eigenvalues of the p(x)-biharmonic operator with indefinite weight

Z. Angew. Math. Phys.c© 2014 Springer Basel

DOI 10.1007/s00033-014-0465-y

Zeitschrift fur angewandteMathematik und Physik ZAMP

Eigenvalues of the p(x)-biharmonic operator with indefinite weight

Bin Ge, Qing-Mei Zhou and Yu-Hu Wu

Abstract. In this article, we consider the nonlinear eigenvalue problem:

{Δ(|Δu|p(x)−2Δu) = λV (x)|u|q(x)−2u, in Ω

u = Δu = 0, on ∂Ω,

where Ω is a bounded domain of RN with smooth boundary, λ is a positive real number, p, q : Ω → (1, +∞) are continuous

functions, and V is an indefinite weight function. Considering different situations concerning the growth rates involved inthe above quoted problem, we prove the existence of a continuous family of eigenvalues. The proofs of the main results arebased on the mountain pass lemma and Ekeland’s variational principle.

Mathematics Subject Classification. 35G30 · 35J35 · 35P30 · 46E35.

Keywords. Fourth order elliptic equation · Eigenvalue problem · Variable exponent · Sobolev space · Ekeland’s variational

principle.

1. Introduction

The study of differential equations and variational problems with nonstandard p(x)-growth conditions hasreceived more and more interest in recent years. It possesses a solid background in physics and originatesfrom the study on electrorheological fluids (see Ruzicka [1]) and elastic mechanics (see Zhikov [2]). Italso has wide applications in different research fields, such as image processing model (see e.g. [3,4]),stationary thermorheological viscous flows (see [5]) and the mathematical description of the processesfiltration of an idea barotropic gas through a porous medium (see [6]).

In this paper, we discuss the existence of the weak solutions for the following nonhomogeneous eigen-value problem: {

Δ(|Δu|p(x)−2Δu) = λV (x)|u|q(x)−2u, in Ω,u = Δu = 0, on ∂Ω,

(P)

where Ω is a bounded domain of RN with smooth boundary, λ is a positive real number, p, q are continuous

functions on Ω, and V is an indefinite weight function. This means that V can change sign in Ω.The study of nonhomogeneous eigenvalue problems involving operators with variable exponents growth

conditions has captured a special attention in the last few years. This is in keeping with the fact thatoperators which arise in such kind of problems, like the p(x)-Laplace operator (i.e., Δ(|Δu|p(x)−2Δu),

Supported by the National Natural Science Foundation of China (nos. 11126286, 11201095), the Youth Scholar Back-bone Supporting Plan Project of Harbin Engineering University, the Fundamental Research Funds for the Central Univer-sities, China Postdoctoral Science Foundation funded project (no. 20110491032), and China Postdoctoral Science (Special)Foundation (no. 2012T50325).

B. Ge, Q.-M. Zhou and Y.-H. Wu ZAMP

where p(x) is a continuous positive function), are not homogeneous and thus, a large number of techniqueswhich can be applied in the homogeneous case (when p(x) is a positive constant) fail in this new setting.

Ayoujil and Amrouss [7], studied the same nonhomogeneous eigenvalue problem in the particular casewhen V (x) = 1.

In the case when maxx∈Ω

q(x) < minx∈Ω

p(x) it can be proved that the energy functional associated to

problem (P ) has a nontrivial minimum for any positive λ (see Theorem 3.1 in [7]).In the case when min

x∈Ωq(x) < min

x∈Ωp(x) and q(x) has a subcritical growth Ayoujil and Amrouss [7] used

the Ekelands variational principle in order to prove the existence of a continuous family of eigenvalueswhich lies in a neighborhood of the origin.

In the case when maxx∈Ω

p(x) < minx∈Ω

q(x) ≤ maxx∈Ω

q(x) < Np(x)N−2p(x) , by Theorem 3.8 in [7], for every λ > 0,

the energy functional Φλ corresponding to (P ) Mountain Pass type critical point which is nontrivial andnonnegative, and hence Λ = (0,+∞).

The same problem, for V (x) = 1 and p(x) = q(x) is studied by Ayoujil and Amrouss [8]. The authorsestablished the existence of infinitely many eigenvalues for problem (P ) by using an argument based onthe Ljusternik-Schnirelmann critical point theory. Denoting by Λ the set of all nonnegative eigenvalues,they showed that sup Λ = +∞ and they pointed out that only under special conditions, which aresomehow connected with a kind of monotony of the function p(x), we have inf Λ > 0 (this is in contrastwith the case when p(x) is a constant; then, we always have inf Λ > 0).

Inspired by the above-mentioned papers, we study problem (P ) in two distinct situations. In this newsituation we will show the existence of a continuous family of eigenvalues for problem (P ).

The paper is organized as follows. In Sect. 2, we recall the definition of variable exponent Lebesguespaces, Lp(x)(Ω), as well as Sobolev spaces, W k,p(x)(Ω). Moreover, some properties of these spaces willbe also exhibited to be used later. In Sect. 3, we give the main results and their proofs.

