Infinitely many solutions for a class of p ( x )-Laplacian equations in R N

In this paper, we study the existence of infinitely many solutions for a class of p(x)-Laplacian equations in RN , where the nonlinearity is sublinear. The main tool used here is a variational method combined with the theory of variable exponent Sobolev spaces. Recent results from the literature are extended.


Introduction
In this paper, we consider the following p(x)-Laplacian equation in where the p(x)-Laplacian operator is defined by ∆ p(x) u = div(|∇u| p(x)−2 ∇u), p : R N → R is Lipschitz continuous and 1 < p − := inf R N p(x) ≤ sup R N p(x) := p + < N, V is the new potential function, f obeys some conditions which will be stated later and W 1,p(x) (R N ) is the variable exponent Sobolev space.
In recent years, the study of various mathematical problems with p(x)-growth condition has attracted more and more attention because these problems possess a solid background in physics and originate from the study on electrorheological fluids (see [1]) and elastic mechanics (see [2]).They also have wide applications in different research fields (see e.g.[3][4][5] and the references therein) and raise many difficult mathematical problems.In particular, the presence of the p(x)-Laplacian operator together with the appearance of the potential function V make its mathematical analysis more difficult than the corresponding p-Laplacian L. Duan and L. Huang equations.Therefore, the mathematical results on the p(x)-Laplacian equations are far from being perfect.
To go directly to the theme of the present paper, we only review some former results which are closely related to our main results (a complete literature on p(x)-Laplacian equation is beyond the scope of this paper, interested authors are referred to [1,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], and the references therein.When V(x) is radial (for example V(x) ≡ 1), Dai studied the following problem in [9]: by means of a direct variational approach and the theory of variable exponent Sobolev spaces, sufficient conditions ensuring the existence of infinitely many distinct homoclinic radially symmetric solutions are established.Based on the theory of variable exponent Sobolev spaces, Avci in [8] studied the existence of infinitely many solutions of problem (1.2) with Dirichlet boundary condition in a bounded domain.Fan and Han in [11] discussed the existence and multiplicity of solutions to problem (1.2).Fu and Zhang in [13] also obtained that problem (1.2) possesses at least two nontrivial weak solutions.For p(x) = p, problem (1.1) reduces to The existence of ground states of problem (1.3) with a potential which is periodic or has a bounded potential well is studied in [21] by Liu.Liu and Zheng in [22] studied problem (1.3) with sign-changing potential and subcritical p-superlinear nonlinearity, by using the cohomological linking method for cones, an existence result of nontrivial solution is obtained.Li and Wang in [23] proved that problem (1.3) has at least a nontrivial solution by using variational methods combined with perturbation arguments.Recently, Alves and Liu in [7] established the existence of ground state solution for problem (1.1) via modern variational methods under some hypotheses on the potential V and the nonlinear term f , particularly, the nonlinearity is superlinear.However, one of the remaining cases is that V is nonradial potential and f (x, u) is sublinear at infinity in u and to the best of our knowledge, no results on this case have been obtained up to now.Based on the above fact and motivated by techniques used in [24,25], the main purpose of this paper is devoted to investigate the existence of infinitely many solutions for problem (1.1) when the nonlinearity is sublinear in u at infinity.Our analysis is based on the variable exponent Lebesgue-Sobolev space theory and variational methods.
We are now in a position to state our main results.
Theorem 1.1.Suppose that the following conditions are satisfied.
N−p(x) , and s(x) p * (x) means that ess inf Infinitely many solutions for a class of p(x)-Laplacian equations in R N

3
Then problem (1.1) possesses infinitely many solutions {u k } satisfying Remark 1.2.From the variational viewpoint, the main difficulty in treating problem (1.1) in R N arises from the lack of compactness of the Sobolev embeddings which prevents from checking directly that the energy functional associated with problem (1.1) satisfies the Palais-Smale condition.To overcome this difficulty, we use a Bartsch-Wang type compact embedding theorem for variable exponent spaces established by Alves and Liu in [7].
Remark 1.3.In this paper, we consider the case that the nonlinearity is sublinear and obtain infinitely many small negative-energy solutions of problem (1.1), which complement and extend previously known results in [7,8,11,13,21,22].
The structure of this paper is outlined as follows.In Section 2, some preliminary results and the variational tools we used are presented.In Section 3, the proof of the main result is given.
Notations: Throughout this paper, we denote a generic positive constant by C which may vary from line to line.If the dependence needs to be explicitly pointed out, then the notations C i (i ∈ Z + ) are used.

