radially symmetric solutions to a superlinear …

transactions of theamerican mathematical societyVolume 315, Number 1, September 1989

RADIALLY SYMMETRIC SOLUTIONS TO

A SUPERLINEAR DIRICHLET PROBLEM IN A BALL

WITH JUMPING NONLINEARITIES

ALFONSO CASTRO AND ALEXANDRA KUREPA

Abstract. Let p, <p : [0, T] —» R be bounded functions with <p > 0. Let

g: R —> R be a locally Lipschitzian function satisfying the superlinear jumping

condition: (i) limu->-ao(g(u)/u) e R, (ii) limu^<x(g(u)/ul+f) = oo forsome

p > 0 , and (iii) limu^oo(w/g(u))Nl2(NG(Ku) - ((N - 2)/2)u • g(u)) = oo for

some k 6 (0,1] where G is the primitive of g . Here we prove that the number

of solutions of the boundary value problem Au + g(u) = p(\\x\\) + c<p(\\x\\) for

x e R* with ||;c|| < T, u(x) = 0 for ||x|| = T, tends to +oo when c tends

to +00 . The proofs are based on the "energy" and "phase plane" analysis.

1. Introduction

In this paper we consider the existence of solutions to the Dirichlet problem

Au + g(u) = p(\\x\\) + ctp(\\x\\), xe£l,

u = 0, xeôïl,

where Q is the ball of radius T in R centered at the origin, g : R —» R is

a locally Lipschitzian function, p e L (Q), c e R, and <p: [0, T] —> R is a

differentiable function with

(1.2) <p>0 on[0,r].

In addition we assume that the problem is superlinear with jumping nonlinear-

ities, i.e., that there exist real numbers Af and p > 0 such that

(1.3) lim ^p- = M,

(1.4) lim ^ = oo.

For the sake of simplicity of the proofs we assume, without loss of generality,

that

#(0) = 0 and g is strictly increasing, M > 0,

pe L°°(Q), and 1 < <p(t) < 2 for all t e [0,T].-

Received by the editors June 8, 1987 and, in revised form, March 29, 1988.1980 Mathematics Subject Classification (1985 Revision). Primary 35J65; 34A10.Key words and phrases. Dirichlet problem, superlinear jumping nonlinearity, singular differential

equation, radially symmetric solutions.

©1989 American Mathematical Society0002-9947/89 $1.00+ $.25 per page

353

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

354 ALFONSO CASTRO AND ALEXANDRA KUREPA

In order to state our main results we introduce the following notations:

(1.6) L(zc, u) = NG(ku) - ((N - 2)/2)ug(u),

(1.7) L(k)= lim L(K,u)(u/g(u))m,u—yoo

where G(u) = /0" g(v) dv and k e (0,1].

Our main result is

Theorem A. If (1,2)-(l.5) hold and L(k) = oo for some k e (0,1], then there

exists a positive integer J and an increasing sequence {c¡ : j = /, J + 1, ...}

tending to oo such that for c > c¡ the equation (1.1) has a radially symmetric

solution Uj with w(0) < 0, and u¡ has j interior nodal surfaces. In particular

if c > Cj then the equation (1.1) has j - J radially symmetric solutions with

u(0)<0.

Theorem A is in the spirit of studying boundary value problems for which the

interval (hmu_>_oo(g(u)/u), limu_too(g(u)/u)) contains at least one eigenvalue

of -A with Dirichlet boundary conditions. Such a problem is called superlin-

ear (resp. sublinear) if limuôo(g(u)/u) = oo (resp. limuôo(g(u)/u) < oo).

The one-dimensional superlinear version of Theorem A is given in [5], and it

motivated our result (see also [4 and 13]). For studies on the sublinear case we

refer the reader to [11] and references therein.

The proof of Theorem A is based on the shooting method that we have also

used in [3]. We study the singular initial value problem

,.-, u" + ^û' + g(u)=p(t) + cy>(t), te[0,T],(1.8) i

u(0) = d, u'(0) = 0,

where d e R. A simple argument based on the contraction mapping principle

shows that ( 1.8) has a unique solution u(t, d, c) on the interval [0, T] depend-

ing continuously on (d,c) (see [3, Lemma 2.1]). Radially symmetric solutions

of (1.1) are the solutions of (1.8) satisfying

(1.9) u(T,d,c) = 0.

We analyze the energy of the corresponding solutions, i.e., we analyze the func-

tion

(1.10) E(t,d,c)={U'{t'^C)) +G(u(t,d,c)).

In order to count the number of zeros of the solutions to (1.8) we show that

(1.11) E(t, -cC,c)>0

for C> (N+l)/(N+4) and c sufficiently large. In turn ( 1.11 ) implies that in the

(u,u) plane, for c sufficiently large and for d < _c(Ar+1)/(A,+4) ) a continuous

argument function y/(t,d,c) can be defined. The function y/(t,d,c) is such


JUMPING NONLINEARITIES 355

that y/(t,d,c) = in + n/2 (i an integer) if and only if u(t,d,c) = 0. We

show that (see Lemma 4.1 and Lemma 2.7)

(1.12) limsupy/(T, d,c) = S,—d—yoo

and that

(1.13) liminfy/(T,-ci,c) = oo,c—yoo

where S eR and £ € [(N + l)/(N + 4), 1). The intermediate value theorem

and (1.13) imply that given any integer ; there exists c such that if c > Cj

then

(1.14) y/(T,-ci,c)>jn + n/2.

