, "Maximum Principles in

Differential Equations," Springer-Verlag, Berlin, 1984.

[16] P. W. SCHEAFER, On a maximum principle for a class of

fourth-order semilinear elliptic equations, Proc. Royal Soc.

Edinburhg, 77A (1977), 319-323.

[17] P. W. SCHEAFER and W.WALTER, On pointwise estimates for

Metaharmonic functions, J. Math. Anal. Appl., 69 (1979),171-179.

[18] J. SNYDERS and M. ZAKAI, On nonnegative solutions of the

equation AD+DA'=-C, SIAM J. Appl. Math., 18 (1970), 704-714.

[19] R. SPERB, "Maximum Principles and Their Applications," Academic

.._.--_. - --- -----

Press, New York, 1981.

34

[20] G. STRANG, "Linear Algebra and Its Applications," Second Edition,

Academic Press, New York, 1980.

[21] P. SZEPTYCKI, Existence theorem for the first boundary problem

for a quasilinear elliptic system, Bull. Acad. Polon. des

Sciences, 7 (1959), 419-424.

[22] J. WASOWSKI, Maximum principles for a certain strongly elliptic

system of linear equations of second order, Bull. Acad. Polon.

Sci., 18 (1972), 741-745.

[23] M. J. WINTER and PUI-KEI WONG, Comparison and maximum theorems

for system of quasilinear elliptic differential equations, J.

Math. Aanl. Appl., 40 (1972), 634-642.

[24] T. YOSHIZAWA, "Stabil ity Theory by Liapunov's Second Method," The

Mathematical Society of Japan, 1966.

PART II

SCHAUDER ESTIMATES AND EXISTENCE THEORY FOR

ENTIRE SOLUTIONS OF LINEAR FOURTH ORDER ELLIPTIC EQUATIONS

35

1. INTRODUCTION

We investigate here solutions in ~n of the linear, fourth order,

elliptic, variable coefficient, nonhomogeneous partial differential

equation

36

(1.1)

n.- L a. °klrP.. k 1-1 IJ x.xoxkx II,J,,- IJ

n+ L b.okrP.. k IJ x.x.xkI,J, =1 1 J

n n+ L c.. ¢ + L d .¢ + e¢ = f,

o • 1 IJ X.X. . 1 1 X.1, J= 1 J 1= 1

and the related homogeneous equation,

(1.2) up = O.

Following the prevailing terminology in the literature, we refer to

solutions of these equations defined in all of ~n as entire solutions.

We shall develop Schauder-type a priori estimates for entire solutions

with prescribed behavior required at infinity. These estimates, of

perhaps independent interest themselves, lead to an existence and

uniqueness theory for entire solutions with certain behaviour required

at infinity whenever the operator L can be separated as

(1. 3)

Here

n fi n aL .- a .D2

+ b .D + c .- Lao ° a a + Lb. ~ + c (s=1,2)s s s so. 1 s 1 J x 0 x 0 • 1 s1 VlI.. S1,J= 1 J 1= 1

are two second order elliptic operators.

The coefficients a , b , c of L (s = 1, 2) are assumed to bes s s s

Holder-continuous in ~nU{oo}, and, as x ~ 00, the matrices a approachs

the n x n identity matrix I, the vectors b approach the 1 x n zeros

37

vector, and the scalars cs approach zero. Thus Ls (s = 1, 2) approach

the Laplace operator ~ and L = L2L1

approaches the biharmonic operator

~2 near infinity. We also assume the operators L are uniformlys

elliptic in ~n, and c ~ 0 if n ~ 3, c =0 if n = 2 (s = 1, 2). Withs s

these assumptions, and for n ~ 5, we shall establish a one-to-one

correspondence between the entire solutions of (1.1) that are O(lxl m)

near infinity and the polynomial solutions to the biharmonic equation

of degree no greater than m; when f =0, this result is an extension

of a Liouville-type theorem.

For dimension n ~ 4 we have similar results, but, like the case

of n = 2 in [2, 9] for second order equations, there are complications

which make the situations in four, three and two dimensions more and

more difficult. As Friedman, Begehr and Hile showed in [9, 2], when

n = 2 is the order of the equation, there need not exist an entire

bounded solution of Ll~ = f, even when L1

is the Laplace operator and

f has compact support in ~2. Begehr and Hile also proved that there

exists a unique entire solution of L1¢ = f with some growth

restriction at infinity and the condition ~(x) - 1logixl ~ 0 as

x ~ 00, for some constant 1. We obtain a similar result for the

~ .nsr(x) ~ 0 as x ~ 00, for some constant numbers

that ~(x) -

fourth order equation (1.1) when n ~ 4; our condition at infinity is

4-n1

s=o

~o' constant vector 71

and constant symmtric matrix ~2' Here r is a

fundamental solution of the biharmonic equation.

Several authors, for example [2, 5, 9, 10, 13, 15], have

investigated existence and/or uniqueness questions for entire

38

solutions of linear second order elliptic equations and systems.

There has also been considerable work on entire solutions of nonlinear

and quasilinear second order elliptic equations and systems, as for

example in [3, 12, 14, 16, 18-20]. The usual result obtained for a

linear or nonlinear second order equation is the traditional statement

of Liouville's Theorem, namely, a bounded entire solution is

necessarily constant. For the higher order case, Douglis and

Nirenberg proved a Liouville-type theorem for certain homogeneous

systems with constant coefficients in [4]. Liouville Theorems

for polyharmonic and some metaharmonic functions can be found in

[1, 6, 8],

Results analogous to the ones presented here for fourth order

equations appear to be most closely related to those of Begehr and

Rile [2] for second order equations. The Schauder estimates for

entire solutions are obtained from analogous Schauder estimates for

bounded domains as found in [4].

2. NORMS AND A SCHAUDER ESTIMATE FOR ENTIRE SOLUTIONS

For vectors v, w in ~n we let VeW denote the dot product of v and

w, and Ivl the Euclidean norm of v.

= (b .. ) in ~nxn we defineIJ

For n x n matrices A = (a .. ), BIJ

i ,j1 a .. b .. ,

IJ IJIAI = (AeA)I/2 = { 1 a~.}1/2 .

. . IJ1, J

) . nxnxnSimilarly, for tensors A = (a. 'k)' B = (b" k In ~ and C =IJ IJ(c

i j kl),D = (d

i j kl)in ~nxnxnxn we define

------ ---- -----------

and

A.B = 1 a. 'kb"k. . k 1J 1J1, J ,

I I ~ 2} 1/2A = { LJ a i jki ,j,k

39

C.D =

Let

~ I I 2 1/2LJ Cijkldijkl' e = { 1 ci j kl}i,j,k,l i,j,k,l

(The inequalities IA.al ~ IAIIal and IC.DI ~ lei IDI hold.) We let x

= (xl' x2' ... , xn)T denote a point in ~n, n ~ 2, and n a domain in

~n. For ~ = ~(x), a real valued function defined in n, we let D¢,

D2~, D3~ and D4~ represent the n X 1 vector of first derivatives of ~,

the n X n matrix of second derivatives of ~, the n X n X n tensor of

third derivatives of ~, and the n x n X n X n tensor of fourth

derivatives of ~,respectively. Thus,

D¢ .- (~x ' ~x ' ... ~x )T, D2~:= (~x.x.)nxn '1 2 n 1 J

D3~ ._ (~) D4~ := (~ ) .x.x'Xk nxnxn x.x,xkx I nxnxnxn

1 J 1 Ja=(a.· k l) ,b=(b..k) ,c=(c .. ) ,d=(d.) ,eIJ nxnxnxn IJ nxnxn 1J nxn 1 lxn

be functions defined in n, with values in ~nxnxnxn, ~nxnxn, ~nxn, ~n,

and ~,respectively. The tensors a, b and the matrix c are always

assumed to be symmetric; i.e.,

a i j kl = au(i)u(j)u(k)u(l) , for any permutation u of {i, j, k, I},

bi jk = b ikj = bj i k = bj ki = bki 1 = bkj i ' and a i j = a j i

We consider the fourth order linear partial differential operator L,

defined by

(2.1) ~ := a.D4~ + b.D3~ + c.D2~ + d.~ + e~

n n= 1 a. 'kl~ + 1 b. 'k~. , k 1-1 1J x .x ,xkx1 .. k 1 1 J x .x .XkI,J, , - 1 J I,J, = 1 J

n n+ 1 c ..~ + 1 d. ~ + e~ .

. . 1 IJ X.X. 1'--1 1 Xl'I,J= 1 J

Associated with the operator L are the homogeneous equation,

40

(2.2) up =0,

and the nonhomogeneous equation

(2.3) up = f ,

where the right-hand side f is assumed to be a real valued function

in n.

Using the notation of [11], for points x, y in 0, we define

dx := dist(x, an), dx,y:= min(dx' dy)'

and for u = u(x) a real valued, vector valued, matrix valued, or

tensor valued function in 0 we define the following norms (where ff E

JR, 0 < 0: ::; 1):

(2.4)

(2.5)

(2.6)

lul~~~ := sup dUlu(x)1 ,, XEO x

(o) [uCx) - u (y) I[u]O 0:'0'- sup (dx y)U+O: -----0:-

" x, YEO' Ix - y Ilui (e') := lui (U) + Iu l (o) .

0,0:;0 0;0 0,0:;0

By a classical solution of up = f, we mean a solution ~ with ~,

D¢, D2~, D3~, D4~ continuous. It is known (see [7], Chapter 9) that

if a, b, c, d, e and f are Holder-continuous with exponent 0: in n,

o < 0: < 1, and ~ is a solution of (2.3), then~, D¢, D2~, D3~ and D4~

are all Holder-continuous with the same exponent 0: in O. So, under

the condition that all cofficients and f are Holder-continuous, any

solution of (2.3) is a classical solution. Douglis and Nirenberg have

obtained a Schauder estimate for certain higher order elliptic systems

(Theorem 1, [4]). Applying their result to the single equation (2.3),

41

we get the following basic Schauder estimate:

LEMMA 2.1 ([4]). Let 0 be a bounded domain in ~n, n ~ 2, and let

~ E C4(0), with 1~1~?h finite, and with ~ a classical solution of the,nonhomogeneous equation (2.3) in O. Assume 0 < a < 1, that If\(4)

o,a;O

is also finite, and that for some nonnegative constant A,

I 1(0) I 1(1) I 1(2) I 1(3) I 1(4)

(2.7) a o,a;O' b o,a;O' c o,a;O' d o,a;O' e o,a;O ~ AAlso assume the symmetric tensor a satisfies the ellipticity

condition

n(2.8) E aijkl(x)eiejekel ~ Alel

4,for all x E 0, eE ~n,

i,j,k,I=1

for some positive constant A. Then for some universal nonnegative

constant M= M(n, a, A, A) (independent of ~) we have the bound

(2.9) I~(x) 10(?A + IDr/> I (~) + ID2~1 (~) + ID3~1 (~) + ID4~1 (4).,I' 0,0 0,0 0,0 0,0:,0

~ M(n, a, A, A){I~I~~n + Ifl~~~;o}.

(Throughout this chapter we denote generic constants by M( ), where

inside the parentheses are listed the entities that determine M.)

We will derive an analogue of Lemma 2.1 for the case n = ~n.

For this purpose it is useful to define the following norms, where

again U E ~, 0 < a ~ 1, and now u js a function defined on all of ~n.

Let

Ilullu := sup (l + Ix I)-ulu(x) I,x~n

[utx) - uf y) I(1 + [x] )-a-o: ----­

Ix _ ylasup

nx,y~

o<lx-yl~(1+lxl)/2

Ilull := lIull + Ilull( )o ,« a U ,0:

Ilull(u,a) .-(2.11)

(2.12)

42

The norms (2.10)-(2.12) were first defined by H. Begehr and G. N. Hile

in [2], and may be viewed as analogous to the norms (2.4)-(2.6),

where the distance dx' vanishing as x approaches an, has been replaced

by the quantity (1 + Ixl)-l, which likewise vanishes as x approaches

the boundary point at infinity of Rn. The condition 0 < Ix - yl ~

(1 + Ixl)/2 of (2.11) is a technical convenience, ensuring that y

approaches infinity along with x, and more or less in the same

direction.

DEFINITION 2.2. We say that a real valued, vector valued, matrix

valued, or tensor valued function u defined on all of Rn is in the

space

(i) Bu if and only if lIullu is finite,

(ii) B if and only if lIull is finite.u,a u,a

We see that if u E Bu' then lu(x)1 = O(lxl u) as x ~ 00. The

space Bo consists of bounded functions in IRn, with lIullo reducing to

the ordinary supremum norm. If u E B then the Holder quotient,u,a

lu(x)-u(x)l/lx-yla, is O(lxl u-a) (with the restriction 0 < Ix-YI ~

(1+lxl)/2). In particular, functions in B are uniformly Holder-u,a

continuous with exponent a on compact subsets of Rn.

