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INFINITELY MANY SMALL SOLUTIONS FOR THE p(x)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH

  • Zhou, Chenxing (College of Mathematics, Changchun Normal University) ;
  • Liang, Sihua (College of Mathematics, Changchun Normal University)
  • Received : 2013.04.03
  • Accepted : 2013.06.26
  • Published : 2014.01.30

Abstract

In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$. We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$, where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.

Keywords

1. Introduction

In this paper we deal with quasilinear elliptic problem of the form

where Ω ⊂ ℝN (N ≥ 3) is a bounded domain with smooth boundary and p(x), q(x) are two continuous functions on where denote by p(x) ≪ q(x) the fact that infx∈Ω(q(x) − p(x)) > 0. λ is a positive parameter, Δp(x)u := div(|∇u|p(x)−2∇u) is the p(x)-Laplacia operator. On the exponent q(x) we assume that is the critical exponent in the sense that is the critical exponent according to the Sobolev embedding. Our goal will be to obtain infinitely many small weak solutions which tend to zero for (1) in the generalized Sobolev space for the general nonlinearities of the type ƒ(x, u).

The study of differential equations and variational problems involving variable exponent conditions has been a very interesting and important topic. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, image processing and so on. For example, Chen, Levin and Rao [4] proposed the following model in image processing

where p(x) is a function satisfies 1 ≤ p(x) ≤ 2 and ƒ is a convex function. For more information on modelling physical phenomena by equations involving p(x)-growth condition we refer to [1,19,28,30]. The appearance of such physical models was facilitated by the development of variable Lebesgue and Sobolev spaces, Lp(x) and W1,p(x), where p(x) is a real-valued function. On the variable exponent Sobolev spaces which have been used to study p(x)-Laplacian problems, we refer to [5,21,29]. On the existence of solutions for elliptic equations with variable exponent, we refer to [2,6,7,8,9,10,11,16,17,31].

In recent years, the existence of infinitely many solutions have been obtained by many papers. When p(x) ≡ p = 2 (a constant) with Dirichlet boundary condition, Li and Zou [23] studied a class of elliptic problems with critical exponents, they obtained the existence theorem of infinitely many solutions under suitable hypotheses. He and Zou [20] proved that the existence infinitely many solutions under case the general nonlinearities. When p(x) ≡ p ≠ 2. Ghoussoub and Yuan [18] obtained the existence of infinitely many nontrivial solutions for Hardy-Sobolev subcritical case and Hardy critical case by establishing Palais-Smale type conditions around appropriate chosen dual sets in bounded domain. Li and Zhang [24] studied the existence of multiple solutions for the nonlinear elliptic problems of p&q-Laplacian type involving the critical Sobolev exponent, they obtained infinitely many weak solutions by using Lusternik-Schnirelman’s theory for Z2-invariant functional.

On the existence of infinitely many solutions for p(x)-Laplacian problems have been studied by [2,7,9,31], but they did not give any further information on the sequence of solutions. Moreover, these papers deal with subcritical nonlinearities. Very little is known about critical growth nonlinearities for variable exponent problems [14,15], since one of the main techniques used in order to deal with such issues is the concentration-compactness principle. This result was recently obtained for the variable exponent case independently in [12,13]. In both of these papers the proof are similar and both relates to that of the original proof of P.L. Lions [25,26].

Recently, Kajikiya [22] established a critical point theorem related to the symmetric mountain pass lemma and applied to a sublinear elliptic equation. But there are no such results on p(x)-Laplacian problem with critical growth (1).

Motivated by reasons above, the aim of this paper is to show that the existence of infinitely many solutions of problem (1), and there exists a sequence of infinitely many arbitrarily small solutions converging to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya [22]. In order to use the symmetric mountain-pass lemma, there are many difficulties. The main one in solving the problem is a lack of compactness which can be illustrated by the fact that the embedding of W1,p(x)(Ω) into Lp*(x)(Ω) is no longer compact. Hence the concentration-compactness principle is used here to overcome the difficulty. The main result of this paper is as follows.

Theorem 1.1. Suppose that f(x, u) satisfies the following conditions:

Then there exists λ∗ such that for any λ ∈ (0, λ∗), problem (1) has a sequence of non-trivial solutions {un} and un → 0 as n → ∞.

Remark 1.1. If without the symmetry condition (i.e. ƒ(x,−u) = -ƒ(x, u)) in Theorem 1.1, we get an existence theorem of at least one nontrivial solution to problem (1) by the same method in this paper.

Remark 1.2. In this paper, we use concentration-compactness principle due to [12] which is slightly more general than those in [13], since we do not require q(x) to be critical everywhere.

Remark 1.3. There exist many functions ƒ(x, t) satisfy condition (H1)-(H3), for example, ƒ(x, u) = u(p--1)/3, where p− > 1.