2. Preliminaries

Firstly, we introduce some theories of Lebesgue-Sobolev space with variable exponent. For more detailson the basic properties of these spaces, we refer the reader to Kovacik, Rakosuik [9], Fan and Zhao [10],Fan and Zhang [11].

Set

C+(Ω) = {h ∈ C(Ω) : h(x) > 1 for any x ∈ Ω}.

Define

h− = minx∈Ω

h(x), h+ = maxx∈Ω

h(x) for any h ∈ C+(Ω).

For p(x) ∈ C+(Ω), we define the variable exponent Lebesgue space:

Lp(x)(Ω) =

{u : u is a measurable real valued function

∫Ω

|u(x)|p(x)dx < +∞}

,

with the norm

|u|Lp(x)(Ω) = |u|p(x) = inf

{λ > 0 :

∫Ω

∣∣∣∣u(x)λ

∣∣∣∣p(x)

dx ≤ 1

},

and define the variable exponent Sobolev space

W k,p(x)(Ω) = {u ∈ Lp(x)(Ω) : Dαu ∈ Lp(x)(Ω), |α| ≤ k},


with the norm ‖u‖W k,p(x)(Ω) = ‖u‖k,p(x) =∑

|α|≤k |Dαu|p(x).We recall that spaces Lp(x)(Ω) and W k,p(x)(Ω) are separable and reflexive Banach spaces. Denote by

Wk,p(x)0 (Ω) the closure of C∞

0 (Ω) in W k,p(x)(Ω).Denote by Lp′(x)(Ω) the conjugate space of Lp(x)(Ω) with 1

p(x) + 1p′(x) = 1, then the Holder type

inequality ∫Ω

|uv|dx ≤(

1p− +

1p′−

)|u|p(x)|v|p′(x), u ∈ Lp(x)(Ω), v ∈ Lp′(x)(Ω) (2.1)

holds. Moreover, if p1, p2, p3 ∈ C+(Ω) and 1p1(x) + 1

p2(x) + 1p3(x) = 1, then for any u ∈ Lp1(x)(Ω),

v ∈ Lp2(x)(Ω) and w ∈ Lp3(x)(Ω) the following inequality holds (see [12], Proposition 2.5):∫Ω

|uvw|dx ≤(

1p−1

+1

p−2

+1

p−3

)|u|p1(x)|v|p2(x)|w|p3(x). (2.2)

Furthermore, if we define the mapping ρ : Lp(x)(Ω) → R by

ρ(u) =∫Ω

|u|p(x)dx,

then the following relations hold

|u|p(x) < 1(= 1, > 1) ⇔ ρ(u) < 1(= 1, > 1), (2.3)

|u|p(x) > 1 ⇒ |u|p−

p(x) ≤ ρ(u) ≤ |u|p+

p(x), (2.4)

|u|p(x) < 1 ⇒ |u|p+

p(x) ≤ ρ(u) ≤ |u|p−

p(x), (2.5)

|un − u|p(x) → 0 ⇔ ρ(un − u) → 0. (2.6)

We recall also the following proposition, which will be needed later:

Proposition 2.1. ([13]) Let p and q be measurable functions such that p ∈ L∞(Ω) and 1 < p(x)q(x) ≤ ∞,for a.e. x ∈ Ω. Let u ∈ Lq(x)(Ω), u = 0. Then

|u|p(x)q(x) ≤ 1 ⇒ |u|p+

p(x)q(x) ≤ ∣∣|u|p(x)∣∣q(x)

≤ |u|p−

p(x)q(x),

|u|p(x)q(x) ≥ 1 ⇒ |u|p−

p(x)q(x) ≤ ∣∣|u|p(x)∣∣q(x)

≤ |u|p+

p(x)q(x).(2.7)

Definition 2.1. Assume that spaces E, F are Banach spaces, we define the norm on the space X := E⋂

Fas ‖u‖X = ‖u‖E + ‖u‖F .

In order to discuss problem (P ), we need some theories on space X := W1,p(x)0 (Ω)

⋂W 2,p(x)(Ω).

From the definition 2.1, we can know that for any u ∈ X, ‖u‖ = ‖u‖1,p(x) + ‖u‖2,p(x), thus ‖u‖ =|u|p(x) + |∇u|p(x) +

∑|α|=2

|Dαu|p(x).

In Zang and Fu [14], the equivalence of the norms was proved, and it was even proved that the norm|Δu|p(x) is equivalent to the norm ‖u‖ (see [14, Theorem 4.4]).