By combining (1.12), (1.14) and the intermediate value theorem we see that if

c is sufficiently large and j > J := [(S - n/2)/n] + 1, then there exist numbers

dj < dj+\ < — <dj with

(1.15) y/(t,di,c) = in + n/2, i e {J,J + I, ... ,j}.

Hence

(1.16) ui(T,di,c) = 0, ie{J,J+l,...,j}.

Thus if c is sufficiently large, then (1.1) has j — J radially symmetric solutions.

Remarks, (i) If Q is a ring of the form {x e RN : a < \\x\\ < b}, then the

equation in (1.8) is no longer singular and thus the problem reduces to the one

studied in [5].

(ii) Condition (1.2) can be considerably weakened. For example, it is easy to

verify that if y> > 0 on [0, e) and <p = 0 on [e, T], then Theorem A holds.

2. PHASE-PLANE ANALYSIS

Let (u(t,d,c),u'(t,d,c)) ¿ (0,0) for all te[0,t_). By defining r2(t,d,c)

= u (t,d,c) + (u(t,d,c)) we see that for d < 0 there exists a unique contin-

uous argument function y/(t,d,c), t e [0,z), such that

u(t ,d ,c) = -r(t ,d ,c) cos y/(t ,d ,c),

(2.1) u'(t,d,c) = r(t,d,c)siny/(t,d,c),

y/(0,d,c) = 0.

An elementary calculation shows that

y/'(t,d,c) = sin y/(t,d,c)

(g(u(t,d,c)) + ^¿(t,d,c)- c<p(t) -p(t))cosy/(t,d,c)

r(t,d,c)

From this formula follows:



Remark 2.1. If r(t,d,c) > 0 for all re [0,71, and y/(i,d ,c) = (2k + l)n/2

for some t e [0, T] and some integer k, then y/(t,d,c) > (2k + l)n/2 for all

t > t. In particular, y/(t,d,c)> -n/2 for all te[0,T].

Now let y be such that

(2.2) yx := 1 - 12e < y < 1 - 10e := y2,

where e is defined by

^2'3^ fi = 4(6N + 6Np + 8 + 8/z) '

Elementary calculations show that

(a) 2y>2y + (y-l)A>27 + |(y-l)A>l + T^,

(2.4) (b) y + e < 1, (c) y > e +1

! + />'

(d, 3,-2-£>^, (e) 2y - 1 >JV+1

A + 4'

Let u(t,d,c) := u(t) be a solution to (1.8). Since the following constant

appears often in this section we let

[\/2Ñ+5l2N]2(2.5) Ö := ß(A0 := 2NThroughout the paper k will denote a nonnegative integer.

Lemma 2.2. There exists Cx such that if c> Cx, E(tx) = c y, u(tx) > -ac y~

and y/(tx) e [2kn,2kn + n/2], for some tx e (0, T] and a e [l,64N], then

there exists t2 e [tx, tx + 2a\/Ncy~ ], such that

y/(t2) = 2kn + n/2,

and

7zz particular E(t2) = c2y' with y e [y - (In 128)/(21nc), y + (In Q)/(21nc)].

Proof. Let t > t x be such that u(s) < 0 for all s e(tx,t]. Therefore, we have

u'(t) = t~N+X [if-V(i,) + j'sN-x(ctp(s)+p(s) - g(u(s)))ds

/V-V-ii/ziu^Jt,

> t-N+\

c-\\PlN

MlA'-l

t>£zM*(í_íl).

N

Thus, for c sufficiently large we infer

(2.6)u(t)>-ac2y-X + C~

2N-(t-t.y

ï-^-' + ^-h)2



This shows that for some z2 e [r,, tx + 2a\/Ñcy~ ] we have

(2.7) u(t2) = 0,

and u'(s) > 0 for all í e (tx ,t2). In particular

(2.8) y/(t2) = 2kn + n/2.

By the continuity of u there exists x < tx such that u'(x) = 0. Since

u(tx) > -ac2y~x, then from (1.3) for u sufficiently large, we have that \g(u)\ <

(M + l)\u\. Therefore, for c sufficiently large

(2.9) G(u(tx))<[(A

Thus for c sufficiently large we have

(2.9) G(u(tx))<[(M + 2)/2]a2c4y 2.

(2.10) u'(tx)>cy.

Hence,

j < u'(tx) = t'xN+X j" sN-X(c<p(s) +p(s) - g(u(s))) ds < lc(i, - T) ,

which implies that

(2.11) tx -x>Ncy X/4.

Therefore,

"'('2) = t2N+l j2 S" ' (C<P(S) + P(S) - S(U(S))) ds

>C-\\P\\oo(t _T)>V-1- N [ 2 ' ~ 8

Thus

(2.12) E(t2)>^c2y.

ll >l2-

Í lu'(tx)+ f'sN x(ctp(s)+p(s)-g(u(s)))dsJtl

Now, for t e (tx, t2] we have

u'(t) = t

<V2cy + rN+x f sN-X(2c + \\p\\oo + (M+l)\u(tx)\)dsJ'\

(2.13) <V2cy + rN+X i'sN-X(2c + \\p\\oo + (M+l)ac2y-X)dsJti

< V2cy + (2c + ll/zIL + (M+ l)ac2y-X)2a^-cy-X

. /x y , . 2q y-i V^Â + Sa y< v2c' + 4c-=c <-7=—c ,

s/N - \ÍÑ



where we have used (1.3), the hypothesis of the lemma and the fact that t2 e

[tx ,tx +2a\fÑcy~ ]. Since E(t2) = (u'(t2))2/2 we see that for c> Cx we have

(2.14) £„2)<(^±i£>>,

where C, is such that (2.6), (2.9) and (2.10) hold. This together with (2.8) and

(2.12) proves the lemma.