We gether some miscellaneous observations regarding the spaces Bu

and B into the following lemma. The proof of the lemma can beu,a

found in [2].

LEMMA 2.3. Let u, T E R, 0 < a, ~ ~ 1, and let u and v denote

functions defined on IRn (both scalar valued, both vector valued, both

43

matrix valued, or both tensor valued). Then

U s r , 0 < {J s a s 1 ~ B C B (J' Ilull (J s lIull ;a ,« T,,., T,,., o ,«

(c) u E Bu ' v E B". ~ UeV E BU+T ' lIuevllu+T ~ lIullu IIvll". ;

(d) u E B , v E B ~ UeV E B ,u,a T,a U+T,a

(a) U ~ ". ~ B... C B , lIull ~ Ilull ;v r r U

(b)

Moreover,

(e) if ~ is a real valued function in Cl(~n), with ~ E B ,U-l

then 1I~II(u,1) is finite, and in fact

1I~II(u,l) s M(u)II~llu_l

Finally,

~ K for all m, for some nonnegative constant K.

that we have a uniform bound, Ilu IIm u,a

Then there

(f) let {um}: =1 be a sequence of functions in Bu,a ' with all

n nxn nxnxnum's having values in the same space ~, ~ , ~ , ~ , or

nxnxnxn~ , and suppose

exists a subsequence {u }, and a function u in B , withmk u,a

also Ilull ~ K, such that u ~ u in the norm of any spaceu,a mk

B or B (J with". > u, 0 < {J < a.T T,,.,

We now present the following modification of Lemma 2.1, where nnis replaced by ~ and the norms (2.4)-(2.6) by newly-defined norms

(2.10)-(2.12).

THEOREM 2.4 (Schauder Estimate in ~n). Let ~ E C4(~n), n ~ 2, with

II~II finite for some U E ~, and with ~ a classical solution in ~n ofU

the nonhomogeneous equation (2.3). Assume 0 < a < 1, that IIfll isu-4,a

also finite, and that for some nonnegative constant A,

44

(2.13)

Also assume the tensor a is symmetric and satisfies the ellipticity

condition

n 41 a. ·kl(x)e.e.ekel ~ Aiel , for all x, e E ~n,

. . k 1-1 1 J 1 J1, J, , -

for some positive constant A. Then for some nonnegative constant M=

M(n, a, u, A, A) we have the bound

(2.14) 114>llu , l + 1ID4>lIu_1 , 1 + IID24>lIu_2 , 1 + IID3

4>lIu_3 , 1 + IID44> lIu_4 ,a

:s; M(n, a, o; A, A){1I4>11 + IIfll 4 }.(J U- ,a

Proof: First, we consider the case Ixl :s; 5. We apply Lemma 2.1 to

the domain

o := {z E ~n : Iz I < 9}.

The restrictions Ixl :s; 5 and 0 < Ix - yl :s; (1 + Ixl)/2 imply that

(2.15) 1 :s; dx' dy' dx,y :s; 9.

In order to check that conditions (2.7) hold in 0, we observe first

that

and hence

C2.18) I I (4)e 0 .n ~ MCa)A.,a,

45

In a similar manner we derive the estimates

C2.19) Idl~3~.n ~ MCa)l\dll_3 a ~ MCa)A,, , ,

Icl~~~;n s MCa)l\ cll_2,a s MCa)A,

Ibl~l~.n s MCa)l\bll_1 a ~ MCa)A,, , ,

lal~~~;n s MCa)llallo,a ~ M(a)A;

C2.20) It/>I~~n ~ MCu)IIt/>lIu' IfI6~~;n ~ M(a,u)llfllu_4,aTherefore Lemma 2.1 applies, and from C2.9), C2.20) we obtain, for

x, YEn, an estimate of the form

C2.21)

But for

1 ~ 1 +

C2.22)

It/>Cx) \ + d I~Cx)1 + d2ID2t/>Cx)I + d3ID3t/>Cx)I + d4\D4t/>CX)\x x x x

4 ID4t/>Cx) - D4t/>Cy) I

+ dx+~ a ~ MCn,a,u,>.,AHIIt/>lIu + II fl\u_4 a}', [x - yl '

Ixl ~ 5, 0 < Ix -y! ~ C1 + Ixl )/2, we have C2.15) and also

Ixl ~ 6, and therefore we may write C2.21) as

C1+lxl)-ult/>Cx)I + C1+lxl)1-ulDt/>Cx)I

+ C1+lxl)2-uID2~Cx)1 + C1+lxl)3-uID3~Cx)1

ID4~Cx) _ D4~Cy)1+ C1+lxl )4-u ID4tjJCx) I + C1+lxl )4-u+a ------­

Ix _ yla

~ MCn,a,u,>.,A){II~1I + IIrll 4 i.U u- ,a

We next consider the case Ixl ~ 5. We define

C2.23) p := u + Ix I)/6, Ix I = 6p - 1,

and we now apply Lemma 2.1 to the domain

n := {z E ~n : p < Izi < lOp}.

We again check conditions C2.7) in n. For z, ~ E n we have

d ,d~, d ~ ~ 5p, P ~ l+lzl, 1+1~1 ~ 11p.z ~ z, ~

As in (2.16) and (2.17) we derive

1t;16~n:::; 54 sup O+lzl)4Ie(z)1 :::; 5411elL4 :::; 54A,

[e](4) (\ s (5p)4+a(lIe ll 4 (J' sup O+lzl )-4-ao,a;u (- , ) zen

+ 2-a!le1L4 sup (1+ Iz I)-4-a)

zen

s M(a)lIeIL4 a s M(a)A.,

A similar analysis shows that (2.19) continues to hold, but in place

of (2.20) we obtain

46

+

(2.24) ItPl6?n = sup 1¢(z)1 :::; (llp)(J' sup O+lzl)-(J'ltP(z)I :::; M((J')p(J'lltPll ,, ZEn zen (J'

IfI6~~;n :::; M(a)lI fll_4,a

~ Mca.UlpU{ sup Cl+lzll,-ulf(zl!

supZ,~En

0<lz-~I:::;(1+lzl)/2

:::; M(a,(J')p(J'lI fll(J'_4,a .

We now apply (2.9) to our particular point x, also using (2.24), to

obtain, for yEn,

(2.25) \tP(x) I + d 1D¢(x)I + d2 ID2tP(x)I + d3 ID3tP(x)Ix x x

4 4 4 a ID4¢(x)

- D4

tP(y ) I+ d ID tP(x) I + d + a

x x x,y [x _ yl

:::; M(n,a,(J',>.,A)p(J'{II¢1I + IIfll 4 }.(J' (J'- ,a

Now we have 1 + Ixl = 6p, and if 0 < Ix - yl :::; (1 + Ixl)/2, then

yEn and p :::; dx' dy ' dx,y :::; 5p. Thus (2.25) implies (2.22) also for

Ixl ~ 5, 0 < Ix - yl :::; (1 + Ixl)/2.

Upon taking supremums in (2.22), we conclude that

(2.26) 114>lIu + 1ID4>lIu_1 + IID24>llu_2 + IID34>lI

u_3 + IID44>lIu_4 a,

:;:; M(n,a,u,.x,AHII4>lIu + II f llu_4 aL,By Lemma 2.3(e) we have 114>11 ,.. 1 :;:; M(u)IID4>1I I' which implies we may

(v, ) u-

add the term 114>11 to the left-hand side of (2.26). SinceU,I

D24>E B 2' we may also apply Lemma 2.3(e) to each of the derivativesu-

ux. and conclude that 1ID4>II(U_I,I) s M(u)IID24>llu_20 Similarly, since1

D34>E B 3 and D44>

E B 4' we also obtain thatu- u-

IID24>II(U_2,1) :;:; M(u)IID34>lIu_3 and IID34>II(U_3,1):;:; M(u)IID

44>lIu_4'

Thus (2.26) implies our desired inequality (2.14). I

3. BOUNDS FOR ENTIRE SOLUTIONS OF THE NONHOMOGENEOUS BIHARMONIC

EQUATION

Before further discussing entire soluitons of the general fourth

order elliptic equation, it is useful both as motivation and as a

mathematical aid to study entire solutions of the nonhomogeneous

biharmonic equation,

47

(30U2/}. 4> = f.

In Rn, let r denote the fundamental solution of the biharmonic

equation,

(3.2) rex)

and for a real

1 IxI 4-

n , 1°f n"" 5 32(2-n) (4-n)w ~ or n = ,n

._ - _1_ l og[x] if n = 4,4w

4

~ Ixl 2(loglxl - 1), if n = 2;w2

valued function f on Rn define the Z-potential of f as

(3.3) Zf(x) := J r(x-y)f(y)dy.IRn

48

(3.4)

Here w denotes the surface area of the unit ball in IRD.n

We have the following estimates on the Z-potential.

LEMMA 3.1. let f be a real valued and measurable function on IRn

(n ~ 2), in the space B ,T > 4. Then-T

(a) for x E IRn the integral Zf(x) exists and

(1+IX I)4- 7" . f, 1 n~5, r-cn ,

(1+lx!)4-nl og(2+lxl), if n~5, T=n,

IZf(x) I :;;; M(n,T)lIfll O+lxl )4-n , if n=3 or ~5, T>n,-7"

log(2+lxl) , if n=4,

O+lxl )2 l og(2+lxl) ,if n=2;

(b) for Ixl ~ 1 we have,

(i) if n = 4, 4 < T < 5,

(3.5)"10

= IZf(x) + 4w4l0glxl I

:;;; M(7")llfll (1+lx\)4-T l og(2+lx\);-7"(ii) if n = 3, 4 < T < 5,

(3.6) IZf(x) - "Ior(x) + "Ilenr(x)I = \Zf(X) + :~ Ixl - ~I

:;;; M(T)lIfll O+lxl)4-7";-T

(iii) if n = 2, 4 < 7" < 5,

(3.7) IZf(x) - "Ior(x) + "Ilenr(x) - t "I2en2r(x)I

"10 2 "I IOX

= Zf(x) - -- Ixl (loglxl-1) + ---- (2loglxl-1)8~ 8~

lcglxl'Y2- -.811"

Ixl2

~ M(T)lIfli (1+lxl)4-T l Og(2+lxP;-T

where

loglxl

49

(3.8)

'Yo := J~n f(y)dy, "11 := J~n yf(y)dy,

'Y2 := Jn yyTf(Y)dY = J n (Y~ Yl~2Jf(Y)dYIR ~ lY1Y2 Y2

are a constant scalar, vector and matrix, respectively.

Proof: (a) By applying the estimate

to the integral (3.3), we obtain an inequality

(3.10)

where we define

(3.11)

(3.12)

(3.13)

:= J Ir(x-y)l(l+lyl)-Tdy,ly-xl~(1+lxl)/2

:= J [I'(x-y) I (1+lyl)-Tdy,Iy-x I,=2(1+ Ix I)

:= J Ir(x-y)I (1+lyp-Tdy.

(1+lxl)/2~ly-xl~2(1+lxl)

In I1(x),

we observe that

(3.14) Iy-xl ~ (1+lxl)/2 ~ (1+lxl)/2 ~ l+lyl ~ 3(1+lxl)/2,

which gives

(3.15)

{

(1+IX\) 4- r ,~ M(n,r) 4 r

(l+lxl) - log(2+lxl),

if n = 3 or n ~ 5,

if n = 2 or 4.

50

[y-x] ~ 2(1+lxl) =::;> [x] ::; Ix-YI/2 ::; IYI,

and therefore

if n = 2 or 4.

if n = 3 or n ~ 5,

(3.17) I2

(x) s M(r)J Ir(x-y) Ilx-yl-rdyly-xl~2(1+lxl)

~ M(r)J Ir(z)llzl-rdzIzl~l+lxl

{

( 1+ IX I)4-r ,

::; M(n,r) 4 r(l+lxl) - log(2+lxl),

In 13

( x ) ,

(3.18) (1+lxl)/2 ~ !y-xl ~ 2(1+lxl) =::;> Iyl ::; 2+3Ixl,

and moreover, if n = 3 or n ~ 5,

(3.19)

(3.21)

(3.20)

and if n = 4,

-Iog2 ~ log ~::; loglx-yl = 4w4Ir(x-y)I

~ log 2(1+lxl) ::; 2Iog(2+!xl),

Ir(x-y)I ::;~ Iog(2+lxl);w4

and if n = 2, applying (3.18) and (3.20),

Therefore, for n = 3 or n ~ 5,

(3.23) I3

(x ) ::; M(n)(1+lxl)4-nJ (l+lyl)-rdyIYI::;2+3Ixl

{

(1+lxl)4-r , if n ~ 5, r < n,

::; M(n,r) (1+lxl)4-n1og(2+lxl), if n ~ 5, r = n,

(1+lxl)4-n , if n = 3 or n ~ 5, r > n.