Let us choose on X the norm defined by ‖u‖ = |Δu|p(x). Note that, (X, ‖ · ‖) is also a separable andreflexive Banach space. Similar to (2.3), (2.4), (2.5) and (2.6), we have the following, define mappingρ : X → R by

ρ(u) =∫Ω

|Δu|p(x)dx,


then the following relations hold

‖u‖ < 1(= 1, > 1) ⇔ ρ(u) < 1(= 1, > 1), (2.8)

‖u‖ > 1 ⇒ ‖u‖p− ≤ ρ(u) ≤ ‖u‖p+, (2.9)

‖u‖ < 1 ⇒ ‖u‖p+

1 ≤ ρ(u) ≤ ‖u‖p−, (2.10)

‖un − u‖ → 0 ⇔ ρ(un − u) → 0. (2.11)

Hereafter, let

p∗(x) =

⎧⎪⎪⎨⎪⎪⎩

Np(x)N − 2p(x)

, p(x) <N

2,

+ ∞, p(x) ≥ N

2.

Remark 2.1. If h ∈ C+(Ω) and h(x) < p∗(x) for any x ∈ Ω, by Theorem 3.2 in [8], we deduce that Xi iscontinuously and compactly embedded in Lh(x)(Ω).

3. The main result and proof of the theorem

We say that λ ∈ R is an eigenvalue of problem (P ) if there exists u ∈ X\{0} such that∫Ω

|Δu|p(x)−2ΔuΔvdx − λ

∫Ω

V (x)|u|q(x)−2uvdx = 0,

for all u ∈ X. We point out that if λ is an eigenvalue of problem (P ), then the corresponding eigenfunctionv ∈ X\{0} is a weak solution of problem (P ).

In this section we prove two theorems for problem (P ). First, we prove the existence of a continuousfamily of eigenvalues for problem (P ) in a neighborhood of the origin.

Theorem 3.1. Suppose that the following conditions hold:H1(p,q, s) : 1 < q(x) < p(x) < N

2 < s(x), ∀x ∈ Ω.

H1(V) : V ∈ Ls(x)(Ω) and there exists a measurable set Ω0 ⊂ Ω of positive measure such that V (x) >0, ∀x ∈ Ω0.

Then there exists λ0 > 0 such that any λ ∈ (0, λ0) is an eigenvalue of the problem (P ).

Proof. In order to formulate the variational problem (P ), let us introduce the functionals F, G, ϕλ : X →R defined by

F (u) =∫Ω

1p(x)

|Δu|p(x)dx, G(u) =∫Ω

V (x)q(x)

|u|q(x)dx

and

ϕλ(u) = F (u) − λG(u).

Denote by s′(x) the conjugate exponent of the function s(x) and put α(x) := s(x)q(x)s(x)−q(x) . From

H1(p,q, s), we have s′(x)q(x) < p∗(x), ∀x ∈ Ω, α(x) < p∗(x), ∀x ∈ Ω. Thus, by remark 2.1, theembeddings X ↪→ Ls′(x)q(x)(Ω) and X ↪→ Lα(x)(Ω) are compact and continuous.

The proof is divided into the following four steps.Step 1. We will show that ϕλ ∈ C1(X, R).We only need to prove that G ∈ C1(X, R), that is, we show that for all h ∈ X,

limt↓0

G(u + th) − G(u)t

= 〈dG(u), h〉,


and dG : X → X∗ is continuous, where we denote by X∗ the dual space of X.For all h ∈ X, we have

limt↓0

G(u + th) − G(u)t

=d

dtG(u + th)|t=0

=

⎛⎝ d

dt

∫Ω

V (x)q(x)

|u + th|q(x)dx

⎞⎠ |t=0

=∫Ω

d

dt

(V (x)q(x)

|u + th|q(x)

)|t=0dx

=∫Ω

V (x)|u + th|q(x)−2(u + th)h|t=0dx

=∫Ω

V (x)|u|q(x)−2uhdx

= 〈dG(u), h〉.The differentiation under the integral is allowed for t close to zero. Indeed, for |t| < 1, using inequalities(2.2), (2.7) and condition H1(p,q, s), we have

∫Ω

|V (x)|u + th|q(x)−2(u + th)h|dx ≤∫Ω

|V (x)||u + th|q(x)−1|h|dx

≤∫Ω

|V (x)|(|u| + |h|)q(x)−1|h|dx

≤3|V |s(x)

∣∣|u| + |h|∣∣qi−1

q(x)|h|α(x)

< + ∞,

where i = + if∣∣|u| + |h|∣∣

q(x)> 1 and i = − if

∣∣|u| + |h|∣∣q(x)

≤ 1. Since X ↪→ Lα(x)(Ω), X ↪→ Lq(x)(Ω) and

V ∈ Ls(x)(Ω).On the other hand, by remark 2.1, we have X ↪→ Lα(x)(Ω)(compact embedding). Furthermore, there

exists c1 such that |h|α(x) ≤ c1‖h‖. Therefore, by condition H1(p,q, s), we have

|〈dG(u), h〉| =∣∣∣∫Ω

|V (x)|u|q(x)−2uhdx∣∣∣

≤∫Ω

|V (x)||u|q(x)−1|h|dx

≤( 1

s− +q+

q+ − 1+

1α−

)|V |s(x)

∣∣|u|q(x)−1∣∣

q(x)q(x)−1

|h|α(x)

≤( 1

s− +q+

q+ − 1+

1α−

)|V |s(x)|u|qi−1

q(x) |h|α(x)

≤ c1

( 1s− +

q+

q+ − 1+

1α−

)|V |s(x)|u|qi−1

q(x) ‖h‖,

for any h ∈ X.