Lemma 2.3. If c > C2, y/(x) = 2kn, u(x) = -c2y~x, u'(x) = 0, with x e

[0, T], then there exists t2 e (x, x + 2\fÑcy~x) such that

y/(t2) = 2kn + n/2,

and

l-Tc2-'<E(t2)<^:8)2c2y<Qc2y.16N - K2' - 2N

In particular E(t2) = c2/ with y e [y - (In 16N)/(2 In c), y + (In Q)/(2 In c)].

Proof. Using the same arguments leading to the proof of (2.7) in Lemma 2.2

we get the existence of t2e[x,x + 2\fÑcy '] satisfying u(t2) = 0. Since for

u < 0 sufficiently large \g(u)\ < (M + l)\u\ < (M + l)cy x , on [t,Z2] we have

u'(t) < (3c + (M+ l)c2r~l)^t < P(t - x).

Thus,

u(t)<-c2y-x + ^c(t-x)2.

Hence, t2 - x > \/Ñcy~x/\/2, which leads to z/(/2) > c}'/2\fTÑ, and

2J - 16N "

On the other hand by replacing tx with x in (2.11) and using the fact that

(2.15) E(t2)>

On the other hand 1

u'(x) = 0, we obtain

(V2N + 8)2 2y(2.16) E(t2)< vv 2N > c

Hence, the lemma is proven.

Lemma 2.4. There exists C3 such that if c > C3, E(t2) = cy, and y/(t2) e

[2kn + n/2,2kn + n) for some t2 e [Ncy~x/4\[Q, T], then there exists t3 e

(t2,t2 + 2c7~x) such that y/(t3) = 2kn + n, and

¡c2y<E(t3)<2c2y.

In particular E(t3) = c2y' with y' e[y+ (ln\)/(2lnc) ,y+ (ln2)/(2lnc)].

Proof. Without loss of generality we can assume that u(t2) < g~ (4c). For

/ > t2 such that u(t) < ^"'(4c), and y/(s) e [2kn + n/2,2kn + n], for all



E(t) = E(t2)+ / (-u'(s) + c<p(s)+p(s))u'(s)ds

< c2y + 3cu(t).

s e(t2,t), we have

(2.17)

Since

(2.18) G(u(t)) < (4c)g'X(4c) < I6cx+{X/{X+P))

for c sufficiently large, from (2.17) using (2.4) we see that there exists a positive

constant R independent of (c,y) suchthat (u'(t)) <k cy. On the other hand

2y _ 4(N- l)y/QlC J+(l/(l+p)) 3 2y-C N - 4 '

for c sufficiently large (see (2.4)). From (2.18) and (2.19) we have

{u(t)f>\c2y.

Thus

(2.20) (4c)X,{X+p)>u(t)>^cy(t-t2).

Hence, there exists t* e (t2, t2 + 6c{x/{x+p))~y) such that

(2.21) u(t*) = g~X(4c) and u'(t*)>^-cy.

For t > t* with u'(s) > 0 for all s e[t*, t] we have

""(') = -^T-^'W + Cf(t)+p(t) - g(u(t)) < -C .

Hence,

(2.22) u'(t) < u'(t*) - c(t - t*) < ^j-cy - c(t - t*).

Thus, there exists t3 e (t* ,t* + \ßcy~x¡2) c (t2,t2 + 2cy~x), (see (2.4)), such

that

(2.23) u\t3) = 0

for c sufficiently large, and u'(t) > 0 for all t e[t* ,t3). In particular

(2.24) y/(t3) = 2kn + n.

Moreover, for t e [t2, t3] from (2.17) we have

(2.25) G(u(t)) <E(t)< c2y + 3cu(t).



Thus from (1.4) we infer

(2.26) (u(t))2+p - 3cu(t) < c2y.

Now, if (u(t))2+p < 6cu(t), then

(2.27) u(t) < 6cx/{x+p).

For (u(t))2+p > 6cu(t) from (2.26) we see that

(2.28) u(t) < \c2y~x.

By using (2.4), (2.27), and (2.28) for c sufficiently large we infer

(2.29) u(t) < \c2y~x.

Thus, replacing (2.29) in (2.25), for t e [t2, t3] we obtain

(2.30) E(t) < 2c2y.

Hence, from (2.30) we have

(2.31) u'(t)<cy.

Imitating the arguments leading to (2.19) it is easy to see that

(2.32) E(t3)>\c2y,

for c > C3, where C4 is assumed to be such that (2.19), (2.23), (2.29) and

(2.32) hold. Thus, (2.24), (2.30) and (2.32) prove the lemma.

Lemma 2.5. There exists C4 such that if c > C4, E(t3) = c y, and y/(t3) e

[2kn + n,2kn + \n] for some t3 e [Ncy~x/%\/Q,T] then there exists t4 e

[t3, t3 + 2c"e/2] such that

y/(t4) = 2kn + \n.

Moreover,

\c2y<E(U)<c2y.

In particular E(t4) = c2y with y e[y + (In |)/(21nc), y].

Proof. Since y/(t3) e [2kn + n, 2kn + fzr], we have u(t3) > 0 and u'(t3) < 0.

Hence either

(2.33) u(t3)>g-x(2c + \\p\\oo + cy+£),

or

(2.34) M(/3)<^-,(2c-r||p||00 + c''+e).