Preliminaries
In this section, we first recall some preliminary results about Lebesgue and Sobolev variable exponent spaces, which are useful for discussing problem (1.1).We refer the reader to [26][27][28][29] and the references therein for a more detailed account on this topic. Set In this paper, for any p ∈ C + (R N ), we will denote and denote by p 1 p 2 the fact that ess inf x∈R N (p 2 (x) − p 1 (x)) > 0. Denote by S(R N ) the set of all measurable real-valued functions defined on R N .Note that two measurable functions in S(R N ) are considered as the same element of S(R N ) when they are equal almost everywhere.
Let p ∈ C + (R N ), the variable exponent Lebesgue space is defined by and the variable exponent Sobolev space is defined by L. Duan and L. Huang equipped with the norm Proposition 2.1 ([27]).The spaces L p(x) (R N ) and W 1,p(x) (R N ) are separable and reflexive Banach spaces.
Now, let us introduce the modular of the space L p(x) (R N ) as the functional ρ p(x) (u) : for all u ∈ L p(x) (R N ).The relation between modular and Luxemburg norm is clarified by the following propositions.
we equip it with the norm Then (E, • ) is continuously embedded into W 1,p(x) (R N ) as a closed subspace.Therefore, (E, u ) is also a separable reflexive Banach space.In addition, defining the modular for all u ∈ E, in a similar way to Proposition 2.2, the following proposition holds.
Proposition 2.3.Let u ∈ E and let {u m } be a sequence in E, then Infinitely many solutions for a class of p(x)-Laplacian equations in R N 5 (5) Lemma 2.4 (Hölder-type inequality [12]).The conjugate space of L p(x) (R N ) is L q(x) (R N ), where Lemma 2.6 ([6,11]).Let q, s ∈ C + (R N ) with q(x) ≤ s(x) for all x ∈ R N and u ∈ L s(x) (R N ).Then, or there exists a number q ∈ [q − , q + ] such that The following Bartsch-Wang type compact embedding will play a crucial role in our subsequent arguments.Lemma 2.7 ([7, Lemma 2.6]).If V satisfies (H 1 ), then (i) we have a compact embedding E → L p(x) (R N ), 1 < p − ≤ p + < N; (ii) for any measurable function s(x) : R N → R with p < s p * , we have a compact embedding E → L s(x) (R N ).
Remark 2.8.By virtue of Lemma 2.7, we know that there exists a constant C 1 > 0 such that |u| p(x) ≤ C 1 u for any u ∈ E.
V(x) > 0 and there exists r > 0 such that for all M > 0, where µ denotes the Lebesgue measure on R N , then a similar compact embedding has been established by Ge et al. in [14].
In the following, we present the variational tools named the variant fountain theorem established by Zou [31], which will be used to get our result.
Let E be a Banach space with the norm • and E = ⊕ j∈N X j with dim X j < ∞ for any j ∈ N. Set L. Duan and L. Huang Consider the following C 1 functional I λ : E → R defined by where A, B : E → R are two functionals.
Theorem 2.10 ([31, Theorem 2.2]).Suppose that the functional I λ defined above satisfies the following conditions: (C 1 ) I λ maps bounded sets to bounded sets uniformly for λ ∈ Then there exist as n → ∞.In particular, if {u(λ n )} has a convergent subsequence for every k, then I 1 has infinitely many nontrivial critical points In order to discuss the problem 1.1, we need to consider the energy functional I : E → R defined by Under our conditions, it follows from Hölder-type inequality and Sobolev embedding theorem that the energy functional I is well-defined.It is well known that I ∈ C 1 (E, R) and its derivative is given by for each u ∈ E. It is standard to verify that the weak solutions of problem (1.1) correspond to the critical points of the functional I.

Proof of main result
In order to apply Theorem 2.10, we define the functionals A, B and I λ on the working space E by . We choose a completely orthogonal basis {e j } of E and define X j := Re j , and Z k , Y k defined as (2.4).Now, we show that I λ has the geometric property needed by Theorem 2.10.
Proof.It is obvious that B(u) ≥ 0 from the definition of the functional B and (H 2 ).Next, we claim that on any finite dimensional subspace of E. First, for any finite dimensional subspace F ⊂ E, there exists δ > 0 such that Otherwise, for any positive integer n, there exists u n ∈ F \ {0} such that Since dim F < ∞, we know from the compactness of the unit sphere of F that there exists a subsequence, say {v n }, such that v n → v 0 in F, In view of the equivalence of the norms on the finite dimensional space F, we obtain By Lemma 2.4, (2.1) and (3.4), we have Then there exist α 1 , α 2 > 0 such that If this is not true, then, for all positive integer n, one has and hence one easily checks that v 0 = 0.This is a contradiction with (3.3) and therefore (3.6) holds.Now let and , we have for all positive integer n.Let n be large enough such that for all sufficiently large n, which is a contradiction to (3.5).Therefore, (3.1) holds.Second, for the δ given in (3.1), let Then by (3.1), Combining (H 2 ) and (3.8), for any u ∈ F \ {0}, we have on any finite dimensional subspace of E. The proof is completed.
Obviously, ρ k → 0 as k → ∞.Combining this with (3.9), straightforward computation shows that Furthermore, by (3.9), for any u ∈ Z k with u ≤ ρ k , we have u q( u ) , and therefore 0 ≥ inf The proof is completed.
for u = r k < ρ k sufficiently small.The proof is completed.Now we are in a position to prove Theorem 1.1.In our proof of Theorem 1.1, we will consider A as a functional on (E, • ).We say that an operator Proof of Theorem For simplicity, we denote u(λ n ) by u n for all n ∈ N. We will show that {u n } is bounded in E.
Finally, we show that there is a strongly convergent subsequence of {u n } in E. Indeed, in view of the boundedness of {u n }, passing to a subsequence if necessary, still denoted by {u n }, we may assume that u n u 0 in E, in view of Lemma 2.7, we have Moreover, by (2.5), direct calculation produces It is clear that By virtue of (H 2 ), Remark 2.8, Lemma 2.6 and (3.13), one can deduce that |u n | q(x)−1 s(x) q(x)−1 → 0, as n → ∞.Since A is of (S + ) type (see [7,11]), we obtain u n → u in E. Now from the last assertion of Theorem 2.10, we know that I = I 1 has infinitely many nontrivial critical points.Therefore, problem (1.1) possesses infinitely many nontrivial solutions.The proof of Theorem 1.1 is completed.

10 L.
Duan and L. Huang

Lemma 3 . 3 .
Under the assumptions of Theorem 1.1, for the sequence {ρ k } k∈N obtained in Lemma 3.2, there exist 0