Infinitely many solutions for elliptic problems in R N involving the p ( x )-Laplacian

We consider the p(x)-Laplacian equations in RN . The potential function does not satisfy the coercive condition. We obtain the existence of infinitely many solutions of the equations, improving a recent result of Duan–Huang [L. Duan, L. H. Huang, Electron. J. Qual. Theory Differ. Equ. 2014, No. 28, 1–13].


Introduction and main results
Let us consider the following nonlinear elliptic problem: − p(x) u + V(x)|u| p(x)−2 u = f (x, u), in R N , (P) where 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, and the nonlinear term f is sublinear with some precise assumptions that we state below.We emphasize that the operator −∆ p(x) u = div(|∇u| p(x)−2 ∇u) is said to be p(x)-Laplacian, which becomes p-Laplacian when p(x) ≡ p (a constant).The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian, for example, it is inhomogeneous and in general, it does not have the first eigenvalue.The study of various mathematical problems with variable exponent growth condition has received considerable attention in recent years.These problems appear in a lot of applications, such as image processing models (see e.g.[6,21]), stationary thermorheological viscous flows (see [2]) and the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium (see [3]).We refer to [4,[14][15][16][17][18][23][24][25] for the study of the p(x)-Laplacian equations and the corresponding variational problems.
It is well known, the main difficulty in treating problem (P) in R N arises from the lack of compactness of the Sobolev embeddings, which prevents from checking directly that the energy functional associated with (P) satisfies the C-condition.To overcome the difficulty of the noncompact embedding, Dai in [7,8] proves that the subspace of radially symmetric functions of W 1,p(x) (R N ), denoted further by W 1,p(x) r (R N ), can be embedded compactly into L ∞ (R N ) whenever 2 ≤ N < p − ≤ p + < +∞.Later, Alves-Liu [1] (who use the conditions (V 1 ) and (V 2 )), Ge-Zhou-Xue [19,20] (who assume that conditions (V 1 ) and (V 3 ) hold) also establish new compact embedding theorems for the subspace of W 1,p(x) (R N ) when 1 < p − ≤ p + < N. Furthermore, the authors make the following assumptions on the potential function V.
where µ is the Lebesgue measure on R N .We emphasize that in our approach, no coerciveness hypothesis (V 2 ) and not necessarily radial symmetry will be required on the potential V. To the best of our knowledge, there are only a few works concerning on this case up to now.Inspired by the above facts and the aforementioned papers, the main purpose of this paper is to study the existence of infinitely many solutions for problem (P) when F(x, u) is sublinear in u at infinity.Our tool used here is a variational method combined with the theory of variable exponent Sobolev spaces.
We are now in the position to state our main results.
Then problem (P) possesses infinitely many solutions {u k } satisfying The rest of this paper is organized as follows.In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the nonsmooth critical point theory of the locally Lipschitz functionals.In Section 3, the proof of the main results is given.

Preliminaries
In order to discuss problem (P), we need some facts on space W 1,p(x) (R N ) which are called variable exponent Sobolev spaces.For a deeper treatment on these spaces, we refer to [10][11][12]22].Write 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 L p(x) (R N ) is defined by Then we define the variable exponent Sobolev space With these norms, the spaces L p(x) (R N ) and W 1,p(x) (R N ) are separable reflexive Banach spaces; see [12] for the details.
Next, we consider the case that V satisfies (V 1 ).On the linear subspace Then (E, Proposition 2.4 ([11]).Let p : R N → R be Lipschitz continuous and satisfy 1 < p − ≤ p + < N, and q : R N → R be a measurable function.
loc (R N ).Proposition 2.5 ([10]).Let p(x) and q(x) be measurable functions such that p(x) ∈ L ∞ (R N ) and In particular, if p(x) = p is a constant, then |u| p q(x) = |u| p pq(x) .Set We know that (see [5]), I ∈ C 1 (E, R) and p(x)-Laplacian operator −∆ p(x) u is the derivative operator of J in the weak sense.We denote L= I : Proposition 2.6 ([13]).Set E and L as above, then (i) L : E → E * is a continuous, bounded and strictly monotone operator; (ii) L is a mapping of type (S + ), if u n u (weak) in E and lim sup n→∞ In order to assure the existence of infinitely many solutions for the problem (P), our main tool will be the variant fountain theorem [26, Theorem 2.2], which will be used in our proof.
Let X be a Banach space with the norm • and Consider the following C 1 -functional ϕ λ : E → R defined by where A, B : X → R are two functionals.
Lemma 2.7.Suppose that the functional ϕ λ defined above, and satisfies the following conditions.
(3) There exist Then there exist Particularly, if {u(λ n )} has a convergent subsequence for every k, then ϕ 1 has infinitely many nontrivial critical points In order to discuss the problem (P), we need to consider the energy functional ϕ : E → R defined by Under our hypotheses, it follows from a Hölder-type inequality and Sobolev's embedding theorem that the energy functional ϕ is well-defined.It is well known that ϕ ∈ C 1 (E, R) and its derivative is given by It is standard to verify that the weak solutions of problem (P) correspond to the critical points of the functional ϕ.

Proof of the main results
In this section, for the notation in Lemma 2.7, the space X = E, and related functionals on So the perturbed functional which we will study is . We choose a completely orthogonal basis {e i } of E and define X i := Re i , and Z k , Y k defined as (2.1).We shall prove that ϕ λ satisfies the conditions of Lemma 2.7.Following along the same lines as in [9], we can obtain that • B(u) ≥ 0, and B(u) → ∞ as u → ∞ on any finite dimensional subspace of E (see [9,Lemma 3.1]).
• There exists a sequence For further details, we refer the reader to Duan-Huang [9, Lemma 3.2], • There exists a sequence for details, see [9,Lemma 3.3].
Obviously, condition (1) in Lemma 2.7 have been satisfied.Thus, conditions (1), ( 2) and (3) in Lemma 2.7 have been verified.Therefore, we know from Lemma 2.7 that there exist For simplicity, we denote u(λ n ) by u n for all n ∈ N. Claim: The sequence {u n } is bounded in E. By virtue of hypothesis H( f ), Proposition 2.2, Proposition 2.3 and Proposition 2.5, we have for some M 0 > 0. Since 1 < q − ≤ q + < p − , (3.3) implies that {u n } is bounded in E. Next, 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 weakly in E.
From the choice of the function b ∈ L s(x) s(x)−q(x) (R N ), for any given number ε > 0, we can choose R ε > 0 such that loc (R N ), and hence, lim Also from Proposition 2.1 and (3.5), there exists n 0 ∈ N such that Now it is easily seen that We will estimate the right-hand side of (3.8).One clearly has Moreover, we have where Using the Proposition 2.5, Hölder's inequality and (3.7), we have On the other hand, using Proposition 2.4, Proposition 2.5, Hölder's inequality and (3.6), we have (3.12) Similarly, we also have that and Lu n − Lu 0 , u n − u 0 → 0 as n → +∞, (3.16) which implies u n → u 0 in E from Proposition 2.6 (ii).Thus, from the last assertion of Lemma 2.7, we know that ϕ = ϕ 1 has infinitely many nontrivial critical points.Therefore, problem (P) possesses infinitely many nontrivial solutions.The proof of Theorem 1.1 is completed.