If (2.33) holds then for t > t3 with u(s) > g'x(2c + Wp^ + cy+e) for all

ie[i3,z) we have

u'(t) = rN+X [if-V(i3) + psN-x(cy>(s)+p(s) - g(u(s)))ds

(2.35) <rN+l f'sN-x(-cy+E)ds

N

h

-[t-



Suppose that for all t e(t3,t3(l+c"E'2)) we have u(t) > g~x(2c+\\p\\oo+cy+e).

Hence, now for all X e (—e, - e/2), by (2.35) we infer

u'(t3(l+cx)) < -irt3[(l+cx)-(l+cYN+x]

(2'36) <l£lJL.r*V< N 2y->- 2N-X %y/Q° C - 2N+2JQ ■

Thus

u(t3(l +c~E/2)) < u(t3(l +c'c))+ i' -c2y~X[N/(2N+2y/Q)]ds(2.37) 2 Jh{x+C'c)

^ t* \ N 3y-2/ -e/2 -e.

Since E(t3) = c2y, we have that u(t3) < G~x(c2y). By (1.4) we see that for

c sufficiently large g~X(c2y) < c y'^+p). Hence, there exists C4 > HpH^ such

that for c> C4

(2.38) u(t3)<c2y/(2+p).

Also, since e > 0 and (2.4) holds, we can assume C4 to be such that c~£' -

c~e > c~e' ¡2 for c > C4, and that there exists a constant kx > 0 such that

for c> C4

(2.39) «(*3(1 +c-£/2)) < c2y/(2+/>) -kxc3y-2-{£/2) < 0,

which is a contradiction. Thus for c > C4 there exists /' e (í3,í3(l + c~e/2))

such that

(2.40) u(t') = g-X(2c + \\p\\oo + cy+£).

This and (2.34) show that there exists t" e [t3 ,t3(l + c~e/2)] with

(2.41) u(t")<g-x(2c + \\p\\00 + cy+e).

Now we estimate E(t) for t > t3 with u'(s) < 0 for all s e[t3,t]. Since

(2.42)

dE(t) N - I . ,, ..2 , , , ,../,.-gp«-—("(0) + to(0+/>('))" (0

<(c-I1p|U«'w<o,we have

(2.43) £(i)<c2l\

Therefore,

(2.44) |M'(Z)|<v/2cy.

Hence,

(2.45) -?Lz±u\t) + c<p(t)+p(t)<4c,



for c sufficiently large. If in addition u(t) > 0 then

E(t) = E(t3)+ j (-^-^u(s) + C(p(s)+p(s)\u'(s)ds

['-2A6^ > c2y + 4c(u(t) - u(t3)) > c2y - 4cu(t3)

>c2y-4cx+(2yl{2+p))>lc2y,

where we have used (2.4) and (2.39). Thus from (2.43) and (2.44) we obtain

(2.47) \c2y < E(t2) < c2y.

Now, for / > t" we have

,248, G(u(t))<ug(u)<g-X(4c)4c

[ " } <l6cx+(x'(x+p))<\c2y,

(see (2.4)). Combining (2.47) and (2.48) we get

(2.49) u'(t) < -cy

for all / > /" such that u(s) > 0 and u'(s) < 0 with 5 e [t", t]. Hence

(2 U{t) - g~l{2C + I|P|I~ + CÏ+£) ~ C"{t ~ ^

[ ' ) <4cx/{X+p)-cy(t-t"),

(see (2.4)) for c sufficiently large. From (2.50) we see that for some f4 e

[t", t" + c{x,{X+p))-y] c [t3, t3 + 2c~e/2] (see (2.4)) we have

(2.51) u(t4) = 0,

in particular y/(tA) = 2kn + \n for c > C4 , where C4 is in addition assumed

to be such that (2.46)-(2.50) hold. This and (2.47) prove the lemma.

Lemma 2.6. There exists C5 suchthat ifc> Cs, y/(tA) e[2kn+\n,2(k + \)n],

E(t4) = c2y, with t4 e [Ncy~x/S^/Q,T], and u(t4) > -ac2y~x, a e [1,64A],

then for some t5 e [t4, t4 + \/2cy~x] we have y/(t5) = 2(k + l)n. Moreover,

-(a + 4)c2y~X < u(L) <-~-c2y~X.- K5)- 64v/£+16

Proof. Let now t > t4 be such that u'(s) < 0 for all je(i4,(). The existence

of such t 's follows from the fact that y/(t4) e [2kn + \n, 2(k + l)n]. Since

u"(s) =-— u'(s) + ctp(s) + p(s) - g(u(s)) > § ,

we have

u'(t) > u(t4) + §(/ - t4) > -J2E(t4) + §(i - t4)

> -y/2cr + Ut - r4).



Therefore, there exists t5 e [t4,t4 + 2\/2cy~x] with u'(ts) = 0 and u'(s) < 0

for all s e[t4,t5]. Thus

(2.53) y/(t5) = 2(k + l)n,

and y/(s) e [2kn + \n,2(k + l)n], for all s e[t4, tA . Furthermore,

"(*0 = u(t.) + l u'(s)ds > -ac2y~X - \f2cy2s/2cy~X(2.54) 7(4

>-(a + 4)c2y~X.

By (1.3) there exist constants Mx > 0, and M2 such that G{u) <Mxu + M2

for u < 0. Let C5 be such that for c > C5 we have

(2.55) G(-(a + 4)c2y~X) <Mx(a + 4)V~2 + M2 < C— .

Thus

(2.56) G(u(t4)) < G(u(t5)) < Mx(u(t5))2 + M2<—.