(3.25)

For n = 4,

For n = 2,

13(x)

s .!.(1+!xl)2 l 0g(2+\Xj)J (l+lyj}-rdyw lyl~2+3Ixl

~ M(r)(1+lxl)2 l og(2+lxl).

Now combining (3.10), (3.15), (3.17), (3.23)-(3.25), we obtain the

required estimate (3.4).

51

(b) In order to derive (3.5)-(3.7) we write

Zf(x) - ~or(x) = J 4[r(x-y) - r(x)]f(y)dy (n = 4),IR

Zf(x) - 1or(x) + 11.Dr(x) = J 3[r(x-y) - rex) + Dr(x).y]f(y)dy (n=3) ,IR

J 1 2 T= 1R2[r(x-y) - rex) + Dr(x).y - 2 D r(x).(yy )]f(y)dy (n = 2),

and apply (3.9) to obtain

(3 .26) IZf(x)-~l(x) I ~ II f l'-r [ 11(x)+1 2(x)+Jo(x)+Ko(x)+N1(x l l (n=4) ,

(3.27) IZ(x)-1or(x)+~1.Dr(x)1

~ II f ll_r [ 11 (x)+1 2(x) +J0 (x)+J1(x)+Ko(x) +K1(x)+N2(x) ] (n=3) ,

(3.28) IZf(x)-~or(x)+11.Dr(x)-12.to2r(x)1

~ II f lLr [ 11(x)+1 2(x)+Jo(x)+J1(x)+J2(x)+Ko(x)+K1(x)+K2(x)+N3(x)]

(n=2) ,

where 11(x), 12(x) are as defined in (3.11) and (3.12), and we futher

define

52

J. (x)1

K. (x)1

.- Inir<x)IJ lyli(1+lyp-rdy, (i = 0,1,2),ly-xl~(l+lxl)/2

.- Inif(x)IJ IYli(l+IYI)-rdY, (i = 0,1,2),ly-xl~2(1+lxl)

:= J If(x-y) - fCx)l(l+lyl)-rdy,(1+lxl)/2~ly-xl~2(1+lxl)

:= J IfCx-y)-f(x)+yenf(x)l(l+lyl)-rdy,(1+lxl)/2~ly-xl~2(1+lxl)

N3

(X) .-

J If(x-y)-f(x)+yenfCx)-io2f(x)eyyTIC1+lyl)-rdy.C1+lxl)/2~ly-xl~2C1+lxl)

By the definition of fCx) in (3.2), we have

(3.29) {-~ ,

nfCx) =8~ (2loglxl - 1),

if n = 3,

if n = 2;

and

loglxl[

l og [xI2 1

D I'(x) = 411"(3.30)

Ixl2

In JiCx), i = 0,1,2, we have (3.14); thus,

J.(x) ~ MCr)lnir<x)IC1+\xl)i-rJ . dy ,1 lyl~3(1+lxl)/2

which gives

{ (1+lxlj'-Tlxl'(lloglxll+l1, if n = 2,

(3.31) Jo(x) S; M(n,r) C1+lxl )3-r lx l , if n = 3,

C1+lxp4-rlloglxll , if n = 4;

{ (I+Ixll'-T [x] (lJogl xll+l) , if n = 2,(3.32) J

1(x)~ M(n,r) 4 r

(1+ Ix I) - if n = 3;

(3.33) J 2(x) ~ M(r)(1+lxl)4-r(lloglxl 1+1), when n = 2.

Applying (3.16) to Ki(x), i =0,1,2" we obtain

Ki(x) ~ Inif(x)IJ (~)i-rdYly-xl~2(1+lxl) 2

~ Inif(x)IJ 2nlzli-rdz,Izl~l+lxl

which yields

53

(3.34)

(3.35)

(3.36)

{

(1+lxl)2-rlxI2l0g(2+lxl), if n = 2,

Ko(x) ~ M(n,r) O+lxl)3-rlxl , if n = 3,

O+lxl )4-rlIoglxllif n = 4;

{

(1+lxl)3-rlxllog(2+lxl), if n = 2,KI(x) ~ M(n,r) 4

(l+lxl) -r if n = 3;

K2(x) ~ M(r)(1+!xl)4-r l og(2+lx\), when n = 2.

To estimate NI(x) (n = 4), we write, considering f as a function of

Ixl rather than x,

f(x-y) - f(x) = - 4~ t-I(lx_yl - Ixl),4

where t is some positive number detween Ixl and Ix-yl. Since (3.18)

holds in NI(x), we have

(3.37) Ixl ~ Ixl/2, Ix-yl ~ Ixl/2 =* t, Ix-syl ~ Ixl/2 for s E [0,1],

and hence

[I'(x-y) - rex) I ~ Mlxl-IIYI (n = 4).

Substituting into NI(x), again noting (3.18), we have

NI(x) S; Mlxl-IJ lyl(l+lyl)-rdy (n = 4),lyl~2+3Ixl

which gives

(3.38)

54

We substitute (3.15), (3.17), (3.31), (3.34), (3.38) into (3.26),

requiring that Ixl ~ 1, and obtain the desired estimate (3.5).

To estimate N2(x) (n = 3), and N3(x ) (n =2), we define

f(t) := T(x-ty), t E~.

By Taylor's Theorem,

f(l) - f(O) - f'(O) = ~ f"(s), for some s E [0,1],..and

f(l) - f(O) - f'(O) - ~ f"(O) = ~ fffl(s) , for some s E [0,1],2 6

and therefore, we have

(3.39) T(x-y) - rex) + nT(x)ey = - 8~ {Ix-yl - Ixl + lit}=-1~1I"{ lyl2 _[(x-sy)ey]2} (n=3),

Ix-sy I 1x-sy 13

and

into

loglxl

(x-sy)ey } (n = 2).Ix-syl2

by substituting (3.39)

1 2 T(3.40) r(x-y) - rex) + nr(x)ey - 2 D r(x)eyy

=.; {lx-YI'(]oglx-YI-ll-lxl'(]og\xl-ll+(2]OgIXI-1)XOY

21 Xl

[

loglxl - 2 +~

x1x2

Ixl2

= __1__ {2 [(X-Sy)ey]3 _ 31yl22411" Ix-syl4

Since (3.18) and (3.37) hold in N2(x),

N2(x), we obtain

N2(x) ~ Ml xl-1J lyI2(1+lyl)-rdy (n = 3),lyl:::;2+3Ixl

which gives

(3.41)

55

We substitute (3.15), (3.17), (3.31), (3.32), (3.34), (3.35), (3.41)

into (3.27), requiring that Ixl ~ 1, and obtain the desired estimate

(3.6).

Now substituting (3.40) into N3(x),

again noting (3.18) and

(3.37), we have

N3(x)

~ Mlxl-1J IYI3(1+IYI)-Tdy (n = 2),lyl~2+3Ixl

which gives

(3.42) N3(x)

~ M(T)lxl-1(1+\xl)5-T, if 4 < T < 5, n = 2.

Finally, we substitute (3.15), (3.17), (3.31)-(3.36), (3.42) into

(3.28), requiring that Ixl ~ 1, and obtain the desired estimate

(3.7). I

From the definition of rex) in (3.2), it is easy to derive the

following estimates for the first, second, third and fourth

derivatives of r:

I~. rex) I1

if n ~ 3,

if n = 2;

if n ~ 3,

if n = 2;

if n ~ 2.

\ax.g:.axk

r(x)1 ~ ~:3 \xI 1-n

, if n ~ 2;1 J n

laxiax~~kaxl rex) I ~By using the proofs of [11, Lemma 4.1, Lemma 4.2], the following

56

lemma concerning the Z-potential on bounded domains can be proved from

the above estimates.

LEMMA 3.2. Let f be bounded and locally Holder continuous (with

exponent 0 < a < 1) in a bounded domaim O. Then

(3.43) z(x) := J r(x-y)f(y)dyn

(3.44)

is in C4(0); and for any x E 0,

a~x) =J ar~-y) f(y)dy, p2 z(x) - J a2r(x-y)

f(y)dy,uA i 0 i OxiOx j - 0 OxiOx j

3 3a Z(x) _ J a r(x-y) f()dOx.Ox.Ox

k- n Ox.Ox.ax

ky y,

1 J u 1 J

a4z (x ) J a4r (x- y) )

Ox Ox Ox Ox = Ox ax a a [f(y -f'(x l Idyi j k I 0

0i j xk xl

( J a3r(x-y) )- f x) a a Ox "i (y dS.an x . x , k Y

o 1 J

Here i,j,k,l = 1,2, ... ,n, v = (VI' V2' ... , vn) is the outward

normal direction, 00

is any domain containing 0 for which the

divergence theorem holds, and f is extended to vanish outside n.

From Lemma 3.2 we have, for any x E n,

if n = 2,

if n ~ 3,

~Z(x) =J ~ r(x-y)f(y)dyn x

= { (2-~)WnJn -1-x~;;;"'y-l-n---2 f(y)dy,

__1__ Jloglx-ylf(y)dyW2 n

which is the well known Newtonian potential of f on the domain n.

(3.45)

Therefore, we also have, for x E n,

(3.46)

57

LEMMA 3.3. Let f be real valued on Rn and in some space B , with-r,o:

(3.47) a(zf) = Sf,

and

(3.48) ~2(Zf) = ~(Sf) = f

if n = 2

if n z 3,

where

(3.49 ){

(2-~)W J n Ix-yI2-nf(y)dy,n R

Sf'( x) :=

2~ JR

2 loglx-ylf(x)dy

is the well known Newtonian potential of f on Rn . Moreover,

. TOnIn Il'. ,

(a) if n ;:: 5, then Zf vanishes at infinity;

(b)'Yo

ifn = 4, then Zf + --4--loglxl vanishes at infinity;w4

'Yo 'Y eX(c) 1 . hif n = 3, then Zf + 81l"l x l - 8iTiT vanIS es at infinity;

(d) if n = 2, then Zf + ~1l" [ -'Yo lx I2( l og lx l- 1) + (210glx!-1) xe'Y

1

Xe'Y x ]- (loglxl - +)tr('Y2) - Ix~2 vanishes at infinity.

Remark Being aware of the fact that Dir(x) vanishes at infinity when

n ;:: 5-i, (i = 0,1,2), we can write (a)-(d) of Lemma 3.3 in the

following uniform way: for n ;:: 2,

(3.50)

Proof of Lemma 3.3: By choosing r smaller if necessary, we may

prescribe 4 < r < n if n ;:: 5, and 4 < r < 5 if n = 2, 3, 4. Then

(a)-(d) follow from Lemma 3.1.

58

In order to discuss differentiability of Zf, we let p denote

some large radius and form the decomposition

if n ~ 3,

Zf(x) =J r(x-y)f(y)dy + J r(x-y)f(y)dyIyl~p Iyl~p

= Z (x) + w (x)p p

Lemma 3.2 gives that zp(x), the first integral of (3.51), is 04 in the

region 0p = {xElRn : IAI < p}, with (3.45), (3.46) valid if we replace

z(x) and °by zp(x) and 0p' It is clear that wp(x) , the second

integral of (3.51), is COO in 0p with

{

JIYI~p Ix_yI2-n(1+lyl)-Tdy,

(3.52) IAWp(X) I s M(n)lI f ILT

J Iloglx-yll(l+lyl)-Tdy, if n = 2,Iyl~p

(3.51)

and

in np

From (3.52), it follows easily

(3.53) a2wp

(x ) == 0

Since p is arbitrary, Zf is 04 in Rn.

that aw (x) ~ 0, ~s p ~ 00, for any (fixed) x ERn. By lettingp

p ~ 00 in (3.52)-(3.53), and also in (3.45)-(3.46) with z(x) and n

replaced by zp(x) and 0p' we obtain (3.47) and (3.48). I

Douglis and Nirenberg have proved a Liouville-type theorem for

certain homogeneous elliptic systems with constant coefficients and

homogeneous order (Theorem 3, [4]). Applying their result to a single

equation of fourth order, we obtain the following:

LEMMA 3.4. (Liouville-Type Theorem) Let ~ be a 04 solution of

a.D4~ = 0 in Rn,

where a is a constant tensor in ~nxnxnxn and satisfies the elliptic

condition (2.8), and assume that, for some nonnegative integer m,

~(x) = o(lxl m) at infinity.

Then ~ =0 if m = 0, and ~ is a polynomial of degree ~ m-1 if m > O.

Particularly, the result holds for entire solutions of the biharmonic

equation (in which case, D4 = ~2).

The following basic result for the nonhomogeneous biharmonic

equation will serve as our model for the general fourth order elliptic

equation.