Thus there exists c2 = c1

(1

s− + q+

q+−1 + 1α−

)|V |s(x)|u|qi−1

q(x) such that

|〈dG(u), h〉| ≤ c2‖h‖.

Using the linearity of dG(u) and the above inequality we deduce that dG(u) ∈ X∗

Note that map Lq(x)(Ω) � u �→ |u|q(x)−2u ∈ Lq(x)

q(x)−1 (Ω) is continuous. For the Frechet differentiability,we conclude that G is Frechet differentiable. Furthermore,

〈G′(u), v〉 =∫Ω

V (x)|u|q(x)−2uvdx,

for all u, v ∈ X. Similarly, we can also prove that F ∈ C1(X, R). The Step 1 is completed.It is clear that (u, λ) is a solution (P ) if and only if F ′(u) = λG′(u) in X∗.Step 2. There exists λ0 > 0 such that for any λ ∈ (0, λ0) there exist τ, a > 0 such that ϕλ(u) ≥ a > 0

for any u ∈ X with ‖u‖ = τ.

Since the embedding X ↪→ Ls′(x)q(x)(Ω) is continuous, we can find a constant c3 > 0 such that

|u|s′(x)q(x) ≤ c3‖u‖, ∀u ∈ X. (3.1)

Let us fix τ ∈ (0, 1) such that τ < 1c3

. Then relation (3.1) implies |u|s′(x)q(x) < 1, for all u ∈ X with‖u‖ = τ . Thus, ∫

Ω

V (x)|u|q(x)dx ≤ |V |s(x)

∣∣|u|q(x)∣∣s′(x)

≤ |V |s(x)|u|q−

q(x)s′(x), (3.2)

for all u ∈ X with ‖u‖ = τ.Combining (3.1) and (3.2), we obtain∫

Ω

V (x)|u|q(x)dx ≤ cq−3 |V |s(x)‖u‖q−

, (3.3)

for all u ∈ X with ‖u‖ = ρ.Hence, from (3.3) we deduce that for any u ∈ X with ‖u‖ = τ < 1, we have

ϕλ(u) =∫Ω

1p(x)

|Δu|p(x)dx − λ

∫Ω

V (x)q(x)

|u|q(x)dx

≥ 1p+

‖u‖p+ − λcq−3

q− |V |s(x)‖u‖q−

=1

p+‖u‖p+ − λcq−

3

q− |V |s(x)τq−

= τ q−(

1p+

τp+−q− − λcq−3

q− |V |s(x)

).

Putting

λ0 =τp+−q−

2p+

q−

cq−3 |V |s(x)

,

then for any λ ∈ (0, λ0) and u ∈ X with ‖u‖ = τ , there exists a = τp+−q−

2p+ , such that

ϕλ(u) ≥ a > 0.

Step 3. There exists φ ∈ X such that φ ≥ 0, φ = 0 and ϕλ(tφ) < 0, for t > 0 small enough.


In fact, assumption H1(p,q, s) implies q(x) < p(x), ∀x ∈ Ω0. In the sequel, we use the notationp−0 = inf

Ω0p(x), q−

0 = infΩ0

q(x) and q+0 = sup

Ω0

q(x). Thus, there exists ε0 > 0 such that q−0 + ε0 < p−

0 .

Since q ∈ C(Ω0), there exists an open set Ω1 ⊂ Ω0 such that

|q(x) − q−0 | < ε0, ∀x ∈ Ω1.

Thus, we deduceq(x) ≤ q−

0 + ε0, ∀x ∈ Ω1. (3.4)

Take φ ∈ C∞0 (Ω0) such that Ω1 ⊂ suppφ, φ(x) = 1 for x ∈ Ω1 and 0 < φ < 1 in Ω0. Without loss of

generality, we may assume ‖φ‖ = 1, that is∫Ω0

|Δφ|p(x)dx = 1. (3.5)

Using (3.4) and (3.5), for all t ∈ (0, 1), we get the estimate

ϕλ(tφ) =∫Ω

tp(x)

p(x)|Δφ|p(x)dx − λ

∫Ω

tq(x)

q(x)V (x)|φ|q(x)dx

=∫Ω0

tp(x)

p(x)|Δφ|p(x)dx − λ

∫Ω0

tq(x)

q(x)V (x)|φ|q(x)dx

≤ tp−0

p−0

∫Ω0

|Δφ|p(x)dx − λ

q+0

∫Ω1

tq(x)V (x)|φ|q(x)dx

≤ tp−0

p−0

− λtq−0 +ε0

q+0

∫Ω1

V (x)|φ|q(x)dx.

Then, for any t < τ1

p−0 −q

−0 −ε0 with 0 < τ < min

{1,

λp−0

q+0

∫Ω1

V (x)|φ|q(x)dx}

, we conclude that

ϕλ(tφ) < 0.