Since E(t4) = c y, from (2.56) we have

(2.57) u'(t4)<-cy,

for c sufficiently large. By the continuity of u we see that there exists / e

(t4,t5) such that

(2.58) M'^ = -y-

Since \u'(t4)\ < \f2cy, and \g(u)\ < (M + l)\u\ for u < 0 sufficiently large,then for t e [t4, t] we have

(2.59) u"(t)< S("N~})^V2cy + 3c + (M+l)(a + 4)c2y~X <(l6^Q + 4)c,cy N

for c sufficiently large. Thus by integrating (2.59) on [Z4 , /], and using (2.57)

and (2.58) we have that t-t4> cy~x/2(l6s/Q + 4). Hence

u(t5) < u(t) = u(t4) + / u'(s) ds

(2.60) '4 , ,cy,~ , c2y~x

<0-T(t-t4)<-2 4 64y/Q+l6'

From (2.54) and (2.60) we see that if c > C5, where C5 is in addition assumed

to be such that (2.59) holds, then the lemma is proven.

The next lemma summarizes the results of this section.



Lemma 2.7. Given any positive integer j there exists a real number m. such

thatifomj, E(t,d,c) > 0 for all t e[0,T], E(tx,d ,c) = c2y, for some

tx e [Ncy~x/4,T], y e [(5yx + 3y2)/8,(3y, + 5y2)/8], and either

(A) y/(tx,d,c)e[2kn + i,2kn + \n],or

(B) y/(tx,d,c)e[2kn,2kn + %)\J(2kn + \n,2(k+l)n], and u(tx,d,c) >-c2y~x,

then y/(T,d,c) > jn + n/2.

Proof. We split the proof into four cases depending on the quadrant where

y/(tx ,d,c) lies.

Case I. Suppose that y/(tx ,d,c) e [2kn,2kn + n/2]. By (2.4) there exists C5

such that for c > C5 we have that (2>/2 + 20^ + 2)cn~x < c~E/2. Thus

by applying consecutively Lemmas 2.2 (or 2.3), 2.4, 2.5, 2.6, we see that if

c > C := max{C,,... , C5} then there exist t2< t3< t4 < t5 such that

(2.61) t5e[tx,tx+k*c~{£/2)]

and

(2.62) y/(t5,d,c) = 2(k+l)n.

Moreover, from Lemma 2.2. we have

(2.63) E(t2,d,c) = c2/,

with

,..„ In 128 ^ / lnß

(2-64) -TTnT + ̂ ^y+2ln7-

Also, by combining (2.64) with the results from Lemmas 2.4 and 2.5 we see

that E(t4,d,c) = c2y" , with

..... 21n(3/4) + lnl28 ^ „ , ln2 + lnQ

(2'65) y-2hVc-*' ^y + ^m-c~-

Now by Lemma 2.6 (since in this case a = 0) we have that u(t5) = -c ?~

with

or

(2.67)

Let now

-k'c2y~X := -%Qc2y~x < u(t5) < -[72(64^Q + I6)]c2y '

:=-K'c2y-X

„,0, i ( 4j\lnk'\ \ ( 4j\lnK'\\ ,.,^2/e '(2.68) m,:=max exp ¡J(-—) , exp (_-L \,(j/T) ,C



From (2.67) and (2.68) it follows that

(2.69) 5yi + 3y2 _ 3y2-3y, < _ < 3yx + 5y2 + 3y2 -3yx ^8 87 8 87

Also, from (2.68) we have

(2.70) t5-tx<T/j.

Now, we observe that t5 = x, and y = y satisfy the hypothesis of Lemma 2.3.

Thus, iterating the above argument j times we see that there exist t5 < t6 <

■•• < tj+3 < T, with V(tj+3,d,c) = 2(k + j)n. Hence, by Remark 2.1. we have

that

(2.71) y/(T,d,c)> 2(k + j)n - § > 2jn - § > jn + §,

and that proves Case I.

Case II. If y/(tx ,d,c) e [2kn + n/2, (2k-+- 1)tt] , then by applying Lemmas 2.4,

2.5, 2.6. we see that there exists t5 that satisfies (2.61), (2.62), (2.67), (2.69),

and thus, we are in Case I.

Case III. If y/(tx,d,c) e [(2k + l)n,(2k + l)n + n/2], then we can apply

Lemmas 2.5 and 2.6, and Case III reduces to Case I.

Case IV. If y/(tx ,d ,c) e [(2k + l)n + n/2,2(k + l)n], then by applying Lemma2.6 we are in Case I, and that concludes the proof of Lemma 2.7.

3. Energy analysis

Throughout this section we use K, k', K', k, kx, k2 to denote various

constants independent of (c, y).

Let now u(t) := u(t,d,c) be a solution to (1.8). Suppose that there exists

t e [Ncy~ ' /4, T], with ye[yx,y2], such that

(3.1) E(i,d,c) = c2y, E(t)>0 forallí€[0,f]andM(£)>-c2!'"1.

Arguing as in the proof of Lemma 2.7 (using Lemmas 2.2, 2.4 and 2.5 if

y/(t,d,c) e [2kn,2kn + n/2], Lemmas 2.4 and 2.5 if y/(t,d,c) e [2kn +

n/2,(2k+ l)n], Lemma 2.5 if y/(t,d,c) e [(2k + \)n,(2k+ l)n + n/2], and

Lemmas 2.6, 2.3, 2.4 and 2.5 if y/(t,d,c) e [(2k + l)n + n/2,2(k+ l)n]), we

see that there exists xx e [i,i + A"c_(£/2)] such that u(xx) = 0, u'(xx) < 0, and

(3.2) k'c2y <E(xx,d,c)<K'c2y.

