Multiple small solutions for p ( x )-Schrödinger equations with local sublinear nonlinearities via genus theory

In this paper, we deal with the following p(x)-Schrödinger problem: { −div(|∇u|p(x)−2∇u) + V(x) |u|p(x)−2 u = f (x, u) in RN ; u ∈W1,p(x)(RN), where the nonlinearity is sublinear. We present the existence of infinitely many solutions for the problem. The main tool used here is a variational method and Krasnoselskii’s genus theory combined with the theory of variable exponent Sobolev spaces. We also establish a Bartsch–Wang type compact embedding theorem for the variable exponent spaces.


Introduction
In this paper, we consider the following p(x)-Schrödinger equation in R N −div(|∇u| p(x)−2 ∇u) + V(x) |u| p(x)−2 u = f (x, u), in R N ; u ∈ W 1,p(x) (R N ), (P) where N ≥ 2, p : R N → R is Lipschitz continuous, 1 < p − := inf x∈R N p(x) ≤ p + := sup x∈R N p(x) < N and V ∈ C(R N , R) 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.
These interests are stimulated mainly by the development of the studies of problems in elasticity, electrorheological fluids, flow in porous media, calculus of variations, differential equations with p(x)-growth (see [1,4,10,20,27,28]).We refer to the p(x)-Laplace operator ∆ p(x) u := div(|∇u| p(x)−2 ∇u), where p is a continuous non-constant function.This differential operator is a natural generalization of the p-Laplace operator ∆ p u := div(|∇u| p−2 ∇u), where p > 1 is a real constant.However, the p (x)-Laplace operator possesses more complicated nonlinearity than p-Laplace operator, due to the fact that ∆ p(x) is not homogeneous.This fact implies some difficulties; for example, we cannot use the Lagrange multiplier theorem in many problems involving this operator.Among these problems, the study involving p (x)-Laplacian problems via variational methods is an interesting topic.Many researchers have devoted their work to this area (see [2,3,13,15,17,26]).
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 is proved.The case of p, when V are radially symmetric on R N with 1 < p − ≤ p + < N and V − > 1 was discussed by Ge, Zhou and Xue in [19].The existence of at least two nontrivial solutions has been established.In [30], Zhou and Wang studied the existence of infinitely many solutions for a class of (P) when the potential function does not satisfy the coercive condition.
For p(x) = p, problem (P) reduces to Liu [24] studied that the existence of ground states of problem (P 0 ) with a potential which is periodic or has a bounded potential.Liu, Wang [23] discussed the problem (P 0 ) with sign-changing potential and subcritical p-superlinear nonlinearity, by using the cohomological linking method for cones, and obtained an existence result of nontrivial solution.Recently, Alves, Liu [3] established the existence of ground state solution for problem (P) via modern variational methods on the potential function V under following hypothesis , where µ(.) denotes the Lebesgue measure in R N .By using a variational method combined with the theory of variable exponent Sobolev spaces, Duan, Huang [13] and Wang, Yao, Liu [29] studied the existence of infinitely many solutions for a class of (P) equations in R N , which the potential V satisfies hypothesis (V 0 ) and f (x, u) is sublinear at infinity in u.Moreover, the authors proposed new assumptions on the nonlinear term to yield bounded Palais-Smale sequences and then proved that the special sequences converge to critical points respectively.The main arguments are based on the geometry supplied by the fountain theorem.Consequently, they showed that the problem (P) under investigation admits a sequence of weak solutions with high energies.Now, p(x) and V(x) satisfy the following assumptions.
(p 1 ) The function p : R N → R is ipschitz continuous and 1 < p − ≤ p + < N; > 0 and there are constants r > 0, α > N such that for any b > 0, where B r (y) = x ∈ R N : |x − y| < r and µ(•) denotes the Lebesgue measure.
Remark 1.1.Condition (V 1 ) which is weaker than (V 0 ), is originally introduced by Bartsch, Wang, Willem [6] to guarantee the compact embedding of the working space.There are functions V satisfying (V 1 ) and not satisfying (V 0 ).
In the present paper, we concern with the existence of infinitely many solutions for (P) in R N without any growth conditions imposed on f (x, u) at infinity with respect to u and to the best of our knowledge, no results on this case have been obtained up to now.The main tool used here is a variational method and Krasnoselskii's genus theory combined with the theory of variable exponent Sobolev spaces.We also prove a Bartsch-Wang type compact embedding theorem for variable exponent spaces.We emphasize that in our approach, no coerciveness hypothesis (V 1 ) and not necessarily radially symmetric will be required on the potential V .Based on the above fact and motivated by techniques used in [24,29,30], the main purpose of this paper is devoted to investigate the existence of infinitely many solutions for problem (P) when the nonlinearity is sublinear in u at infinity.
Assume that f : R N × R → R satisfies Carathéodory and the following conditions.
where g : R N → R + is a positive continuous function such that g ∈ L q(x) p * (x) = N p(x) N−p(x) , and q(x) p * (x) means that ess inf x∈R N (p * (x) − q (x)) > 0.
( f 2 ) There exist an x 0 ∈ R N and a constant r > 0 such that lim inf for all t ∈ R and for all x ∈ R N .