By Step 2, we haveinf

v∈∂Bρ(0)ϕλ(v) > 0. (3.6)

On the other hand, by Step 3, there exists φ ∈ X such that ϕλ(tφ) < 0 for t > 0 small enough. Using(3.3), it follows that

ϕλ(u) ≥ 1p+

‖u‖p+ − λcq−3 |V |s(x)‖u‖q−

, ∀u ∈ Bρ(0).

Thus,

−∞ < cλ := infv∈Bρ(0)

ϕλ(v) < 0.

Now let ε be such that 0 < ε < infv∈∂Bρ(0)

ϕλ(v)− infv∈Bρ(0)

ϕλ(v). Then, by applying Ekeland’s variational

principle to the functional

ϕλ : Bρ(0) → R,


there exist uε ∈ Bρ(0) such that

ϕλ(uε) ≤ infv∈Bρ(0)

ϕλ(v) + ε,

ϕλ(uε) < ϕλ(u) + ε‖u − uε‖, u = uε.

Since

ϕλ(uε) ≤ infv∈Bρ(0)

ϕλ(v) + ε ≤ infv∈Bρ(0)

ϕλ(v) + ε < infv∈∂Bρ(0)

ϕλ(v),

we deduce that uε ∈ Bρ(0).Now, we define Tλ : Bρ(0) → R by

Tλ(u) = ϕλ(u) + ε‖u − uε‖.

It is clear that uε is an minimum of Tλ. Therefore, for small t > 0 and v ∈ B1(0), we have

Tλ(uε + tv) − Tλ(uε)t

≥ 0,

which implies that

ϕλ(uε + tv) − ϕλ(uε)t

+ ε‖v‖ ≥ 0.

As t → 0, we have

〈dϕλ(uε) + ε‖v‖ ≥ 0, ∀v ∈ B1(0).

Hence, ‖ϕ′λ(uε)‖X∗ ≤ ε. We deduce that there exists a sequence {un}∞

n ⊂ Bρ(0) such that

ϕλ(un) → cλ and ϕ′λ(un) → 0. (3.7)

It is clear that {un}∞n is bounded in X. Thus, there exists u in X such that, up to a subsequence, un ⇀ u

in X.Step 4. We will show that un → u in X.Claim:

limn→∞

∫Ω

V (x)|un|q(x)−2un(un − u)dx = 0.

In fact, from the Holder type inequality, we have∫Ω

V (x)|un|q(x)−2un(un − u)dx

≤ |V |s(x)

∣∣|un|q(x)−2un(un − u)∣∣s′(x)

≤ |V |s(x)

∣∣|un|q(x)−2un

∣∣q(x)

q(x)−1|un − u|α(x)

≤ |V |s(x)|(1 + |un|q+−1q(x) )|un − u|α(x)

Since X is continuously embedded in Lq(x)(Ω) and {un}∞n is bounded in X, so {un}∞

n is boundedin Lq(x)(Ω). On the other hand, since the embedding X ↪→ Lα(x)(Ω) is compact, we deduce that |un −u|α(x) → 0 as n → +∞. Hence, the proof of Claim is complete.

Moreover, since dϕλ(un) → 0 and {un}∞n is bounded in X, we have

|〈dϕλ(un), un − u〉|≤ |〈dϕλ(un), un〉| + |〈dϕλ(un), u〉|≤ ‖dϕλ(un)‖X∗‖un‖ + ‖dϕλ(un)‖X∗‖u‖,


that is,

limn→+∞〈dϕλ(un), un − u〉 = 0.

Using the previous Claim and the last relation we deduce that

limn→+∞

∫Ω

|Δun|p(x)−2ΔunΔ(un − u)dx = 0. (3.8)

From (3.8) and the fact that un ⇀ u in X it follows that

limn→→+∞〈F ′(un), un − u〉 = 0,

and by Proposition 2.5 (ii) in Ayoujil and Amrouss [7] we deduce that un → u in X. Thus, in view of(3.7), we obtain

ϕλ(u) = cλ < 0 and ϕ′λ(u) = 0. (3.9)

The proof is complete. � �

Next we will prove second main theorem for problem (P ). Firstly, for the case of a non-constant signweight V we define the following:

W+ =

⎧⎨⎩u ∈ X :

∫Ω

V (x)|u|q(x)dx > 0

⎫⎬⎭ ,

W− =

⎧⎨⎩u ∈ X :

∫Ω

V (x)|u|q(x)dx < 0

⎫⎬⎭ ,

λ∗ = infu∈W+

F (u)G(u)

, λ∗ = infu∈W+

∫Ω

|Δu|p(x)dx

∫Ω

V (x)|u|q(x)dx, (3.10)

μ∗ = supu∈W −

F (u)G(u)

, μ∗ = supu∈W −

∫Ω

|Δu|p(x)dx

∫Ω

V (x)|u|q(x)dx.

So we have:

Theorem 3.2. Suppose that the following conditions hold:H2(p,q) : p(x) < N

2 , 1 < p− ≤ p+ < q− ≤ q+ ≤ p∗(x), ∀x ∈ Ω and q+ − 12 < q−.