Now we prove by induction

Lemma 3.1. Let t be as in (3.1) and let i < T, < tx < r2 < ••• < x¡ < f. <

T/+1 < • • • < t denote the zeroes of u(t) for t > t, such that E(s,d,c) > 0 for

î6[t,,/). T/Tor some k e (0,1 ) L(k) = oo, then there exists c* such that if

c > c* then

(33) k'c2y+^N<xfE(xi,d,c)

<x?[k"c2y + (i-l)c3y-x]<K'c2y.

In particular E(t,d,c)>0 for all te[0,T].



Proof Since xx > i > cy ' from (3.2) we have (3.3) for i = 1. Also, from

(3.2) we see that

(3.4) k'c2y+(y-X)N <E(xi,d,c)<K'c2y.

Thus, as in (2.54) we obtain the existence of a e [x¡,x¡ + Kcy~x] such that

u'(a) = 0 and u(a) > -(a + 4)c2y~x . Furthermore, since u"(t) < Kcx~ycy +

7,c + (M+ l)\u(a)\ < Kc for t e [x^a] we infer 0 = u'(a) < -c7+('"ip/2) +

Kc(a - t;) . Therefore a - x¡ > Kcy-x+{y-x)[N/2).

Also, by the continuity of u we see that there exists â e [t; , a] such that

u'(a) = \u(r¡). Since u(a) = z/(t,) + /Tfl[u"(s)ds < u(ri) + Kc(â — xi) we

have â-T,>^-1+(),-1)W2).Thus

u(a) < u(â) = u(x,) + u'(â)(â - x¡) < -Kc2y~x+(y~x)N.

Arguing as in (2.6)-(2.7) we see that f; < ez + Kcy~x . In particular there exists

a unique b (u is convex on (x¿, x¡)) suchthat u(b) = -cxl(X+p). Since

u(b) = u(a) + f" u'(s) ds < -Kc2?-l^-V" + {b- a)Kcy,Ja

we infer

(3.5) b-a>Kcy-XHy'X)N,

where we have also used the fact that 2y + (y-l)N-l > 1/(1+ p). Hence, as in

(2.12) we obtain that E(xt,d,c) > Kc2y+{-y~X)N . Imitating the arguments used

in Lemma 2.4 and Lemma 2.5 we show the existence of x¡ < x\ < x* < x'[ < xi+x

such that u(x\) = u{x") = Kg~x(4c), u(x'¡) > 0, u(x") < 0, and u'(x*) = 0.

In addition, we obtain that

(3.6) T;.-Ti.<jvc[1/(1+",1-y-(y-1)W2),

and E(x* ,d,c) > Kc2y+(y~X)N . Therefore, as in Lemma 2.4 we infer

Jl/('+Z')l-)'-(}'->)(Ar/2)(3.7) xi+x-x¡<Kcl

Since [\/(\ + p)]-y-(y- 1 )( A/2) < y + (y - 1 ) N - 1 (see (2.4a)) we have

rb

(3.8)/Ja

Kcr u(r)dr > i7[T, ,T¡]U[T¡',T/+

Kcr u(r)dr]

Multiplying (1.8) by r u and integrating over [t(,t/+1] we obtain

xME(xM,dtc)-xt E(xl,d,c)

(3.9)

'T/+l \N-2 N-\, I, ,,2 .7 N-\„, , ,,-^^r (u(r)) -Nr G(u(r))

-r■n- dr

[c<p(r)+p(r)]rNu'(r)dr,



where we have integrated by parts the term /tT,+l r [u"(r)u'(r) + (G(u(r)))'] dr.

Similarly, multiplying (1.8) by rN~xu and integrating over [x¡,x¡ ,] we infer

(3.10)

/ r (uJtí

(r)) dr

= f'V X{g(u(r))u(r)-[c<p(r)+p(r)]u(r)}dr.Jx¡

By replacing (3.10) in (3.9) we obtain

h+xE(xi+x,d,c)-xNiE(xi,d,c)

(3.11)= r rN-x

Jr,NG(u(r)) - !LJLg<u(r))u(r)) dr

l[ctp(r)+p(r)]>, s tf-2 , ,

™ (r) + —3—u(r) dr.

Integrating by parts the last term we get

(3.12)xf+xE(xi+l,d,c) -x?E(xt,d,c)

r,+i n-\ \ n-2> j rN \NG(u(r)) - i-j-g(u(r))u(r) dr

s7[T,,T;]u[T;',r,+1]

+ P rN~x ¡NG(u

fb- kc r u(r)dr.

Ja

NG(u(r)) - IL^Lg(u(r))u(r) - kcu(r) dr

N-2(r))-—g(u{r))u(r) - kcu(r) dr

Since g is an increasing function for u > 0 we have

rKU rU

G(u)= I g(s)ds+ i g(s)ds>G(Ku) + (l-K)ug(Ku).Jo Jku

Thus, from the assumption that L(k) = oo, there exists kx > 0 such that for

u > kx (kx is chosen so that NG(ku) - ((N - 2)/2)g(u)u > 0 for zc« > ze,)

NG(u) - ^-^g(u)u > NG(ku) + N (^—^) Kug(Ku) - ^-^g(u)u

(3.13) > N (LJ^J KUg(KU) > N (L-Ji] G(KU)

> l—^J -^—g(u)u = k2g(u)u.