59

THEOREM 3.5. Let f be real valued on ~n and in some space B with-T,O:

T > 4, 0 < 0: < 1. Then there exists exactly one entire C4 solution ~

of the equation ~2~ = f, namely the solution ~ = Zf, such that

(a) when n ~ 5, ~ vanishes at infinity; if moreover 4 < T < n,

then for this solution we have the bound

(3.54) 1IC/>1I 4_T,1 + 1IDcf>11 3_T, 1 + IID2C/>11

2_T, 1 + IID3~1I1_T, 1 + liD ~II_T, a

~ M(n, a, T)lIfll-T,a

(b) when n = 4, ~(x) - 1or(x) vanishes at infinity for some

constant 10

; moreover, for this solution c/> and any positive

constant € we have the bound

(3.55) 11c/>II E,1 + IIDtI>II E_1,1 + Iln2~IIE_2,1 + IID3~11€_3,1 + IID4~IIE_4,a

s M(a, T, dllfll ;-T,a

(c) when n = 3, c/>(x) - 1or(x) + 11enr(x) vanishes at infinity for

some constant 10

and constant vector 11; moreover, for this

solution we have the bound

(3.56) 114>11 1,1 + 1ID4>1I0,1 + IID24>11_1, 1 + IID34> 1I _2 , 1 + IID44>1I_3,a

:;;; M(a, r)lIfll ;-r,a

(d) when n = 2, 4>(x) - 10r(x) + 11eDr(x) - t 12eD2r(x) vanishes at

infinity for some constant 10, constant vector 11 and constant

symmetric matrix 1 2; moreover, for this solution 4> and any

positive constant E we have the bound

60

(3.57) 114>II E+2,1 + 11D4>II E+1, 1 + IID24>II

E , 1 + IID34>IIE_ 1, 1 + IID44>II

E_ 2,a

:;;;M(a, r , E)llfll .-r,a

The uniquely determined constant 10, constant vector 11 and constant

matrix 12 in (b), (c) and (d) are defined by (3.8).

Proof: (a) By Lemma 3.4 for biharmonic functions (also see [6, 8]),

uniqueness of the solution is clear since the difference of two

solutions would be biharmonic and vanish at infinity, thereby being

identically zero. Lemma 3.3 states that 4> = Zf is a solution

vanishing at infinity. In order to derive the bound on 4>, we apply

(2.14) of Theorem 2.4 to the special case of the biharmonic operator,

with u = 4 - r, obtaining

114>1I 4_r,1 + I1D4>1I 3-r ,1 + IID24>112_r, 1 + IID

34>111_r, 1 + IID44>II_

r ,a

s M(n, a, r) {1I4>11 4_r + II fll_r ,a}'

But Lemma 3.1 implies, for 4 < r < n,

which, upon substitution into the previous inequality, yields (3.54).

(b) For uniqueness, suppose there were two C4 solutions 4>1' 4>2

and real constants 101, 102, with 4>i(x) - 10ir(x) vanishing at

61

infinity, i = 1, 2. Setting ~ = ~1 - ¢2' 10 =101- 1 02, we have

2 1 0~ ~ = 0, and ~(x) + 4w loglxl vanishes at infinity. By Lemma 3.4,

4

~ =constant, and therefore 10 = 0 and ~ =O.

Again, Lemma 3.3 states that ~ = Zf is a solution of the problem,

with 10 defined by (3.8). To obtain the bound on~, we observe that

since

(3.58)

Lemma 3.1 (a) implies the inequality

We also have, since £ - 4 > -r,

Il f ll £_4 a s Ilfll_r 0:', ,

We substitute these inequalities into the right-hand side of (2.14),

as applied to the biharmonic equation with u = E, and obtain (3.55).

(c) For uniqueness, we need to prove that the only solution of

~2~ = 0 on R3, with

1 0 1 1 - x 3(3.59) ~(x)+8~lxl~vanishing at infinity for some 10E R, 11E R ,

is ~ =0, in which case 10 = 0, 1 1 = O.

By Lemma 3.4, we have ~(x) = q_x + p, a polynomial with degree

::;; 1. If 0: ~ 0, we choose a unit vector i such that q-i = 0; then

setting x = ki in (3.59) and letting k ~ 00, we conclude that 10 = 0,

and then q = O. Therefore, we must have ~ =p, 11 = 0 and

(3.60)

Now, if 11

in (3.60),

1 1 -X

P - 81iTiT ~ 0 as x ~ 00.

~ 0, by letting x = k11, k ~ 00, and then x = -k11, k ~ 00

we get p = ~~1111 =- ~~1111. Thus we have proved that

62

¢ = 0, 10 = 0 and 11 = O.

Again, Lemma 3.3 states that ¢ = Zf is a solution of the problem,

with 10 and 11 defined by (3.8). The bound (3.56) follows from (2.14)

of Theorem 2.4 (with u = 1), Lemma 3.1 (a) and Lemma 2.3 (b).

(d) For uniqueness, we must show that the only solution of

2 2/),. ¢ = 0 on IR ,

(3.61) 8~¢(x) - 10IxI2(10glxl-1)

21 Xl

[

loglxl - 2 +~

- 12 -x

1x2

Ixl 2 loglxl 1-2+

~ 0,

where

(3.62)

~ - f 0 ~ EIRE 1R2 symmetrl'c ~2 E 1R2X2,as x --~, or s me '0 ,1 11 ,

is ¢ =0, in which case 10 = 0, 11 = 0 and 12 = O.

By Lemma 3.4, ¢ is a polynomial of degree ~ 2. Therefore, from

(3.61), we must have 10 = O. Thus ¢ = 0(lxI 2) at infinity, and again

Lemma 3.4 implies that ¢ is a polynomial with degree ~ 1. Hence,

again from (3.61) (with now 10 =0), 11 = O. Applying Lemma 3.4

again, we conclude that ¢ =p, a constant, and (3.61) becomes

2

8~ - (loglx I _1. + ~.!_J'"1 - (lOg [xI2 Ix 12) 11

2x1x2- -- 1 ~ 0 as X ~ 00,I X 12 12 '

(1 11 112] := '"1

2112 122

From (3.62), we must have

(3.63)

Now by letting x

we get

(3.64)

and

T= (xl' 0) -4 00 and then X =

63

1 1(3.65) 8~P + 2 1 1 1 - 2 '22 = o.

Solving the linear system (3.63)-(3.65), we obtain ~ =p = 0, 'II =

'22 = 0; then, from (3.62), we also have '12 = O. Thus we have proved

that ~ = 0, , = 0, , = 0 and, = O.o 1 2

Again, Lemma 3.3 states that ~ =Zf is a solution of the problem,

with '0' 'I and '2 defined by (3.8). The bound (3.57) follows from

(2.14) (with u = € + 2), (3.58), Lemma 3.1 (a) and Lemma 2.3 (b). I

4. AN A PRIORI BOUND FOR THE FOURTH ORDER ELLIPTIC EQUATION

We consider the nonhomogeneous equation,

(4.1)

and derive an a priori bound for entire solutions analogous to

Theorem 3.4. The conditions on the coefficients will guarantee that

the operator L approaches the biharmonic operator near infinity at a

certain rate; and the condition of the separability of L (= L2L1

)

will be used to prove the uniqueness of the solution. Let J denote

the nxnxnxn tensor (OijOkl) with J.D4 = a2• We require:

Condition F (i) The operator L can be separated as L = L2L 1

, where

L = a .D2+ b .D + c (s = 1, 2),s s s s

and a , b , c are matrix, vector, scalar valued functions on Rn,s s s

respectively, with aI' bl and· c1

being C2 functions and

(4.2) Cs ~ 0 for n ~ 3, and Cs =0 for n = 2 (s = 1, 2).

For these coefficients, there exist constants 8, a, A, with

°> max{O, 4-n} , 0 < a < 1, A ~ 0, such that

(4.3) lias - IILo a' IIbsll_1 _O a' Il cs ll _2_0 a', , ,

IIDsaIILs_8,a' IIDsbll1_l_s_o,a' IIDsclll_2_s_0,a::;; A, s = 1, 2.

(ii) The matrices as' s = 1, 2, are symmetic, and for some

constant A, A > 0,

64

Remark

(4.4)

From L = L2L1, i.e., from

43222a-D +b.D +c-D +d.D+e = (a 2-D +b2eD+c2)(al eD +bl·D+c1),

and from Lemma 2.3, it follows from condition (4.3) that

(4.3)' lIa - JII_o,a' Ilbll_I _8,a' II cll_2_0,a' IIdll_3_8,a' lIell_4_0,a s M(A).

So, under Condition F, we can use both (4.3) and (4.3)'.

THEOREM 4.1. Suppose Condition F holds, that fEB for some T,-T,a

T > 4, and that a is the same as in Condition F. Let ~ be an entire

C4 solution of (4.1).

(a) For n ~ 5, assume that ~(x) vanishes at infinity. Then

(4.5) 114>11 0 , 1 + IIDt1>II_ 1 , 1 + IID2~11_2,1 + IID3~IL3,1 + IID4~1I_4,a

~ M(n, a, 8, T, A, A)llfll .-r,a

Moreover, for any c such that, € ~ min {o, T-4},

(4.6) lIa24>11_

4 _ € ,a ~ M(a,8, T ,A,A) II fll_T ,a'

and when also 0 < € < 1 and Ixl ~ 1,

(4.7)

65

(b) For n = 4, assume that ~Cx) - ~ologlxl vanishes at infinity

for some constant ~o. Then

(4.8)

and for any E > 0,

C4.9) 1I~IIE,I+ 1ID<PII E_1,1 + IID2~IIE_2,1 + IID3~IIE_3,1 + IID4~IIE_4,a

::; M(a,o,T,E,A,A)lIfll .-T,a

If, moreover, E < 0, E ~ 1"-4, then

C4.10) lIa2~IL4_E a::; MCa,o,T,E,A,A)lI fILT a', ,and if also 0 < E < 1 and Ixl ~ 1, then

C4.11) I~(x)-~ologlxll::; M(a,o,r.E,A,A)llfll_T a(1+lxP-Cl ogC2+lxIL,~lex

Cc) For n = 3, assume that ~(x) - ~olxl +~ vanishes at

ninfinity for some ~o E ~ and ~1 E ~. Then

(4.12) I~ol, h 11 s MCa,o,T,A,A)llfll_T a',

C4.13) 114>11 1,1 + 11D<Pllo,1 + IID2~1I_1,1 + IID34>1I_

2,1 + IID4~1I_3,a

::; MCa,o,1",A,A)llfll .-1",a

Moreover, for any c such that E ::; min {a-I, 1"-4},

C4.14) 11L\24>11_4

_ E a ::; M(a, a, 1", A,A) Il f ll _1" a', ,and when also 0 < E < 1 and Ixl ~ 1,

C4.15) I~Cx) - 'Yolxl + ~le61 ::; MCa,a,1",E,A,A)lIfll Cl+lxl)-E.IXI -1",a

loglxl

(21og!xl-l)'Yl ex

2I Xl

--+--2 [x1

2

Cd) For n = 2, assume that

~Cx) - ~olxI2(IOglxl-l) +

- ._. _._- ----------

vanishes at infinity for some constant 10

, constant vector 11

and

constant matrix 12, Then

66

(4.16)

loglxl

and for any E > 0,

(4 .17) 11~IIE+2,l + IIDt1>II E+1,l + IID2~IIE,l + I\D3~IIE_1, 1 + IID4~I\E_2,0:

:;; M(o:,o,r,E,>',A)llfll .-1",0:If, moreover, E < 0-2, E ~ r-4, then

(4.18) l\a2~IL4_E,0: ~ M(o:,O, r , E,>',A)lIfll_r ,0:'

and if also 0 < E < 1 and Ixl ~ 1, then

(4.19) I~(X) - 1o lx \2( l og lx l- 1) + (2Ioglxl-1)11•x

21 xl

- -+--2 Ixl2

Ixl2

~ M(o:,o,r,E,>',A)lIfll C1+lxl)-E l og(2+lxl).-r,o:

Proof: (a) First assume that (4.5) holds for all C4 solutions ~ such

th&t ~ vanishes at infinity.

(4.20)

where

We rewrite (4.1) as

a2~ = g,

(4.21) g := -(a-J).D ~ - b.D if> - c.D if> - d.D¢ - e~ + f.

If E is any fixed real number with E :;; min{o, r-4}, then from (4.20),

(4.21), Lemma 2.3 (b) and (c), Condition F (with (4.3)'), and (4.5) we

have the bound

(4.22) lIa 2if>1I_4_E,0: s Ila-JII_E,o:IID4if>1I_

4 ,0: + IIbIL1_E,0:1ID3~1'-3,0:

+ I\ cll_2_ E,o:I\D2if>11_2,0: + Ildl'-3_E ,0:1ID¢1'-1 ,0:

"..,u.

+ lIell_4 _ £ a ll4>ll o a + II fll_4 _ £ a" ,

s M(AHIID44>11_

4 , 0: + IID34>IL

s , a + IID24>IL

2, 0:

+ 1ID4>IL 1 0: + 114>11 0 a} + II fLr 0:, , ,~ M(a,o,r,A,A)lIfll ,-r,a

which is (4.6). When also 0 < £ < 1 and Ixl ~ 1, the inequality (4.7)

is a consequence of Lemma 3.1 (a) and (4.22).