H2(V) : V ∈ Ls(x)(Ω) is a changing sign function such that s ∈ C(Ω) and

s(x) > max{

1,Np(x)

Np(x) + 2p(x)q(x) − Nq(x)

}for a.e. in Ω.

Then we have(1) Then, λ∗ and μ∗ are the positive and negative eigenvalue of the problem (P ) respectively, satisfying

μ∗ ≤ μ∗ < 0 < λ∗ ≤ λ∗;(2) λ ∈ (−∞, μ∗) ∪ (λ∗,+∞) is an eigenvalue of problem (P ) while any λ ∈ (μ∗, λ∗) is not an

eigenvalue.

Proof. Note that if λ is an eigenvalue of problem (P ) with weight V , then −λ is an eigenvalue of problem(P ) with weight −V . Hence, it is enough to show Theorem 3.2 only for λ > 0. So the problem (P ) hasonly to be considered in W+. For this case, the proof is divided into the following four steps.

Step 1. We will show that λ∗ > 0.


By relation (3.10) we deduce thatq−

p+λ∗ ≤ λ∗ ≤ q+

p− λ∗. (3.11)

Since p+ < q−, we conclude that 0 ≤ λ∗ ≤ λ∗.Claim:

lim‖u‖→0,u∈W+

F (u)G(u)

= ∞. (3.12)

lim‖u‖→∞,u∈W+

F (u)G(u)

= ∞. (3.13)

In fact, using (2.1) and (2.7) it follows that

|G(u)| =∣∣∣∫Ω

V (x)q(x)

|u|q(x)dx∣∣∣

≤ 2q− |V |s(x)

∣∣|u|q(x)∣∣(s(x))′

≤ 2q− |V |s(x)|u|qτ

q(x)(s(x))′ ,

(3.14)

where τ = + if |u|q(x)(s(x))′ < 1 and τ = − if |u|q(x)(s(x))′ ≥ 1.By condition H2(V), we have 1 < (s(x))′q(x) < p∗(x), that is, X is continuously embedded in

L(s(x))′q(x)(Ω), we ensure the existence of a constant c4 > 0 such that

|G(u)| ≤ 2c4

q− |V |s(x)‖u‖qτ

. (3.15)

For u ∈ W+ with ‖u‖ ≤ 1 by relations (3.15), (2.10) we infer

F (u)G(u)

=

∫Ω

1p(x) |Δu|p(x)dx

∫Ω

V (x)q(x) |u|q(x)dx

≥ q−

2c4p+|V |s(x)‖u‖p+−q−

. (3.16)

Since p+ < q− ≤ q+, we conclude by inequality (3.16) that

F (u)G(u)

→ ∞ when ‖u‖ → 0, u ∈ W+.

So relation (3.12) holds.On the other hand, since q+ − 1

2 < q−, it follows that there exists θ > 0 such that q+ − 12 < θ < q−,

which implies that

q+ − 1 < q− − 12

< θ ⇒ 1 + θ − q+ > 0,

2(q− − θ) ≤ 2(q+ − θ) < 1.(3.17)

Take r(x) be any measurable function satisfying

max{

s(x)1 + θs(x)

,p∗(x)

p∗(x) + θ − q(x)

}≤ r(x)

≤ min{

p∗(x)p∗(x) + θs(x)

,1

1 + θ − q(x)

}, ∀x ∈ Ω,

θ(r+

r− + 1)

< q−.

(3.18)

It is obvious that r ∈ L∞(Ω) and 1 < r(x) < s(x) for any x ∈ Ω.


For u ∈ W+ we have by (2.1) and (2.4)

|G(u)| ≤∫Ω

|V |u|θ||u|q(x)−θdx ≤ ∣∣V |u|θ∣∣r(x)

∣∣|u|q(x)−θ∣∣(r(x))′ . (3.19)

Using again (2.4) we know that

∣∣V |u|θ∣∣r(x)

≤

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1, if∣∣V |u|θ∣∣

r(x)≤ 1,

⎡⎣∫

Ω

|V |r(x)|u|θr(x)dx

⎤⎦

1r−

≤ 2∣∣∣|v|r(x)

∣∣∣1

r−s(x)r(x)

∣∣∣|u|θr(x)∣∣∣

1r−

( s(x)r(x) )

′,

if∣∣V |u|θ∣∣

r(x)> 1,

that is,∣∣V |u|θ∣∣

r(x)≤ 1 + 2

∣∣|v|r(x)∣∣ 1

r−s(x)r(x)

∣∣|u|θr(x)∣∣ 1

r−( s(x)

r(x) )′

≤ 1 + 2(1 + |V |

r+

r−r(x)

)(1 + |u|θ

r+

r−θr(x)( s(x)

r(x) )′

).