Thus if c is large enough and x\, x" are chosen so that g(u(x'¿)) = g(u(x")) =

kc/k2 then

(3.14) P'rN-1 NG(u(r)) - *-lg{u{r))u(r) - cku(r) dr>0.

From (1.3) it follows that L(l,u) is bounded below. Thus, (3.8), (3.14) and

(3.12) yield

(3.15) x^+xE(xi+x,d,c)>x^E(xi,d,c)>Kc2y+{y-X)N.

On the other hand, since <p is differentiable we have

E(xi+X) < E(xt) + f '+\c<p(s)+p(s))u'(s)dsJ T¡

< E(x¡) -c i +1 (p'(s)u(s) ds < E(x¡) + Kc j ' \u(s)\ dsJx¡ Jx¡

<E(xi) + Kcc2y-xcy-x

Hence inductively we obtain

E(xi+x)<E(xx) + (i-l)Kc3y-l

„y-i y-USince tj+1 - t(. > Kc , we see that i < (T/(Kc )). Therefore

(3.16) E(xi+X) < E(xx)+Kc2y < K'c2y,

which together with (3.15) proves (3.3).

Let 7o := (?i + ^2)/2 > where yx and y2 are as in (2.3).

Lemma 3.2. Let x e [0, T] be such that u'(x) > 0 and u(x) < -c2yo~x. There

exist C such that if o C then for some

(1/2)N

Tie (W_^_V1/2) 2vWM^!Y{VZ"\3c + (M+l)\u(x)\) >¿V"{ c )

we have u(xx + x) = 0, u(t) < 0 for all t e [x,x + xx], and E(xx ,d,c) >

c2y°/64N.

Proof. Since g is an increasing function, for t > x we have

u(t)>t 2j s ds^2Ñ^^

Hence u(t) > u(x) + (c/4N)(t - x)2. Therefore, t, < 2v/Ä7(|h(t)|/c)(1/2) exists

such that u(xx + x) = 0. Similarly using that for u < 0 sufficiently large

\g(u)\ <(M+ l)\u\ we have u'(t) < t(3c + (M + l)\u(x)\)/N. Thus, u(t) <

u(x) + t2(3c + (M+ 1)|m(t)|)/(2A) . This proves that

(1/2)

T>^2A\3c + (M

«(T)| V

I+1)\U(X)\J

. m^



Hence,

M'(T ) > C-}P}~V2Ñcy°-X > -±=cy\v v - 4N - 4V/2Â7

Therefore, E(xx ,d ,c)> c2yo/64N, and that concludes the proof of the lemma.

Lemma 3.3. Let C be as in Lemma 3.2, and c> C. If t e[0,T] is the largest

number such that E(t,d,c) = c2y°/64N, and E(t,d,c) > 0 for all t e [0,7]

then u(t) > -c2y°~x.

Proof. Suppose that u(t) < -c2y°~x . From Lemma 3.2 for all t > t with

u(t) < 0 we have

If w'(7) < 0 we let i0 denote the smallest number greater than ? such that

u'(t0) = 0, and if «'(7) > 0 we let i0 = 7. Of course, w(i0) < u(t) < -c2y°~x.

Since for all s > t0 with u < 0 on (í0,j) we have that \u'(s)\ < cy°/4\/2Ñ,

hence «(/) < 0 for t e (tQ, r0 + 4y/2Ñcy°~x). Thus, for t = t0 + 4V/2ÄV0_1 we

have

u'(t0 + 4V2Ñcyo~x) = u'(t0)+ f u"(s)dsJt0

to+4V2Ñcro-

u"(s)ds

»K'-Tr1)^-' >4V2A

which contradicts the assumption that t is the largest number such that

E(t,d,c) = c2yo/64N and E(t,d,c) > 0 for all í e [0,f]. Hence the lemma

is proven.

Now we summarize the above results in the following lemma.

Lemma 3.4. If L(k) = oo for some zc e (0,1), then there exists C* such that

if c>C* ,and d < -c2y°~x then E(t,d,c)>0 for all te[0,T].

Proof. From Lemma 3.2 (taking x = 0) we see that either E(t ,d ,c) >c yo/64N

for all t e [0,T], or there exists t > xx > \c2yo~x such that E(t,d,c) =

c2yo/64N and E(t,d,c) > 0 for all t e [0,7]. Also by Lemma 3.3 we

have u(t) > -c y°~x. Hence, taking y = y0 - (ln64/V)/(21nc) we have

E(t,d,c) = c2yl, and

it\ ^ 2j>0—1 2(y0-y') 2/-1 ¿AxT2y'-l 2y'-iu(t) > -c = -c c = -64Nc = -ac

Let now c be such that if c > c then (ln64A/)/(21nc) < \(y2 - yx). By

applying consecutively the lemmas from section 2, depending on the quadrant

where u(t) lies we obtain the existence of t, e [t,t + Kc~(e,ï)] such that

u(xx) = 0, u'(xx) < 0 and (3.2) holds. Now, directly from Lemma 3.1 by

taking C* = max{c*, C, c) we obtain the proof of the lemma.



4. Separation of zeroes

The following lemma is a version of the Sturm comparison theorem for sin-

gular linear differential equations, and it proves our main estimate on the sep-

aration of zeroes.