Therefore, in order to finish the proof of (a) it is sufficient

to show that (4.5) holds. We choose £ sufficinetly small that

(4.23) 0<£ < min {1, 0/2, r-4}.

Observe that (4.3) implies

(4.24)

lIas ll o,o: ~ Ilas-Illo,o: + 11 1110 , 0: ~ Ilas-III_o,a + 11 111 0 s A + vn,

IIbs ll _1 a ~ IIbsll_1 _0 0: ~ A, II csll_2 a ~ II cs ll _2_0 0: ~ A,, , , ,

IIDsallLs 0: ~ IIDsalll_s_o 0: s A,, ,

IIDsblll_l_s a s IIDsblll_l_s_o 0: ~ A,, ,

IIDscllI_2_s a ~ IIDscllI_2_s_0 0: ~ A (s = 1, 2)., ,Hence, from (4.4) and Lemma 2.3, we have (2.13) with A replaced by

M(A). Thus Theorem 2.4 can be applied with u = 0 to get the estimate

(4.25) 114>11 0 1 + 1ID4>1I_1 1 + IID2¢11_2 1 + IID34>11_

3 1 + IID44>IL4 a, , , , ,

~ M(n,a,r ,A,AHII4>lI o + Ilfll_r a},,Therefore it is sufficient to derive, instead of (4.5), the more

conservative estimate

(4.26) 1I¢llo ~ M(n,o:, 0, r ,A,A)Ilfll_r a',Suppose that (4.26) is false. Then there exist sequences {a },sm

{bsm}, {csm}' {fm}, {4>m}' s = 1, 2; m = 1, 2, 3, '" , such that all

a ,b ,c satisfy Condition F with the same constants 0, 0:, A, A,sm sm sm

such that each f is in B ,each ~m is of class C4 and vanishes atm -1' ,a:

infinity, with

68

(4.27)

and with

3+ b .D ~m m

+ b .D +2m

2+ c .D ~ + d.~ + e ~m m m m mm

c2

)(al

.D2~ + bl.~ +m m m m m

(4.28) lI~ml\o = 1, I\ fmll_T , a: s ~ .Then inequalities (4.24), (4.25) are satisfied by the members of

these sequences; and in particular, for s = 1, 2, the sequences

{asm-I}, {bsm}' {csm}' {Dsa l m} {D

sblm}, {D

sclm}, {fm}' {~m}' {~m}'

{D2~m}' {D3~m}' {D4~m} (s =1,2) are all uniformly bounded with

respect to the appropriate norms in the spaces B r ,B r ,-U,a: -1-u,a:

B c ,B s: ,B 1 r ,B 0 ".J; ,B._", r•.' Bo, l' B_1 , i ' B_2, 1'-2-u,a: -s-u,a: - -s-u,a: -M-U-V,a '.

B B respectively. By Lemma 2.3 (f), there exist functions-3,1' -4,a:'

as-I, bs' cs' f, ~ together with DSal, DSb

l, DScl (s = 1,2), ~, D2~,

D3~, D4~ all contained in the same spaces as the corresponding

sequences such that (noting (4.23)), after passing to subsequences

(preserving (4.28)), we have with respect to the norms of the spaces

involved,

(a -I) ~ (a -1) in B ,and hence a ~ a in B ,sm s -E sm s -E

in B-2-c'

~ DSb in B1 -1-S-E'

(4.29)

b ~ b in B c ~ csm s -1-E' sm sDSa ~ DSa in B ,DSb

rm 1 -S-E i mDSc ~ DSc in B (s =

1m 1 -2-S-E1, 2),

f ~ f in B 4' ~ ~ ~ in B ,m - m E

~ ~~ in B l' D2~ ~ D2~ in BE-2 'm E- m

D3~ ~ D3~ in B 3' D4~ ~ D4~ in BE-4'm E- m

and, from Lemma 2.3, we also have

69

(4.30)

From (4.28) we obtain

IIf ll _4 = lim Ilfmll_4 s lim IIfmll_r s lim (11m) = O.

Therefore, upon passing to the limit in (4.27) we find that ~ is an

entire C4 solution of the homogeneous equation

(4.31) aeD4~ + beD3~ + ceD2~ + deD~ + e~

2 2= (a2eD + b2eD + c2)(al eD ~ + b1·D4>

+ cl~) = O.

Moreover, it is clear that (4.2) and (ii) of Condition F on c and as s

2are fulfilled. Since (4.30) implies that ~ = aleD ~ + ble~ + cl~

vanishes at infinity, the maximum and minimum principles (applied to

arbitrarily large spheres in ~n) for second-order elliptic equations

([11], [17]) may be applied to solutions ~ of the equation (4.31) to

obtain

(4.32)

We need to show that ~ =0, but first we have to investigate the

bahaviour of each ~m near infinity. We again rewrite (4.27) as

(4.33)

where

(4.34)

2(). 4> = g ,m m

g := -(a -J)eD44> - b eD34> + c eD24> + d enq, + e 4> + f .m m m m m m m m m mm m

Then, from Lemma 3.1, Condition F (with (4.3)') and (4.28) for am' bm,

(4.35)

e and 4> , f , we have the estimatem m m

Ilgmll_ 4_€.a ~ II am-JIL2€ ,aIlD4~mll€_4,a + IlbmIL l _ 2f: ,aIID3~mll€_3 ,a

+ IlcmL 2_2€ allD2cPmllc_2 a + IIdmll_3_ 2€ a llD4>mll€-l a, , , ,

+ lI emL4_ 2f: all~mllE a + I fm"_4_t a" ,

(4.37)

70

~ M(A){M(a,€,~,A).2} + 1.

Thus g E B 4 ' and then Theorem 3.5 (a) and Lemma 3.1 (a) implym - -€,a

(4.36) 4Jm = Zgm' l4Jm(x)I s M(n,dllgmll_4_€(l+l xl)-€.

By substituting (4.35) and letting m -4 00, we see that 4J vanishes at

infinity; and then by applying the maximum and minimum principles (to

(4.32», we conclude 4J =O. Hence (4.30) yields ~ -4 0, ~ -4 0,m m

D2~m -4 0, D3~m ~ 0, D4~m ~ 0 in the spaces B , B , B , B ,€ €-1 €-2 €-3

B , respectively. Again from (4.34), (4.3)' and (4.28) for am' bm,€-4

e and ~ , f , we observe thatm m m

IIgmll_4_€ s II am-JII_ uIlD\Pmll€_4 + IlbmLl_uIlD34Jmll£_3

+ IlcmIL2_uIlD24Jmll£_2 + IIdmIL3_ullD4Jmll€_1

+ lIemIL4_ull4Jmll€ + II fmll_r

~ M(AHIID44>m

ll€_4 + IID34>mll£_3 + IID24>mll£_2+ IID4>mll£_1 + l14>mll£} + ;.

Hence Ilg II ~ 0 as m~ 00. But then from (4.36), Lemma 3.1 (a) wem -r

obtain

l14Jmll£ ~ M(n,e) IlgmIL4_£ ~ 0 as x ~ 00,

which contradicts the assumption (4.28).

(b) First assume that (4.9) holds for all £ > 0 and all C4

solutions 4> such that 4>(x) - 1ologlxl vanishes at infinity for some

constant 10

, If € < 8, £ ~ r-4, then from (4.1), Condition F (with

(4.3)'), and (4.9) (with € replaced by 8-£) we obtain the bound

(4.38) 1IL\24J11_4_£ a,

s Ila-JII_8,aIlD44>118_€_4,a + Ilbll_I_8,aIlD34>118_£_3,a

---------------- ------------------------------

+ Il cll_2_o, aIID2if>lIo_E_2, a + Ildll_3_o, allDif>llo-E-1 , a

+ lI e ll_4_o allif>lIo_E a + II f ll_4_E a" ,

s M(AHIID4if>lIo_c_4 a + IID3if>lIo_c_3 1 + IID2if>llo_c_2 1, , ,

+ 11Dif>llo_E_1 , 1 + 1Iif>lIo_c,1] + II fll_r, a

~ M(a,o,r,c,A,A)lIfli ,-r,a

which is (4.10). Moreover, by Theorem 3.5 (b),

(4.39) if> = Z(~2if», 10

= - 4~ J 4(~2if»(Y)dY,4 IR

which gives the bound

(4.40) 110 1 s ~4 JIR411L12if>1I_4_c(l+lyl )-4-Edy s M(c)IIL12if>1I_4 _ c'

We take c = min {O/2, r-4} (thereby removing dependence on c), and

combine (4.40) with (4.38) to obtain (4.8). When also 0 < c < 1 and

Ixl ~ 1, the inequality (4.11) is a consequence of Lemma 3.1 (b) (with

r replaced by 4 + c) and (4.38).

Therefore, in order to complete the proof of (b) it is sufficient

to show that (4.7) holds for all c > O. First we observe that the

left-hand side of (4.7) increases as c decreases. Thus we may take c

smaller, if necessary, so that

71

(4.41) o < c < min {I, O/2}, E ~ r - 4.

The inequalities (4.24) hold as when n ~ 5, and if if>(x) - 1010glxl

vanishes at infinity for some 10, then if> E~c for all c > O. Thus we

may apply Theorem 2.4 (with u = c) and conclude, analogously to

(4.25),

(4.42) 11if>ll c, 1 + 11Dif>ll c_ 1 , 1 + IID2if>llc_2, 1 + IID3if>llc_3,1 + IID4if>llc_4 ,a

~ M(a, e ,A, A){ 11if>llc + II f ll_r a}',

72

Therefore it is sufficient to derive the estimate

(4.43) II¢II s M(o:,O,1",E,A,A)lIfll .E -1",0:

Assuming that (4.43) is false, we obtain sequences {asm}' {bsm}'

{csm}, {fm}, {¢m}, born}, s = 1, 2; m = 1, 2, 3, '" , such that all

a , b , c satisfy Condition F, such that each f is in B , eachsm sm sm m -1",0:

¢ is of class C4 and a solution of (4.27), with ¢ (x) - ~o loglxlm m m

vanishing at infinity, and with

(4.44)

We substitute (4.44) into (4.42)

obtain

II f mll_1" 0: ~ ; .,(applied to all pairs {¢m,fm}) and

(4.45) lIc/>mllE 1 + lID4>mIlE_1 1 + IID2¢mIIE_2 1 + IID3¢mIIE_3 1 + IID4¢mIIE_4 a, , , , ,~ M(0:,O,1",E,A,A)(2).

Since °> °- E > E > 0 (see (4.41» we may pass to subsequences so

that

a -I ~ a -I in B ~, b ~ b in B ~ l'sm s E-u sm s E-u-

C ~ c in B ~ 2' DSa

l ~ DSal

in B ~ ,sm s E-u- m E-u-S

DSb ~ DSb in B ~ , DSc ~ DSc in B ~ (s=1,2),1m 1 E-u-S-l 1m 1 E-u-S-2

(4.46) fm~ 0 in B_4, ¢m ~ ¢ in BO_E

'

D4> ~ D4> in B~ i ' D2¢ ~ D

2¢in B~ 2'm u-E- m u-E-

D3¢ ~ D

3¢ in B~ 3' D4¢~ D

4¢in B~ 4'm u-E- m u-E-

Then ¢ is an entire C4 solution of the equation (4.31) and, from

Lemma 2.3 (f) and (c),

(4.47) ~ := a eD2

c/> + b eDt/> + clc/> E B 2'1 1 E-

We will conclude that c/> =0, but first we must investigate more

carefully the behaviour of each ¢ near infinity. We again havem

(4.33), with gm defined by (4.34). From (4.34), (4.44), (4.45) we

have the estimate (4.35). Thus gm E B_4

_E

a' and Theorem 3.5 (b),

73

implies

4>. = Zg ,m m

By making estimates as in

10m = - 4~ J4 gm(y)dy.4 IR

(4.40), with (4.35) being applied, we see

that the sequence {1 } is a bounded sequence of real numbers. Onom

passing again to subsequences, we may assume 10m~ 10 for some real

number 10' Lemma 3.1 (b) implies that, for Ixl ~ 1,

Ic/>m(x) - 10mloglxll s M(E)llgmL4_/1+l xl )-E l og(2+lxl).

By substituting (4.35) and letting m~ 00, we obtain for Ixl ~ 1,

Ic/>(x) - 10loglxll ~ M(a,E,A,A)(1+lxl)-El og(2+lxl),

and hence

(4.48) c/>(x) - 10l0glxl vanishes at infinity, for some 10 E IR.