(3.20)

Using the same arguments as above, we get

∣∣|u|q(x)−θ∣∣(r(x))′ ≤ 1 + |u|q+−θ

(q(x)−θ)(r(x))′ . (3.21)

Using again (3.20) and (3.21) in (3.19), we deduce

|G(u)| ≤[1 + 2

(1 + |V |

r+

r−s(x)

)(1 + |u|θ

r+

r−θr(x)( s(x)

r(x) )′

)]

×[1 + |u|q+−θ

(q(x)−θ)(r(x))′

]

= 1 + |u|q+−θ(q(x)−θ)(r(x))′ + 2

(1 + |V |

r+

r−s(x)

)|u|θ

r+

r−θr(x)( s(x)

r(x) )′

+ 2(1 + |V |

r+

r−s(x)

)+ 2

(1 + |V |

r+

r−s(x)

)|u|q+−θ

(q(x)−θ)(r(x))′

+ 2(1 + |V |

r+

r−s(x)

)|u|q+−θ

(q(x)−θ)(r(x))′ |u|θr+

r−θr(x)( s(x)

r(x) )′

≤ 1 + |u|q+−θ(q(x)−θ)(r(x))′ + 2

(1 + |V |

r+

r−s(x)

)|u|θ

r+

r−θr(x)( s(x)

r(x) )′

+ 2(1 + |V |

r+

r−s(x)

)+ 2

(1 + |V |

r+

r−s(x)

)|u|q+−θ

(q(x)−θ)(r(x))′

+(1 + |V |

r+

r−s(x)

)[|u|2(q+−θ)

(q(x)−θ)(r(x))′ + |u|2θ r+

r−θr(x)( s(x)

r(x) )′

].

(3.22)

Since r(x) is chosen such that (3.18) is fulfilled then

1 ≤ θr(x)(s(x)

r(x)

)′, (q(x) − θ)(r(x))′ ≤ p∗(x), a.e. x ∈ Ω.


Thus, X ↪→ Lr(x)( s(x)r(x) )

′(Ω) and X ↪→ L(q(x)−θ)(r(x))′

(Ω) are continuous embedding, so there exist positiveconstant c5, c6, M1,M2 such that

|G(u)| ≤ 1 + c5‖u‖q+−θ + 2c6

(1 + |V |

r+

r−s(x)

)‖u‖θ r+

r−

+2(1 + |V |

r+

r−s(x)

)+ 2c5

(1 + |V |

r+

r−s(x)

)‖u‖q+−θ

+(1 + |V |

r+

r−s(x)

)[c5‖u‖2(q+−θ) + c6‖u‖2θ r+

r−]

≤ M1 + M2

(‖u‖2(q+−θ) + ‖u‖2θ r+

r−). (3.23)

In view of (3.23), for any u ∈ W+ with ‖u‖ > 1 we have

F (u)G(u)

≥1

p+ ‖u‖p−

M1 + M2

(‖u‖2(q+−θ) + ‖u‖2θ r+

r−) → ∞, as ‖u‖ → ∞,

because p− > 1 > 2(q+ − θ) ≥ 2(q− − θ) ≥ 2θ r+

r− > 2θ. So relation (3.13) holds.Now, we turn to proving that λ∗ > 0. If this is false, then λ∗ = 0, using again (3.11) we obtain λ∗ = 0.

Let {un} ⊂ W+ \ {0} be such that limn→∞

F (un)G(un) = 0.

On the other hand, like relation (3.16), we have

F (un)G(un)

≥

⎧⎪⎪⎪⎨⎪⎪⎪⎩

q−

2p+|V |s(x)‖u‖p−−q+

, if ‖u‖ ≥ 1;

q−

2p+|V |s(x)‖u‖p+−q−

, if ‖u‖ < 1.

(3.24)

By condition H2(p,q), we have that p− − q+ < 0, p+ − q− < 0. So (3.24) implies that ‖un‖ → ∞, asn → +∞. Using again (3.13), we deduce

F (un)G(un)

→ ∞, as n → +∞.

But this contradicts with the supposition. Consequently, we have λ∗ > 0.Step 2. We will show that λ∗ is an eigenvalue of problem (P ).In fact, let {un} ⊂ W+ \ {0} be a minizing sequence for λ∗, that is

limn→∞

F (un)G(un)

= λ∗ > 0. (3.25)

By (3.13), we have {un} is bounded in X. Since X is reflexive, it follows that there exists u0 ∈ Xsuch that un ⇀ u0 in X. Note that the function F : X → R is convex function, then it is weakly lowersemi-continuous, thus,

limn→∞ F (un) ≥ F (u0). (3.26)

On the other hand, since X is compactly embedded in Lq(x)s′(x)(Ω), then

un → u0 in Lq(x)s′(x)(Ω). (3.27)


Note that |un|s′(x)q(x) → |u0|s′(x)q(x),∣∣|un|q(x)

∣∣s′(x)

→ ∣∣|u0|q(x)∣∣s′(x)

,∣∣|un|q(x)

∣∣s′(x)

is bounded and

|un|q(x) ⇀ |u0|q(x) in Ls′(x)(Ω), so |un|q(x) → |u0|q(x) in Ls′(x)(Ω). Hence, from (3.14) and (3.27) we have

|G(un) − G(u0)| ≤∫Ω

∣∣V (x)q(x)

(|un|q(x) − |u0|q(x))∣∣dx

≤ 2q− |V |s(x)

∣∣|un|q(x) − |u0|q(x)∣∣s(x)′ → 0, as n → ∞,

that islim

n→∞ G(un) = G(u0). (3.28)

In view of (3.26) and (3.28), we obtain

F (u0)G(u0)

= λ∗, if u0 = 0.