Remark 1.4. Theorem 1.1 is new as far as we know and it generalizes results in [3] for p(x)-Laplacian type problem. We mainly follow the way in [3] to prove our main result.

Definition 1.2. We say that is a weak solution of problem (1) if for any

The energy functional corresponding to problem (1) is defined as follows,

then, it is easy to check that as arguments [27] show that J(u) is well defined on and the weak solutions for problem (1) coincides with the critical points of J. We try to use a new version of the symmetric mountain-pass lemma due to Kajikiya [22]. But since the functional J(u) is not bounded from below, we could not use the theory directly. So we follow [3] to consider a truncated functional of J(u). Denote J′ : E → E∗ is the derivative operator of J in the weak sense. Then

Definition 1.3. We say J satisfies Palais-Smale condition ((PS) for short) in which satisfies that {J(un)} is bounded and ∥J′(un)∥ p(x) → 0 as n → ∞, has a convergent subsequence.

Under assumption (H2), we have

which means that, for all ε > 0, there exist a(ε), b(ε) > 0 such that

Hence, for any constants β we have

for some c(ε) > 0.

The remainder of the paper is organized as follows. In Section 2, we shall present some basic properties of the variable exponent Sobolev spaces. In Section 3, we will prove the corresponding energy functional satisfies the (PS) condition. In Section 4, we shall prove our main results.

 

2. Weighted variable exponent Lebesgue and Sobolev spaces

We recall some definitions and properties of the variable exponent Lebesgue-Sobolev spaces Lp(·)(Ω) and W1,p(·)(Ω), where Ω is a bounded domain in ℝN.

Set

For any we define

We can introduce the variable exponent Lebesgue space as follows:

Lp(·)(Ω) = {u : u is a measurable real-valued function such that

for Equipping with the norm on Lp(x)(Ω) by

which is a Banach space, we call it a generalized Lebesgue space.

Proposition 2.1 ([5,11]). (i) The space (Lp(x)(Ω), | · | p(x)) is a separable, uni-form convex Banach space, and its conjugate space is Lq(x)(Ω), where 1/q(x) + 1/p(x) = 1. For any u ∈ Lp(x)(Ω) and v ∈ Lq(x)(Ω), we have

(ii) If 0 < |Ω| < ∞ and p1, p2 are variable exponents in such that p1 ≤ p2 in Ω, then the embedding Lp2(·)(Ω) ,→ Lp1(·)(Ω) is continuous.

Proposition 2.2 ([5,11]). The mapping ρp(·) : Lp(·)(Ω) → ℝ defined by

Then the following relations hold:

Next, we define W1,p(x)(Ω) is defined by

and it can be equipped with the norm

Denote under the norm

We know that if Ω ⊂ ℝN is a bounded domain, ∥u∥ and ∥u∥1 are equivalent norms on

Proposition 2.3 ([5,11]). (i) W1,p(x)(Ω) are separable re exive Banach spaces; (ii) If then the embedding W1,p(x)(Ω) → Lq(x)(Ω) is continuous.

In this paper, we use the following equivalent norm on W1,p(x)(Ω):

Proposition 2.4 ([21,6]). Let I(u) = ƒΩ|∇u|p(x) + |u|p(x)dx. If u, un ∈ W1,p(x)(Ω), then the following relations hold:

 

3. Preliminaries and lemmas

In the following, we always use C and ci(i = 1, 2, · · · ) to denote positive constants. We give the concentration-compactness principle of the variable exponent due to [12,15].

Lemma 3.1. Let q(x) and p(x) be two continuous functions such that

Let {uj}j∈ℕ be a weakly convergent sequence in with weak limit u, and such that |∇uj|p(x) ⇀ μ weakly-∗ in the sense of measures; |uj|q(x) ⇀ v weakly-∗ in the sense of measures. Assume, moreover that Then, for some countable index set I we have

(i) ν = |u|q(x) + Σi∈Iνiδxi, νi > 0;

(ii) μ ≥ |∇u|p(x) + Σi∈Iμiδxi, μi > 0;

(iii)

where {xi}i∈I ⊂ Γ and S is the best constant in the Gagliardo-Nirenberg-Sobolev inequality for variable exponents, namely

In order to prove the functional J satisfies the local (PS)c condition, we take continuous function η(x) satisfies Denote

Lemma 3.2. Assume condition (H2) holds. Then for any λ > 0, there exists positive constant m∗ > 0 such that the functional J satisfies the local (PS)c condition in

in the following sense: if

and J′(un) → 0 for some sequence in Then {un} contains a subse-quence converging strongly in

Proof. First, we show that {un} is bounded in If ∥un∥ p(x) → ∞ as n → ∞. Thus, we may assume that ∥un∥ p(x) > 1 for any integer n.