Since (4.47) implies that f vanishes at infinity, the maximum

and minimum principles for second order elliptic equations may be

applied to solutions f of the equation (4.31) to obtain

(4.49)

Since c/> E BE is a solution of (4.49), a modified theorem of Begehr and

Hile ([2], Theorem 5.3 with B replaced by B , E < min {1,o}) may bem m+E

applied to solutions of the equation (4.49) to get

c/>(x) - p ~ ° as x ~ 00 for some constant p;

but then (4.40) implies that 1 = p = 0, and thus c/> vanishes ato

infinity. Applying the maximum and minimum principles again, we

conclude that c/> =O. Hence (4.46) yields c/> ~ 0, Dc/> ~ 0, D24>. ~m m m

0, D3c/>m ~ 0, D4c/>m ~ 0 in the spaces BO_E

' BO_E_ 1' BO_

E_

2' BO_

E_

3'

B~ ,respectively. Again from (4.34), (4.44), we have an estimate0-£-4

(4.50) IIgmIL4-£ ~ \lam-JIL6I1D44>mIl6_c_4 + Ilbmll_I_61ID34>mIl6_c_3

+ Ilcmll_2_6I1D24>mIl6_£_2 + \Idmll_3_611D4>mI16_c_l

+ lIemll_4_6114>mI16_c + II fmIL4_£

~ M(A)[ IID44>mllo_c_4 + IID3

if>m11o-c-3 + IID24>mllo_c_2

+ lID4>mIl6_c_1 + l14>m11o_cl + ; .

Hence Ilg II ~ 0 as m~ 00. Finally, from Lemma 3.1 (a) we havem -4-c4for x E IR ,

l4>m(x) I ~ M(c)lIgmll_4_clog(2+lxl),

which implies that 1Ir/J II ~ 0 as m~ 00, contradicting (4.44).m c

(c) Again we first assume that (4.13) holds for all C4 solutions

4> such that 4>(x) - ~olxl + ~1.TiT vanishes at infinity for some

constant ~o and constant vector ~1' If c is any fixed real number

with c ~ min{6-1, r-4} , then from (4.1), Condition F (with (4.3)'),

and (4.11) we obtain the bound

(4.51) liD?4>11_4_c a s lIa- JII_1_c a11D24>1'-3 a + IIb ll_2_c aIID34>11_2 a, " , ,+ Il clL3_c aIID24>II_l a + Ildll_4_E a llD4>lIo a, , "+ lI e ll_5 _ E a ll4>11 1 a + II fll_ 4_c a

" ,~ M(A) [IID44>1I_

3 a + IID34> 1I _2 a + IID24>11_1 a, , ,

+ 1ID4>llo a + 114>11 1 a l + II fll_ r a, , ,:;:; M(a,O,r,A,A)llfll ,-r,a

which is (4.14). Moreover, by Theorem 3.5 (c),

2 1f 2 1f 24> = Z(~ 4», ~o = - 8~ 3(~ 4»(y)dy, ~1 = - 8~ 3 y(~ 4»(y)dy.IR IR

Thus, by (4.51), we have the bound

74

(4.52) I'Ysl::; 8;' JIR3IyIS(l+lyP-4-Ellil2C/>11_4_EdY ~ M(E)lIil2C/>1I_4_E

s M(E)lIil2C/>IL4

_ E a::; M(a,6,1",E,A,A)lI fIL T a (s = 0, 1)., ,Choose E = min {6-1, T-4}. Then M in (4.52) is independent of E, and

(4.12) holds. When also 0 < E < 1 and Ixl ~ 1, the inequality (4.15)

is a consequence of Lemma 3.1 (b) and (4.51).

Therefore, in order to finish the proof of (c) it is sufficient

to show that (4.13) holds. We choose E sufficiently small that

75

(4.53) o < E < min {1, (6-1)/2, T-4}.

Since inequalities (4.24) hold as when n ~ 5, Theorem 2.4 (with u = 1)

may be applied to obtain an estimate

(4.54) 1Ic/>11 1 1 + 1IDc/>lIo 1 + IID2c/>1I _1 1 + IID3c/>1I_2 1 + IID4c/>IL 3 a, , , , ,

s M(a,6,T,>',AHIIC/>11 1 + IIf lLT a}·,Hence it is sufficient to derive the estimate

(4.55)

Suppose that (4.55) is false. Then there exist sequences {asm}'

{bsm}' {csm}' {fm}, {c/>m} , {~om}' {~lm}' s = 1, 2; m = 1, 2, 3,

such that all a , b , c satisfy Condition F, each f is insm sm sm m

B , and each c/> is of class C4 and a solution of (4.27), with-1",a m

c/>m(x) - ~omlxl + ~1m.TiT vanishing at infinity and

(4.56) lIc/>mlll = 1, II f mll_r a:;;; ~ .,

Substituting (4.56) into (4.54) (applied to all pairs {c/> ,f }), we getm m

(4.57) Ilc/>mlll,l + IIDc/>m11o,1 + IID2c/>mll_1,1 + IID

3c/>mll_2,1 + IID4c/>mll_3,a

::; M(a,6,T,>',A)(2).

By (4.53), we may pass to subsequences so that

(4.58)

a -I ~ a -I in B ,b ~ b in B_2

_E

,sm s -1-E sm s

csm ~ Cs in B_3

_E

, DSaim~ DSa

l in B_ I_S_E'

DSb ~ DSb in B , DSc ~ DSc in B ,rm 1 -2-S-E im 1 -3-S-E

f ~ 0 in B , ~ ~ ~ in B l'm -4 m E+

~m ~ ~ in Be D2~m ~ D2~ in BE_I'

D3~m ~ D3~ in BE

_2

, D4~m ~ D4~ in BE

_3

(s = 1, 2).

Therefore, ~ is a C4 solution of the equation (4.31), with

2(4.59) ~ = a l-D ~ + bl-~ + cl~ E B_1,

We need to show that ~ =O. We again have (4.33) and (4,34),

and from (4.34), (4.56), (4.57), we also have, analogously to (4.51),

the estimate

76

(4.60) Ilgm1L4_E ex ::; M(A) [IID4~mll_3 a + IID3~mIL2 a + IID2~mll_1 a, , , ,

+ II~mllo a + II~mlll a] + IlfmlLr a, , ,::; M(A){M(a,8,r,A,A)_2} + 1.

Thus g E B , and then Theorem 3.5 (c) impliesm -4-E,a

(4.61) ~m = Zgm' 'om = - 8; S~3 gm(y)dy, 'Im = - 8; S~3 ygm(y)dy.

By making estimates as in (4.52) (for, and g ), with (4.60) beingsm m

applied, we find that the sequences {'om}' {'lm} are bounded sequences

of real numbers and real vectors, respectively. On passing again to

subsequences, we may assume 'om ~ '0 and 'lm ~'1 for some ' 0 E R

and '1 E ~3. Lemma 3.1 (b) implies that, for Ixl ~ 1,

I~m(x) - lom1x l + 11m-TiTI s M(E)llgmll_4_/1+lxl)-E.

By substituting (4.60) and letting m~ 00, we see that

(4.62) ~(x) - 'olxl + 'I-TiT vanishes at infinity,

3for some 'oE ~, 'IE ~ .

Since, from (4.59), 0/ vanishes at infinity, ihe-maximum and

77

minimum principles for solutions ~ of the second order equation (4.31)

imply that

(4.63) 0/ = al.n2~ + bl.~ + cl~ =0 in R3.

Again, since ~ E B1

is a solution of (4.63), Theorem 5.3 of [2] states

that

(4.64) ~(x) - qex - p ~ 0 as x ~ 003for some pER and q E R .

Then, (4.62) and (4.64) imply that

(4.65) qex + p - ~olxl + ~leTiT vanishes at infinity,

3for some p, ~o E R and some q, ~I E R. A proof analogous to that of

Theorem 3.4 (c) may be appied to ensure that p = ~o= 0 and q = ~I= O.

Hence ~ vanishes at infinity, and then ~ =0 follows form the maximum

and minimum principles (for the equation (4.63».

We now have from (4.58) that ¢ ~ 0, D¢ ~ 0, n2¢ ~ 0,m m m

D3¢ ~ 0, D4¢ ~ 0 in the spaces B ,B, B I' B 2' B 3'm m c+1 c c- c- c-

respectively. Again from (4.34), (4.56), we have

IIgmlL4- c s II am-JII_I_uIlD4c/>mllc_3 + IIbmll_2_2E1ID3~mllc_2

+ IIcmll_3_2EIID2c/>mllc_1 + Ildmll_4_2E11D4>mllc

+ lIemll-5_2cll~mllc+1 + Il f ll_4_c

::; M(A) [IID4~mllc_3 + IID3~mllc_2 + IID2c/>m

ll€_11

+ IIThPmll c + lIc/>mllc+l] + m'Hence IIgmll_4-€ ~ 0 as x ~ 00; and then, from Lemma 3.1 (a),

Ilc/>mlli s M( c) IlgmlL4_c ~ 0 as m ---? 00

which contradicts (4.56).

(d) As in the proof of (b), we first assume that (4.17) holds for

all £ > 0 and all C4 solutions ~ such that

(4.66)

loglxl

as x ~ 00, for some 70

E R, 71

E R2, symmetric 7 2

1, £ < 6-2, £ ~ r-4, then from (4.1), Condition F, and (4.17) (with £

replaced by 6-2-£) we obtain the estimate (4.38) which is (4.18).

78

Hence Theorem 3.5 (d) implies that

~ = Z(a2~), 70 = 8; S 2(a2~)(Y)dY,R

71 = 8; SR2 y(a2~)(Y)dy, 72 = 8; SR2

Thus, using (4.18), we have

T 2yy (a ~)(y)dy.

(4.67) 17s 1 ~ 8; J2IYls(1+IYI)-4-£lIa2~1I_4_iYR

~ M(£)lIa2~1I_4_£ ~ M(a,o,r,£,A,A)lI f ll_r a (s = 0, 1, 2).,Choose £ = min {(O-2)/2, r-4}; then this bound is independent of £ and

becomes (4.16). Again, when also 0 < £ < 1 and Ixl ~ 1, the

inequality (4.19) is a consequence of Lemma 3.1 (b) and (4.18).

Therefore, it is sufficient to prove that (4.17) holds for all

£ > o. Without lose of generality, we may assume, analogously to

(4.41),

(4.68) o < £ < min {1, (6-2)/2}, £ ~ r - 4.

The inequalities (4.24) still hold, and if ~ satisfies (4.66), then

~ E B 2 for all £ > O. Thus we may apply Theorem 2.4 (with u = 2+£)£+

and obtain

79

(4.69) 1Iif>II E+2 , 1 + 11Dc/>IIE+1, 1 + IIn 2if>IIE,1 + IIn3if>II

E_1, 1

s M(a:,<5,7",E,>.,AHIlif>1IE+2

+ IIn4if>IIE_2 a:,

+ IIfll }.-7",a:

It is, therefore, sufficient to derive instead of (4.17) the estimate

(4.70) 1Iif>IL 2 :s; M(a:,O,7",E,>',A)llfll .<;+ -7",a:

Suppose that (4.70) is false. Then there exist sequences {a }sm '

{bsm}, {csm}, {fm}, {if>m}, b om}' b 1m}, b 2m}, s = 1,2; such that all

a , b , c satisfy Conditon F (hence each c =0 in this case),sm sm sm sm

each f is in B , each if> is in B and a solution of (4.27) (withm -7", a: m c+2

c =0), with (4.66) holding for all {if> , 10

' 1 ,12m} and withsm m m nn

(4.71) 1Iif>lI c+2 = 1, IlfmIL7",a: :s; +.Therefore, by substituting (4.71) into (4.69) (applied to all pairs

{A f}) we have'I'm' m '

(4.72) IIif>m ll E+2 , 1 + IIDc/>mIIE+1, 1 + IIn2if>mllc, 1 + IIn3if>mIl E_1, 1 + IIn4if>mIlE_2,a:

:s; M(a:,O,7",E,>',A)(2).

Since now <5 > <5 - E > E + 2 (see (4.68)) we may pass to subsequences

to ensure that (4.46) holds. Then if> is a C4 solution of the equation

(4 .31) (with c = c =0 e =0)' and1 2' ,

2(4.73) ~(x)-~if>(x) = a1(x)en

if>(x)+b1(x)oDc/>(x)-aif>(x)

~ 0 as x ~ 00.

Since ~2if> = - (a-J)on4if> - bon3if> - con2¢ - doDc/> E BE

_O_2

C B_4

_E

, by

Theorem 3.5 (d) and Lemma 3.3, we have

and then, by [2, Lemma 3.2], there is a real constant 1,

1 1 J (~2if»(x)dx, such that ~¢(x) - 1loglxl ~ 0 as x ~ 00.= 2'1l" IR2

Hence, from (4.73), we have

80

~(x) - ~loglxl ~ 0 as x ~ 00 for some ~ E ~.

But since ~ solves the second order elliptic equation,

zazeD ~ + bz.Df =0,

the maximum and minimum of ~ on any circle about the origin must occur

on the boundary of this circle. We conclude that we must have ~ = 0,

~ == O.