It remains to show that u0 is nontrivial. Assume by contradiction the contrary. Then un ⇀ 0 in X,un → 0 in Lq(x)s′(x)(Ω). Thus,

limn→∞ G(un) = 0. (3.29)

Taking ε ∈ (0, λ∗) be fixed. By (3.25) we deduce that for n large enough we have

|F (un) − λ∗F (u0)| < εG(un),

which deduce that(λ∗ − ε)G(un) < F (un) < (λ∗ + ε)G(un). (3.30)

Combining (3.29) and (3.30), we have

limn→∞ F (un) = 0. (3.31)

Using (2.11) and (3.31), we get

un → 0 in X, that is, ‖un‖ → 0, as n → ∞.

From this information and relation (3.12), we have

lim‖un‖→0, u∈W+

F (un)G(un)

= ∞

and this is a contradiction. Thus u0 = 0. Hence u0 is an eigenfunction and Step 2 is proved.Step 3. We will show that any λ ∈ (λ∗,+∞) is an eigenvalue of problem (P ).Let λ ∈ (λ∗,+∞) be arbitrary but fixed. Then λ is an eigenvalue of problem (P ) if and only if there

exists uλ ∈ W+\{0} a critical point of ϕλ.It follows from (3.23) that

ϕλ(u) =∫Ω

1p(x)

|Δu|p(x)dx − λ

∫Ω

V (x)q(x)

|u|q(x)dx

≥ 1p+

‖u‖p− − λ

[M1 + M2

(‖u‖2(q+−θ) + ‖u‖2θ r+

r−)]

→ + ∞, as ‖u‖ → ∞,

because p− > 1 > 2(q+ − θ) ≥ 2(q− − θ) ≥ 2θ r+

r− > 2θ.The above The process of proof in the Step 2 enable us to affirm that G is weakly-strongly continuous.

Since F is weakly lower semi-continuous, so by Weierstrass theorem, there exists uλ ∈ W+ a globalminimum point of ϕλ and thus, a critical point of ϕλ. To prove that uλ is nontrivial. Indeed, since


λ∗ = infu∈W+

F (u)G(u) and λ > λ∗ it follows that there exists vλ ∈ W+ such that F (vλ)

G(vλ) < λ, that is, ϕ(vλ) =

F (vλ) − λG(vλ) < 0. Thus

infu∈W+

ϕ(u) < 0

and we conclude that uλ is a nontrivial critical point of ϕλ, and then λ is an eigenvalue of problem (P ).Step 4. We will show that any λ ∈ (0, λ∗) is not an eigenvalue of problem (P ).Indeed, arguing by contradiction, suppose that there exists λ ∈ (0, λ∗) is an eigenvalue of problem (P )

then there exists uλ ∈ W+ such that

〈F ′(uλ), v〉 = λ〈G′(uλ), v〉, ∀v ∈ W+.

Thus, for v = uλ we have ∫Ω

|Δuλ|p(x)dx = λ

∫Ω

V (x)|uλ|q(x)dx.

The fact that uλ ∈ W+ assures that∫Ω

V (x)|uλ|q(x)dx > 0. Since λ < λ∗ the above information yields

∫Ω

|Δuλ|p(x)dx ≥ λ∗∫Ω

V (x)|uλ|q(x)dx

> λ

∫Ω

V (x)|uλ|q(x)dx

=∫Ω

|Δuλ|p(x)dx.

Clearly, the above inequalities lead to a contradiction. Thus, Step 4 is verified. Therefore, the proof ofTheorem 3.2 is now complete. �

Remark 3.1. At this stage we are not able to deduce whether λ∗ = λ∗ (resp. μ∗ = μ∗) or λ∗ < λ∗ (resp.μ∗ < μ∗). In the latter case an interesting question concerns the existence of eigenvalues of problem (P )in the interval [λ∗, λ∗) ∪ (μ∗, μ∗]. We suggest the readers to the study of these open problems.

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Bin Ge

Department of Applied Mathematics

Harbin Engineering University

Harbin 150001

People’s Republic of China

e-mail: [email protected]

Qing-Mei ZhouLibraryNortheast Forestry UniversityHarbin 150040People’s Republic of Chinae-mail: [email protected]

Yu-Hu WuDepartment of MathematicsHarbin University of Science and TechnologyHarbin 150080People’s Republic of Chinae-mail: [email protected]

(Received: January 21, 2014; revised: June 22, 2014)

eigenvalues of the p(x)-biharmonic operator with indefinite weight

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