Then for n sufficiently large, we have

By (4), for any (x, t) ∈ Ω × ℝ, we have

On the other hand, noting that p(x) ≪ q(x), by the Young inequality, for any ε2, ε3 ∈ (0, 1), we get

and

Thus, relations (13)-(16) imply that

where Thus, we choose ε2, ε3 be so small that d1 − c1ε2 > 0 and It follows from (8) and (17) that {un} is bounded in Therefore we can assume that

Note that if I = ∅ then un → u strongly in Lq(x)(Ω). If not, let xi be a singular point of the measures μ and ν, define a function such that ϕ(x) = 1 in B(xi, ε), ϕ(x) = 0 in Ω \ (xi, 2ε) and |∇ϕ| ≤ 2/ε in Ω. As we obtain that

i.e.

On the other hand, by Hölder inequality and boundedness of {un}, we have that

From (18), (19) and (21), we get that

Combing this with Lemma 2.1 (iii), we obtain This result implies that

If the second case νi ≥ SN holds, for some i ∈ I, then by using Lemma 2.1 and selecting ε2, ε3 in (17) such that we have

where This is impossible. Consequently, νi = 0 for all i ∈ I and hence

Since {un} is bounded in we deduce that there exists a subsequence, again denoted by {un}, and such that {un} converges weakly to Note that

On the other hand, we have

Using the fact that {un} converges strongly to u0 in Lq(x)(Ω) and inequality (5), we have

where c1 c2 and c3 are positive constants. Using |un − u0 | q(x) → 0 as n → ∞, we deduce that

By (23) and (24), we obtain

It is known that

where (· , ·) is the standard scalar product in ℝN. Relations (25) and (26) yield

This fact and relation (10) imply ∥un − u0∥ p(x) → 0 as n → ∞. The proof is complete.

 

4. Existence of a sequence of arbitrarily small solutions

In this section, we prove the existence of infinitely many solutions of (1) which tend to zero. Let X be a Banach space and denote

Σ := {A ⊂ X \ {0} : A is closed in X and symmetric with respect to the orgin}. For A ∈ Σ, we define genus γ(A) as

If there is no mapping φ as above for any m ∈ N, then γ(A) = +∞. Let Σk denote the family of closed symmetric subsets A of X such that 0 ∉ A and γ(A) ≥ k. We list some properties of the genus (see [22]).

Proposition 4.1. Let A and B be closed symmetric subsets of X which do not contain the origin. Then the following hold.

The following version of the symmetric mountain-pass lemma is due to Kajikiya [22].

Lemma 4.2. Let E be an infinite-dimensional space and J ∈ C1(E,R) and suppose the following conditions hold.

Remark 4.1. From Lemma 4.2 we have a sequence {uk} of critical points such that J(uk) ≤ 0, uk ≠ 0 and limk→∞ uk = 0.

In order to get infinitely many solutions we need some lemmas. We first point out that we have

Proposition 2.3 (ii) imply that

where c4 > 0.

Next, we focus our attention on the case when For such a u by relation (9) we obtain

Using (3) and (27)-(29), we deduce that

where with ∥u∥ p(x) < 1. If we define

Then

From the definition of Q(s) and the fact that p+ < q+, we konw that there exists λ∗ such that for λ ∈ (0, λ∗), Q(t) attains its positive maximum, that is, there exists

such that

Therefore, for e0 ∈ (0, e1), we may find R0 < R1 such that Q(R0) = e0. Now we define

Then it is easy to see χ(t) ∈ [0, 1] and χ(t) is C∞. Let φ(u) = χ(∥u∥ p(x)) and consider the perturbation of J(u):

Then

where and

From the above arguments, we have the following:

Lemma 4.3. Let G(u) be defined as in (31). Then

Proof. It is easy to see (i) and (ii). (iii) are consequences of (ii) and Lemma 3.2.

Lemma 4.4.Assume that (H3) of Theorem 1.1 holds. Then for any k ∈ N, there exists

Proof. First, by (H3) of Theorem 1.1, for any fixed we have

Next, given any k ∈ N, let Ek be a k-dimensional subspace of We take u ∈ Ek with norm ∥u∥ p(x) = 1, for 0 < ρ < min{R0, 1}, we get

Since Ek is a space of finite dimension, all the norms in Ek are equivalent. If we define

It follows from (32)that

since lim|ρ|→0M(ρ) = +∞. That is,

This completes the proof.

Now we give the proof of Theorem 1.1.

Proof of Theorem 1.1 Recall that

and define

By Lemmas 4.3 (i) and 4.4, we know that −∞ < ck < 0. Therefore, assumptions (C1) and (C2) of Lemma 4.2 are satisfied. This means that G has a sequence of solutions {un} converging to zero. Hence, Theorem 1.1 follows by Lemma 4.3 (ii).

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