Now we need to prove that ~ == O. We again have (4.33), (4.34);

and from (4.34), (4.68), Condition F (with (4.3)'), (4.72), (4.71), we

also have, analogously to (4.35),

(4.74) Igm" _4 - E,a s IIam-JILz_ZE,aIID4~mIlE_z ,a + IIbmILs_z€ ,aIIDs~mIlE_1 ,a

+ II cm"-4_u ,aIlD2~mll€ ,a + IIdmll_5_u ,all~ 1I€+1,a

+ lI em"-a_z€ ,all~mIIE+z, a + II f mIL4 _E

::;; M(A){M(a,6,T,E,A,A)e2} + 1.

Thus gm E B_4

_E

,a' and Theorem 3.5 (d) implies

~m = Zgm' ~Om = 8; J z gm (y )dy ,~

~1m = 8; J z ygm(y)dy, ~zm = 8; J z yyTgm(y)dy.~ ~

By making estimates as in (4.67), with (4.74) being applied, we find

that the sequences {~om}' {~1m}' {~2m} are bounded sequences of real

numbers, vectors, symmetric matrices, respectively. On passing again

to subsequences, we may assume ~om ~ ~O' ~1m ~ ~1' ~2m ~ ~2 for2 • 2x2some ~o E ~, ~1 E ~ , symmetrIc ~z E ~ . From Lemma 3.1 (b) we

have, for Ixl ~ 1,

loglxl

(4.75) I~m(x) - ~omlxI2(loglx!-1) + (210glxl-1)~lm·x2

[

loglxl - ~ +~2 Ixl2

- ~2m·x

lx2

Ixl2

:;;; M(E)l\g 1\ (1+lx\)-E l og(2+lxl).m -4-E

81

By substituting (4.74) and letting m~ 00, we see that (4.66) holds

as x ~ 00.

2Since ~(x) equals zero for all x E ~ , we have

(4.76)

Again, since ~ E B 2 is a solution of (4.76), the same modifiedE+

theorem of [2] implies that

(4.77)

for some ~ E ~ and some harmonic polynomial

(4.78)

Then (4.66) and (4.77) imply that (4.66) holds with ~(x) replaced by

pz(x) + ~loglxl· Since the growth rates of x~, xlx2' x~, xl' x2 '

loglxl, IxI 20oglxl-1), (21cglxl-1)xl and (2Ioglxl-1)X2

are variant as

x ~ 00 from different directions, it is easy to show, from (4.66),

(4.77) and (4.78), that

(4.79) rll

= r1 2 = r 2 2 = ql = q2 = 10 = 0, ~l = 0, 1 = ~11 + ~22'

and

2 2

(4.80) (XII) ( x2 1) - 2~

xlx 2~ 0,P - ~ - 2"111 - ~ - 2"12 2 12

Ixl2

as x ~ (Xl, where [1'11 1'12] := 1'2'1'12 1'22

Now by letting x = (xl' O)T~ (Xl and

have

rX = (X

2'0) ~ (Xl in (4.80), we

82

1P ± 2(1'11 - 1'12) = 0,

and then combining with l' = 1'11 + 1'22' we get

(4.81) 1'11 = 1'22 = 1'/2, P = O.

Thus, from (4.80) (with (4.81) applied), we also have 1'12= O.

Hence, from (4.78), (4.79), (4.81), we have P2 =0; and then,

from (4.77), we see that ~(x) - 1'loglxl vanishes at infinity. But

since ~ solves the second order elliptic equation (4.76), the maximum

and minimum of ~ on any circle about the origin must occur on the

boundary of this circle.

4> =O.

We conclude that ~ = 0 (hence ~ - ~ - 0), '11- '22- ,

Therefore, (4.46) yields DS~ ~ 0 in B~ ,s = 0 1 2 3'f'm v-E-S ""

4. Again from (4.34) and (4.71) we have estimate (4.50). Hence

which implies

as x ~ 00. Finally, from Lemma 3.1 (a),

14> (x l ] :;; M(E)llg II O+lxl)2 I og(2+lxl),m m -4-£

that 114> II 2 ~ 0 as x ~ eo, contradictingm E+(4.71) . I

Theorem 4.1 can be restated as the following uniform way:

Theorem 4.1' . Suppose Condition F holds, that fEB for some T,-T,O:

T > 4, and that 0: is tha same as in Condition F. Assume that ~ is an

4-nentire C4 solution of e4.1) in ~n en~2) such that 4>ex) - L l' .Dsrex)ss=o

vanishes at infinity for some constants l' of the same dimensions ass

-- ---------- --------

83

h I s M(a,6,1",A,A)llfll ,s -1",a

and for any £ > 0 (£ may equal zero whenever n is odd or n ~ 4+1),

4-1

s~o"osq,l!max(0, 4-n)+£-s, 1 + lIo4q,lImax( 0, 4-n)+£-4,a

~ M(a,6,1",£,A,A)l!fl! .-1",a

If, moreover, £ < 8 - max{O, 4-n}, £ ~ 1" - 4, then

I!Li4 / 2q,IL4_£ ,a ~ M(a,6,1",£,A,A)lI fll_1",a;

and if also 0 < £ < 1 and Ixl ~ 1, then

if n is odd or n ~ 4+1,

if n is even and n ~ 4.

5. EXISTENCE OF SOLUTIONS

We mow demonstrate the existence of entire solutions of the

nonhomogenous equation,

(5.1) L<P:= ao0 4q, + bo03q, + co02q, + doD</> + eq,

= L2L1q,

:= (a200

2+ b

200 + c

2)(a1002q, + b

1°oq, + cIq,) = f,

with certain prescribed behaviour at infinity. Again we assume

Condition F on the cofficients of L.

THEOREM 5.1. Suppose Condition F holds on the cofficients of Land

that fEB for some r > 4.-1",a

(a) For n ~ 5, there exists a unique entire C4 solution q, of

(5.1) such that q, vanishes at infinity. Moreover, for this solution q,

the bound (4.5)-(4.7) of Theorem 4.1 (a) hold.

84

(b) For n = 4, there exists a unique entire C4 solution ~ of

(5.1) such that ~(x) - ~ologlxl vanishes at infinity for some constant

~o. Moreover, for this solution ~ the bounds (4.8)-(4.11) of Theorem

4.1 (b) hold.

(c) For n = 3, there exists a unique entire C4 solution ~ of

(5.1) such that ~(x) - 70lxl + 71-TiT vanishes at infinity for some

constant ~o and constant vector 71

, Moreover, for this solution ~ the

bounds (4.12)-(4.15) of Theorem 4.1 (c) hold.

(d) For n = 2, there exists a unique entire C4 solution ~ of

(5.1) such that

(5.2)

loglxl

~ 0,

as x ~ 00, for some constant 70

, constant vector 71

and constant

symmetric matrix 72

, Moreover, for this solution ~ the bounds

(4.16)-(4.19) of Theorem 4.1 (d) hold.

Proof: In all cases (a)-(d), the uniqueness of the solution and

bounds (4.5)-(4.17) are consequences of Theorem 4.1. Therefore, we

only need to establish existence of a solution.

(a) Let E a be fixed number such that

o < E < 1, E ~ min {8/2, T-4}.

We consider a Banach space ~ defined by

~ := {~ E C4(~n) : ~2~ E B , ¢(x) vanishes at infinity}-4-E,o:

with norm

(5.3) 1Iif>1I* := 11112if> 1I_4_ E 0:.,By Theorem 4.1 (a), applied to L =112, for any if> in B we have

(5.4) 11if>llo,1 + 1ID4>1I_1,1 + IID2if>1I_2,1 + IID3if>1I_3,1 + IID4if>11_4,0:

::; M(n,0:,E)1I112if>1I_4_E 0:',

and for Ixl ;a: 1,

(5.5) Iif>(x) I ::; M(n,0:,dIl112if>1I_4_E o:(l+l x l )-E.,

From (5.4)-(5.5), it is easy to verify that B is indeed a Banach

space (see the similar proof in (b)).

We also consider two families of linear operators defined by

85

(5.6)

and

(5.7)

Setting

we have

and

2(5.9) Itif> = L2(a1t·D if> + b1t·D¢ + c1t¢)·

=: (at-J).D4if> + bt.D3if>

+ Ct · D2¢ + dt°D¢ +

We first note that, for 0 ::; t ::; 1,

(5.10) IlascIII_6,0: = tllas-III_6,0: ::; A,

IIDsa

l t 11-s-6 a' IIDsb

l t"-I-s-o Ct', ,

and then, from (5.5), Lemma 2.3, (4.3)'

Ilbstll_I_O,0:' IlcstlL2_0 ,0: s A,

IIDSCltll_2_s_o Ct::; A (s = 1,2),,

and (5.4), if if> E B,

(5.11) III t4>II_4_€,a:;;; MII4>lI o,a + IID4>II_ I,a + IID24>IL2,a

+ IID34>IL3,a + IID44>1I_

4,a] + 1I~24>IL4_E,a

:;;; M(n,a,€,A)II4>II*·

Similarly, if 4> E S,

(5.12) IIPt4>II_4_€ a::; M(n,a,E,A)II4>II*·,Hence, both :Pt and It, 0 :;;; t :;;; 1, map S into B We also get

-4-€,a

from Condition F that

(5.13) Cst = tc s ::; 0 if n ~ 3, and Cst =0 if n = 2 (s = 1, 2);

and, for any x, e ERn,

86

(5.14) ast(x)e.e = (l-t)leI2

+ tas(x)e·e

~ (l-t)leI2

+ tAlel2~ min{I,A} lel 2 (s = 1, 2).

Therefore, Theorem 4.1 can be applied, and for all 4> E Sand 0 ::; t ::; 1

we have estimates

(5.15)

and

(5.16) 114>11* s M(n,a,6,'T,).,A)IIIt4>11 .-'T,a

According to Theorem 5.2 in [11], the Schauder continuation method

applies. Theorem 3.5 (a) and Theorem 4.1 (a) assert that the operator

2:Po = ~ maps S onto B . Hence, from (5.15), any :Pt, 0 ::; t ~ 1,-4-E,a

does the same and, in particular, :PI = L26

= I o maps S onto B .-4-E,a

Then, from (5.16), applying the Schauder continuation method again, we

conclude that II = L2LI = L maps S onto B_4

_E

,a '

B , the proof of (a) is complete.-4-€,a

(b) Let € be chosen and fixed so that

Since B C-'T,a

o < € < 1, E:;;; min {6/2, 'T-4}.

87

We define a now as the Banach space

(5.17)

at infinity for some real constant ~o}'

with the norm (5.3). By Theorem 4.1 (b), applied to L = ~2, for any ~

in a with corresponding ~o we have

(5 .18) I~o I :;:; M(a, e ) 1I~2~1I_4_E a',(5.19) 1I~IIE,l + 1ID4>II E_1,1 + IID2~lIc_2,1 + IID3~IIE_3,1 + IID4~IIE_4,a

s M(a, E) 11~2~IL4_c a',

and for Ixl ~ 1,

(5.20) I~(x) - 'Yologlxll :;:; M(a,E)II~2~11_4_E a(l+lxl )-E l og(2+lxl).,(These results also follow from Theorem 3.5 and Lemma 3.1.) In order

to verify that a is a Banach space, we consider a sequence {~ },m

Cauchy in the norm (5.3) of a, with corresponding associated constants

{~om}· Inequalities (5.18)-(5.20) show that {~m} is also Cauchy in

the norm determined by the left hand side of (5.19), that the sequence

t~om} is a Cauchy sequence of real numbers, and that the limits ~ and

~ of {~ } and {~ } satisfy an inequality similar to (5.20), therebyo m om

implying that ~(x) - ~ologlxl vanishes at infinity. Hence ~ E a, and

we conclude that a is indeed a Banach space with respect to the norm

(5.3).

We again consider the families of operators Pt and It, 0 ~ t ~ 1,

defined by (5.6)-(5.9), with (5.10), (5.13), (5.14) also being valid.

Then, from (5.10), Lemma 2.3 and (5.19), we have, analogously to

(5.11)-(5.12),

88

(5.21) IIPt4>II_4_£,a' III t4>IL4_£,a:;; MII4>II£,a + 11D4>1I£_l,a + IID24>1I £_2,a

+ IID34>1I £_3 a + IID44>1I£_4 a] + II Ll24>IL4_£ a, , ,

:;; M(a, e ,A) IILl24>1I _4_£ a =M(a, e ,A) 114>11*.,This shows that both Pt and It, 0 :;; t :;; 1, map a into B . Then

-4-£,a

Theorem 4.1 (b) (with T = 4 + £) gives the bound, for all 4> E a,

(5.22) 114>11* :;; M(a,6,£,>',A)emin{IIPt4>II_4_£ a' II I t4>1L4_£ a}', ,Theorem 3.4 (b) and Theorem 4.1 (b) assert that Po = Ll2 maps a onto

B .-4-£,a

= L. Since B C B 4 ' the proof of (b) is complete.-T,a - -£,a

(c) Let £ be a fixed number such that

o < e < 1,

Define a as the Banach space

£:;; min {(O-1)/2, T-4}.

a := {4> E c4(~3) : Ll2

4> E B_4_£,a' 4>(x) - ~olxl + ~leTiT vanishes

at infinity for some constant ~o and constant vector ~1}

with the norm (5.3). The rest of the proof is similar to the proofs

that we presented in (a) and (b).