Lemma 4.1. There exists H := 77(c) > 0 such that if u satisfies (1.8),

u(t°) < -77 and u'(t°) = 0, for some í°6[0,í], then u(t) < 0 for all t e

(t°,t° + \(M+l)~x/2). Moreover, if for some (0e[0,7], u'(t°,d,c) = 0 and

u(t°,d,c) < -H, then

y/(T,d,c)< y/(t°,d,c) + n/2 + 2J(t°)n,

where J(t ) is the greatest integer less then 2(M + l)x'2(T - t°). In particular

taking t =0 we have

limsupy/(T,d,c)< y + 2/(0):= S.— d—yoo ¿

Proof. From ( 1.3)—( 1.4) we see that there exists a real number M* < 0 such

that

(4.1) g(s) - (M + l)s > M* for all 5 G R.

Thus, for A(t) := p(t) + ctp(t) - g(u(t)) + (M + l)u(t), we have

(4-2) A(t)<\\p\\oo + c\\y>\\x>-M*:=H.

By rewriting (1.8) we obtain

u"(t) + ^-u\t) + (M+ l)u(t) = A(t),(4.3) t

u(t°) = -h, u'(t°) = 0,

where h e R is such that h > H. Let now

(4.4) F(t) = \[(u'(t))2 + (M+l)u2(t)].

From (4.3) and (4.4) we see that

(4.5) dF/dt < A(t)u'(t) < H(2F(t))x'2.

Integrating (4.5) on [t ,t] we obtain

|«'(0I < (27-(/))1/2 < (2F(Z°))1/2 + H(t - t°)(4.6)

<(M+l)' h + H(t-f).

Let t > t° be such that u(x) = 0, and u(t) < 0 for all t e [t°, x]. Hence, from

(4.6) we have

0 = u(x) = u(t°) + Í u'(s)dsJfi

(4-7) <u^) + {M+l)x'2h(t-t°) + ^2H(t-t0)2

<h[-l + (M+ l)x/2(t - t°) + Ut - z0)2].



Therefore

(4.8) /-í°< i(A7+l)_1/2:=A.

Hence, u(t) < 0 for all t e [t°, t° + A], and thus in particular

(4.9) y/(t° + A,d,c)<y/(t0,d,c) + n/2.

Iterating this argument we see that

(4.10) y/(t° + 2A,d,c)< y/(t°,d,c) + n/2 + 2n,

and

(4.11) y/(t° + iA,d,c) < y/(t°,d,c) + n/2 + (i-l)2n,

with i e {0,1, ... , J(t0)} . In particular

y/(T,d,c) < y/(t° + J(t°)A,d,c) + 2n

(4.12) < ^(z0,e/,c) + 7r/2 + (/(Z°)-l)2^ + 27r

= y/(t°,d,c) + n/2 + J(t°)2n,

and that proves the lemma.

5. Proof of Theorem A

By Lemma 3.4 we see that there exists C* such that if c > C* and d <

-c2y°~x then E(t,d,c)> 0 for all t e[0,T]. Therefore, if c> mj then by

Lemma 2.7 we have that

(5.1) y/(T,-c2yo'X,c)>jn + n/2.

Let J := [(S - n/2)/n] + 1, where [x] denotes the largest integer less than or

equal to x . If j > J then jn + n/2 > S. Hence by Lemma 4.1, (5.1), and the

intermediate value theorem if c > C- = may.{m}, C*} then there exist numbers

dj < dJ+x < ••• <dj, such that

(5.2) y/(T,di,c) = in + n/2, i e {J,J + I, ... J}.

Hence, u¡(x) = u(\\x\\,d¡,c) is a solution to ( 1.1 ) with exactly i interior nodal

surfaces. Thus, Theorem A is proven.

References

1. A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities

between Banach spaces, Ann. Mat. Pura Appl. 93 (1972), 231-246.

2. A. Castro and A. Kurepa, Energy analysis of a nonlinear singular differential equation and

applications, Rev. Colombiana Mat. 21 (1987), 155-166.

3._, Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball,

Proc. Amer. Math. Soc. 101 (1987), 57-64.

4. A. Castro and R. Shivaji, Multiple solutions for a Dirichlet problem with jumping nonlin-

earities, Trends in Theory and Practice of Nonlinear Analysis, V. Lakshmikantham (ed.),

North-Holland, 1985.



5._, Multiple solutions for a Dirichlet problem with jumping nonlinearities II, J. Math. Anal.

Appl. 133(1988), 509-528.

6. A. Castro and A. C. Lazer, On periodic solutions of weakly coupled systems of differential

equations, Boll. Un. Mat. Ital. (5) 18-B (1981), 733-742.

7. D. G. Costa and D. G. De Figueiredo, Radial solutions for a Dirichlet problem in a ball, J.

Differential Equations 60 (1985), 80-89.

8. J. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure

Appl. Math. 28 (1975), 567-597,

9. A. Lazer and P. J. McKenna, On the number of solutions of nonlinear Dirichlet problem, J.

Math. Anal. Appl. 84 (1981), 282-294.

10._, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc.

Roy. Soc. Edinburgh Sect. A 95 (1983), 275-283.

11._, Critical point theory and boundary value problems with nonlinearities crossing multiple

eigenvalues, Comm. Partial Differential Equations 10 (1985), 107-150.

12. D. Lupo, S. Solimini and P. N. Srikanth, Multiplicity results for an o.d.e. problem with even

nonlinearity, Quad. Matematici (II), no. 112, Dicembre 1985.

13. B. Ruf and S. Solimini, On a class of superlinear Sturm-Liouville problems with arbitrarily

many solutions, SIAM J. Math. Anal. 17 (1986), 761-771.

Department of Mathematics, University of North Texas, Dentón, Texas 76203

Department of Mathematics, Texas Christian University, Fort Worth, Texas 76129


radially symmetric solutions to a superlinear …

Documents