Preliminaries
In this section, we recall some results on variable exponent Lebesgue and Sobolev spaces.Over the last decade, the variable exponent Lebesgue spaces L p(x) and the corresponding Sobolev space W 1,p(x) have been a subject of active research area (we refer to [11,14,16,18,21,25] for the fundamental properties of these spaces).Write The set of all measurable real-valued functions defined on R N will be denote by (R N ).Note that two measurable functions in (R N ) are considered as the same element of (R N ) when they are equal almost everywhere.
which is equipped with the norm, so-called Luxemburg norm Moreover, we define the variable exponent Sobolev space by for all u ∈ W 1,p(x) (R N ).The spaces L p(x) (R N ), W 1,p(x) (R N ) are separable and reflexive Banach spaces [11,18,21].Now, let us introduce the modular of the space L p(x) R N as the functional for all u ∈ L p(x) (R N ).The relation between modular and Luxemburg norm is clarified by the following propositions.
Denoted by N G the Nemytsky operator is defined by G, i.e.
Next, we consider the case that V satisfies (V 1 ).On the linear subspace we equip with the norm Then (E, • E ) is continuously embedded into W 1,p(x) R N as a closed subspace.Therefore, (E, • E ) is also a separable reflexive Banach space.
In addition, we define the modular Λ p(x),V : E → R associated with E as follows: for all u ∈ E, in a similar way to Proposition 2.1.The following proposition holds.
Proposition 2.7 ([11, 18, 21]).Let u, u n ∈ E: J is a convex functional, J : E → E * is strictly monotone, bounded homeomorphism, and is of (S + ) type, namely u n u (weakly) and Definition 2.9.We say that the functional J satisfies the Palais-Smale condition ((PS) for short) if every sequence {u n } ∈ E such that contains a convergent subsequence in the norm of E.

Proof of main result
In order to discuss the problem (P), we need to consider the energy functional I : E → R defined by where 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 all u, ϕ ∈ E.
Definition 3.1.We call that u ∈ E \ {0} is a weak solution of (P), if where ϕ ∈ E.
We are now in a position to state our main results.
For more existence and multiplicity results on p(x)-Laplacian equation in R N , we refer to Fan and Han [15].In [15, Theorem 3.2], the similar results are obtained for the case V(x) ≡ 1.
For the proof of Theorem 3.2 we need some preliminary lemmas.The following Bartsch-Wang type compact embedding will play a crucial role in our subsequent arguments.
(ii) for any measurable function q : R N → R with p(x) p − < q(x) p * (x), we have a compact embedding E → → L q(x) (R N ).
Proof.(i).Let {u n } ⊂ E such that u n E ≤ C. We assume up to a subsequence u n u in E.Then, we have ω n E ≤ C and ω n 0 in E, where ω n = u n − u.We need to show ω n → 0 in |x| α > b ∩ B r (y i ).By the Sobolev compact imbedding theorem in bounded domains (see Proposition 2.5), it p − dx, let {y i } i∈N be a sequence of points in R N satisfying such that R N ⊂ ∪ ∞ i=1 B r (y i ) and each point x is contained in at most 2 N such balls B r (y i ).So for all R > 2r, we have On the other hand, choose a number τ ∈ 1, N N−p − arbitrarily, then p(x) < τ p(x) < p * (x).By Proposition 2.4, we have that is continuous, and there are constants C 0 , C 1 , C > 0 such that Applying the Hölder-type inequality, we get sup Setting Now, for any ε > 0 and choose b > 0, we can find a positive constant R > 0 big enough such that This implies p − (R N ) and the proof of Lemma 3.3 (i) is completed.(ii).Since {ω n } is bounded in E and by Proposition 2.4, we get that {ω n } is bounded in L p * (x) (R N ).It is taken into account p(x) p − < q(x) p * (x), we can use the interpolation inequality [21, Corollary 2.2] and obtain where c > 0 is a constant and and Moreover, it follows from Lemma 3.
In the proof, some ideas in [5,6,12] have been followed.Proof of Lemma 3.3 is completed.
Lemma 3.4.Suppose ( f 1 ) holds.Then functional I is coercive and bounded from below.
Proof.Let us assume that there exists a sequence {u n } in E such that From (3.10), we have |I(u n )| ≤ c 0 .Combining this fact with (3.9) implies that for u E large enough.Since m < p − , we obtain 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 in E.
By Lemma 3.3, we obtain the following results: In view of the definition of weak convergence, we have It is clear that |u n − u| q(x) Together (3.12) with (3.13), one deduces from (3.11) that Since J is of (S + ) type (see Proposition 2.8), we obtain u n → u in E. Proof of Lemma 3.5 is completed.
Theorem A ([8]).Let I ∈ C 1 (X, R) be functional satisfying the (PS) condition.Furthermore, let us suppose that (i) I is bounded from below and even; (ii) There is a compact set K ∈ such that γ(A) = k and sup x∈K I(x) < I(0).
Then I possesses at least k pairs of distinct critical points, and their corresponding critical values c k < 0 satisfying c k → 0 as k → ∞ are less than I(0).
Proof of Theorem 3. (2).As I is coercive, by Lemma 3.3 we know that satisfies (PS) condition.By ( f 3 ), I is an even functional.Now, we will show that c k < 0 for every k ∈ N. Because E is a reflexive and separable Banach space for any k ∈ N, we can choose a k-dimensional linear subspace for all υ ∈ S (k) k < 0, we can find t k ∈ (0, 1) and k > 0 such that I(t k υ) ≤ − k for all u ∈ S (k) r k , that is, I(u) ≤ − k , for all u ∈ S (k) t k r k .It is clear that γ S (k) t k r k = k so c k ≤ − k < 0.Moreover, from ( f 3 ), I is even.Finally, by Lemma 3.4, Lemma 3.5 and above results, we can apply Theorem A to obtain that the functional I admits at least k pairs of distinct critical points, and since k is arbitrary, we obtain infinitely many critical points of I.