(d) Let £ be chosen and fixed so that

o < £ < 1, £ ~ min {(O-2)/2, T-4}.

Define a now as the Banach space

a := {4> E c4(~2) : Ll24> E B 4 and (5.2) holds as x ~ 00- -£,a

2 . 2X2}for some ~o E ~, ~1 E ~ , symmetrIC ~2 E ~

with the norm (5.3). Again, the rest of the proof can be obtained by

following the procedure of (a) or (b). I

In order to discuss the existence of entire solutions of (5.1)

with polynomial growth at infinity, we require the following

---- --------------

modification of Condition F.

89

Condition MF The coefficients of L satisfy Condition F, and moreover,

there exists a nonnegative integer m with m' := m - max{O, 4-n} ,

such that

From (5.1) and Lemma 2.3, we see that (5.23) implies the

following condition on the coefficients a, b, c, d and e:

(5.23)' lIa - JILm, -6,0:' IIbll_m, -1-6 ,0:'

IIdll_m' _3_6 0:',

IIcll_m' _2_0 0:',

lIe!!_m' -4-6 0: s M(A).,

After replacing Condition F by the stronger Condition MF, we will

have the following extension of Theorem 5.1.

THEOREM 5.2. Suppose Condition MF holds on the coefficients of L,

and that fEB for some 7 > 4. Then:-7,0:

(i) For any biharmonic polynomial p, with (degree p) ::; m, there

exists a unique entire C4 solution ~ of (5.1) such that

(a) if n ~ 5, ~(x) - p(x) ~ 0 as x ~ 00;

(b) if n =4, ~(x) - p(x) - ~ologlxl ~ 0 as x ~ 00 for some

constant ~o E JR;

(c) if n =3, ~(x) - p(x) - ~ologlxl + ~1.TiT ~ 0 as x ~ 003

for some ~o E JR and ~l E JR ;

(d) if n = 2, ~(x) - p(x) - 10IxI2(loglxl-l) + (2Ioglxl-l)11ex

21 Xl X1X2

[

l og IX I - '2 +~ I 2 ]

- 'Y,. X,X. Log lx ] :11. + X: ----+ 0

Ixl 2 2 Ixl2

as X~ 00 for some 10 E~, 11 E ~2, symmetric '2 E ~2X2.

(ii) If ~ is an entire C4 solution of (5.1), with ~ E Bandm+O'

o ~ 0' < min {I, o-max(O,4-n)}, then there exists a unique

90

biharmonic polynomial p, with (degree p) ~ m, such that (a)-(d)

above hold.

Proof: (i) The proof of the uniqueness of ~ again follows from the a

priori bounds of Thorem 4.1. By Lemma 2.3 (e), we have p E B ,Dp Em,l

B 1 l' D2p

E B 2 l' D3p

E B 3 l' D4p

E B 4 l' Writingm- , m- , m- , m- ,

Lp = A2p + (L - A2)p = = (a-J)eD4p + beDsp + ceD2p + deDp + ep,

and applying Condition MF (with (5.23)' and (4.3)' being applied), we

have, for 0 < E < 0 - max{O, 4-n} ,

(5.24) IILpll_4_E,a:5 lIa-JILm_E,aIID4pllm_4,a + IIbll_m_l_E,aIlDspllm_s,a

+ II cILm_ 2_E allDzpllm_z a + IIdll_m_ 3 _ E a11DPllm_1 a, , "+ Ilell_m_4_E a11pllm a', ,

s M(A) [IID4pllm_4

1 + IIDspllm_s 1 + IIDzpllm_2 1, , ,

+ IIDPllm_1 1 + IIp11m 1]', ,and hence Lp E B . By Theorem 5.1, there exists a unique entire

-4-E,a

C4 solution ~ of the equation

L~ = f - Lp

such that if n ~ 5, ~(x) ~ 0 as x ~ 00; if n = 4, ~(x) - 10loglxl ~

91

x+ "( e'"T::T ~ 0 as1 'XI

o as x ~ 00 for some 10 E~; if n = 3, ~Cx) - 10lxI

x ~ ~ for some 10 E ~ and 1 1 E ~3; and if n = 2,

C5.25) ~Cx) - 10 Ix I2Cloglxl-1) + C2loglxl-1)11ex

2

[

loglxl - i + 1:~2- 1 2 -

x1x2

Ixl2

X1X2

log IXI

I

:

I:

+ -~J ---. 02 IxI2

as x ~ 00 for some 10 E~, 11 E ~2 and symmetric 12 E ~2X2. Setting

~ := ~ + p, we obtain the desired solution ~.

Cii) The proof of the uniqueness of p follows from the a priori

bounds of Theorem 3.5 applied to a2.

In order to show existence of p, we observe that if ~ is a C4

solution of C5.1), with ~ E B ,then Theorem 2.4 implies thatm+u

~ E Bm+u , I ' ~ E Bm+u_1 , 1 ' D2~ E Bm+u_2 , 1 ' D3~ E Bm+u_3 , 1 ' and

D4~ E B 4. We also havem+u- ,a

a2~ ; ~ _ CL-a2)~ = f - Ca-J)eD4~ - b.D3~ - ceD2~ - de~ - e~.

Choose € so that 0 < € < r-4 and € ~ min {1, o-maxCO,4-n)} - u. Then,

from Lemma 2.3 and Condition MF,

lIa2~1I_4_f,a ~ II fll_ 4_f,a + lIa-JII_m_f_u,aIlD4~lIm+u_4,a

+ Ilbll_m_l_€_u,aIlD3~lIm+0"_3,0: + IIcll_m_2_f_0",o:iID2~lIm+0"_2,0:

+ Ildll_m_ 3_€_u,0:11D4>lIm+0"_1 ,0: + Ilell-m-4-€-u,all~lIm+u,a4

s Ilfll_r 0: + MCA) L IIDs~llm+O"_s a', s=o '

which implies a2~ E B 4 . Therefore, by Lemma 3.3 the potential- -€,a

Zca2~) is C4 in ~n, with a 2CZCa2¢)) = a 2¢. Moreover, Ca)-Cd) of Lemma

3.3 are satisfied by Z(~2~). The function

92

(5.26)

is biharmonic in ~n. If n ~ 4, then, since ~ E B ,Z(~2~) E B ,m+O' 0'

Lemma 3.4 asserts that p is a polynomial with (degree p) s m. The

above reasoning applies for the cases of n = 3, m ~ 1 and n = 2, m ~ 2

since Z(~2~) is in Bl and B2

+0' , respectively. For n = 3, m = 0, we

have ~ E BO', with Z(~2~) - 10lxI + 7l .TiT vanishing at infinity, and

then Lemma 3.4 implies p(x) = q.x + Po' a polynomial with degree S 1.

But then

~(x) - q.x - Po - 10l x l + 1 l .TiT ~ ° as x ~ ~,

with ~ E BO', 0' < 1. It follows that 10 = 0, q = 0, and p(x) = Po is

a polynomial of degree zero. For n = 2, m = ° or 1, we have ~ E Bm+O'

C Bl and (5.25) holds with ~ replaced by Z(~2~). Then Lemma 3.4+0'

implies that

2 2p(x) = rllx l + 2r 12xlx 2 + r 22x2 + qlx l + q2x2 + Po'

a polynomial of degree S 2. Hence, from (5.26), (5.25) for Z(~2~),

~(x) - rllx~ - 2r 12xlx2 - r22x~ - qlx l- q2x2 -Po - 70Ix I2Cl og lx l- 1)

2

[

loglxl - ~ +~2 Ixl2

+ (2Ioglxl-1)1l•x - 12• xlx2

Ixl 2

~ ° as x ~ 00.

If m = 1, then ~ E Bl+O', and hence 70 = r l l = r 12 = r 22 = 0, and

p(x) = qlx l + q2x2 + Po is a polynomial of degree S 1; if m = 0, then

~ E BO', and it follows that 11 = 0, ql = q2 = 0, and hence p(x) =Po'

(5.27)

a polynomial of degree zero. Finally, writing ~ - p =Z(~2~) gives

(a)-(d) . I

Remark. Theorem 5.1 and 5.2 can also be restated in a uniform way as

we did for Theorem 4.1. For example, the statements (a)-(d) of

Theorem 5.2 can be writen in the following uniform way:

4-n~(x) - p(x) - L , eDsr(x) ~ 0 as x ~ 00

s=o s

for some constant ~ of the same dimensions as DSr.s

For the special case of the homogeneous equation,

93

(5.28)

Theorem 5.2 implies the following:

COROLLARY 5.3. Suppose Condition ~w holds on the coefficients of L.

Then the space of entire C4 solutions of (5.28) which are in B formsm

a finite dimensional vector space, of dimension the same as that of

the space of biharmonic polynomials of degree no greater tilan m.

There is a one-to-one correspondence between the members of these two

spaces, determined by (5.27) or, equivalently, by statements (a)-(d)

of Theorem 5.2.

REFERENCES FOR PART II

[1] N. ARONSZAJN, T. M. CREESE and L. J. LIPKIN, "Polyharmonic

Functions," Clarendon Press, Oxford, 1983.

[2] H. BEGEHR and G. N. HILE, Schauder estimates and existence

theory for entire solutions of linear elliptic equations, Proc.

Roy. Soc. Edinburgh, 110A (1988), 101-123.

[3] E. BORN and L. K. JACKSON, The Liouville theorem for a

quasi-linear elliptic partial differential equation, Trans.

Amer. Math. Soc., 104 (1962), 392-397.

[4] A. DOUGLIS and L. NIRENBERG, Interior estimates for elliptic

system of partial differential equations, Comm. Pure Appl.

Math., 8 (1955), 503-538.

[5] D. R. DUNNINGER, On Liouville-type theorems for elliptic and

parabolic systems, J. Math. Anal. Appl., 72 (1979), 413-417.

[6] A. FRIEDMAN, On n-metaharmonic functions and functions of

infinite order, Proc. Amer. Math. Soc., 8 (1957), 223-229.

94

[7] A. FRIEDMAN, "Partial Differential Equations of Parabolic Type,"

Englewood Cliffs, N.J., Prentice-Hall, 1964.

[8] A. FRIEDMAN, "Partial Differential Equations," Holt, Rinehart

and Winston, New York, 1969.

[9] A. FRIEDMAN, Bounded entire solutions of elliptic equations,

Pacific J. Math., 44 (1973), 497-507.

[10] D. GILBARG and J. SERRIN, On isolated singularities of solution

of second order elliptic differential equations, J. D'Analyse

Mathematique, 4 (1954-56), 309-340.

[11] D. GILBARG and N. S. TRUDINGER, "Elliptic Partial Differential

Equations of Second Order," Springer-Verlag, Berlin, 1977.

[12] S. HIDEBRANT and K. WIDMAN, Satze vom Liouvilleschen Typ fur

quasi linear elliptische Gleichungen und Systeme, Nachr. Akad.

Wiss. Gottingen Math.-Phys. Kl., II, 4 (1979), 41-59.

[13] G. N. HILE, Entire solutions of linear elliptic equations with

Laplacian principal part, Pacific J. Math., 62 (1976), 127-140.

[14] A. V. IVANOV, Local estimates for the frist derivatives of

solutions of quasilinear second order elliptic equations and

their applications to Liouville type theorems, Sem. Steklov

Math. Inst. Leningrad, 30 (1972), 40-50; Translated in J. SOy.

Math., 4 (1975), 335-344.

[15] Z. NEHARI, A differential inequality, J. Analyse Math., 14

(1965), 297-302.

95

[16] L. PELETIER and J. SERRIN, Gradient bounds and Liouville

theorems for quasilinear elliptic equations, Annali Scuola Norm.

Sup. Pisa Cl. Sci. (IV) 5 (1978), 65-104.

[17] M. H. PROTTER and H. F. WEINBERGER, "Maximum Principles in

Differential Equations," Springer-Verlag, Berlin, 1984.

[18] R. REDHEFFER, On the inequality au ~ feu, Igrad ul), J. Math.

Anal. Appl., 1 (1960), 277-299.

[19] J. SERRIN, Entire solutions of nonlinear Poisson equations,

Proc. London Math. Soc. 24 (1972), 348-366.

96

[20] J. SERRIN, Liouville theorems and gradient bounds for

quasilinear elliptic systems, Arch. Rat. Meek. Anal., 66 (1977),

295-310.