MULTIPLE SOLUTIONS FOR A CLASS OF p(x)-LAPLACIAN PROBLEMS INVOLVING CONCAVE-CONVEX NONLINEARITIES

Using variational methods, we prove a multiplicity result for a class of p(x)- Laplacian problems of the form 8 : −divjr uj p(x)−2 r u ´ = �j uj r(x)−2 u + f(x,u) in ,


Introduction
In this paper, we are interested in the existence of solutions for a class of p(x)-Laplacian problems of the form −div |∇u| p(x)−2 ∇u = g(x, u) in Ω, u = 0, on ∂Ω, where Ω ⊂ R N (N ≥ 3) is a smooth bounded domain, p ∈ C(Ω), 1 < p − ≤ p + < N , and g : Ω × R → R is a continuous function satisfying subcritical growth condition.
In the case when p(x) = p is a constant, problem (1.1) becomes the p-Laplacian problem of the form −∆ p u = g(x, u) in Ω, u = 0, on ∂Ω.
(1.2) Since A. Ambrosetti and P.H. Rabinowitz proposed the mountain pass theorem in 1973 (see [1]), critical point theory has become one of the main tools for finding solutions to elliptic problems of variational type.Especially, elliptic problem (1.2) has been intensively studied for many years.One of the very important hypotheses usually imposed on the nonlinearities is the following Ambrosetti-Rabinowitz type condition ((A-R) type condition for short): There exists µ > p such that 0 < µG(x, t) := µ for all x ∈ Ω and t ∈ R\{0}.This condition ensures that the energy functional associated to the problem satisfies the Palais-Smale condition ((PS) condition for short).Clearly, if the condition (A-R) is satisfied then there exist two positive constants d 1 , d 2 such that This means that g is p-superlinear at infinity in the sense that lim |t|→+∞ G(x, t) |t| p = +∞.
In recent years, there have been many authors considering elliptic problem (1.2) without the (A-R) type condition, we refer to some interesting papers on this topic [11,13,18,19,20,22,23,24,27,28,30,31,32] and the references cited there.In [28], O.H. Using the mountain pass theorem with the (PS) condition in [1], the authors obtained the existence of a non-trivial weak solution.This result was extended to the p-Laplace operator −∆ p u by G. Li et al [23] and to the p(x)-Laplace operator ∆ p(x) u = −div |∇u| p(x)−2 ∇u by C. Ji [19].Especially, in [23], the authors gave a simpler proof for the existence result by using the mountain pass theorem in [13] with the Cerami condition (see Definition 2.3).
Motivated by the papers mentioned above, in this work, we will study the existence of multiple solutions for problem (1.1) in a more general case when g(x, t) is defined by and λ is a positive parameter, the function f : Ω × R → R is continuous and p + -superlinear at infinity but does not satisfy the (A-R) condition (1.3 Using the mountain pass theorem with the Cerami condition in [13] combined with the Ekeland variational principle in [15] we show the existence of at least two non-trivial weak solutions for (1.5) provided that λ ∈ (0, λ * ), λ * > 0 is small enough.In the case when λ = 0, our result is exactly the one introduced in [19] but our arguments in this present work are clearly different from those presented in [19].Regarding some estimates of the constant λ * , we refer the readers to some recent papers [5,6,7,8,9,10,12] in which the authors have studied the existence and multiplicity of weak solutions for elliptic problems involving the p(x)-Laplacian.
We emphasize that the extension from the p-Laplace operator ∆ p u to the p(x)-Laplace operator involved in (1.5) is interesting and not trivial, since the new operators have a more complicated structure than the p-Laplace operator, for example they are non-homogeneous.Finally, it should be noticed that our result is new even in the case when p(x) = p is a constant, see [2,3,4,23,28,33,34].
Our paper is organized as follows.In Section 2, we will recall some useful results on Sobolev spaces with variable exponents and the mountain pass theorem with the Cerami condition.In section 3, we will state and prove the main result of this paper.

Preliminaries
In this section, we recall some definitions and basic properties of the generalized Lebesgue-Sobolev spaces L p(x) (Ω) and W 1,p(x) (Ω) where Ω is an open subset of R N .In that context, we refer to the book of Musielak [29] and the papers of Kováčik and Rákosník [21], Fan et al.
[ 16,17] and the lecture notes by L. Diening et al. [14].Set For any h ∈ C + (Ω) we define For any p(x) ∈ C + (Ω), we define the variable exponent Lebesgue space We recall the following so-called Luxemburg norm on this space defined by the formula Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if 1 < p − ≤ p + < +∞ and continuous functions are dense if p + < +∞.The inclusion between Lebesgue spaces also generalizes naturally: if 0 < |Ω| < +∞ and p 1 , p 2 are variable exponents so that x ∈ Ω then there exists a continuous embedding We denote by L p ′ (x) (Ω) the conjugate space of L p(x) (Ω), where holds true.
An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the L p(x) (Ω) space, which is the mapping ρ p(x) : L p(x) (Ω) → R defined by Proposition 2.1 (see [17]).If u ∈ L p(x) (Ω) and p + < +∞ then the following relations hold (2.1) provided |u| p(x) < 1 and In this paper, we assume that p ∈ C log + (Ω), where C log + (Ω) is the space of all the functions of C + (Ω) which are logarithmic Hölder continuous, that is, there exists R > 0 such that for all x, y ∈ Ω with 0 [14].We define the space W 1,p(x) 0 (Ω) as the closure of C ∞ 0 (Ω) under the norm Proposition 2.2 (see [17]).The space W 1,p(x) 0 (Ω), . is a separable and Banach space when 1 < p − ≤ p + < +∞.Moreover, if q ∈ C + (Ω) and q(x) < p * (x) for all x ∈ Ω then EJQTDE, 2013 No. 26, p. 4 the embedding W 1,p(x) 0 (Ω) ֒→ L q(x) (Ω) is continuous and compact, where In our proof of the main result, we will use the mountain pass theorem with the Cerami condition in [13].For the reader's convenience, we recall it below.Definition 2.3.Let (X, .) be a real Banach space, J ∈ C 1 (X, R).We say that J satisfies the Cerami condition (write (C c ) condition for short) if any sequence {u m } ⊂ X such that Proposition 2.4 (see [13]).Let (X, .) be a real Banach space, J ∈ C 1 (X, R) satisfies the (C c ) condition for any c > 0, J(0) = 0 and the following conditions hold: (i) There exists a function φ ∈ X such that φ > ρ and J(φ) < 0; (ii) There exist two positive constants ρ and R such that J(u) ≥ R for any u ∈ X with Then the functional J has a critical value c ≥ R, i.e. there exists u ∈ X such that J ′ (u) = 0 and J(u) = c.

Multiple solutions
In this section, we state and prove the main result of this paper.We will use the letter C i to denote a positive constant whose value may change from line to line.Let us introduce the following hypotheses: for all x ∈ Ω; (F 1 ) There exists a positive constant t > 0 such that F (x, t) ≥ 0 a.e.x ∈ Ω and all t ∈ [0, t], where F (x, t) t p + = +∞ uniformly in x ∈ Ω, i.e., f is p + -superlinear at infinity; (F 4 ) There exists a constant C * > 0 such that F(x, t) ≤ F(x, s) + C * for any x ∈ Ω and 0 < t < s or s < t < 0, where F(x, t) := tf (x, t) − p + F (x, t).EJQTDE, 2013 No. 26, p. 5 It should be noticed that the condition (F 4 ) is a consequence of the following condition, which was firstly introduced by O.H. Miyagaki et al. [28] for problem (1.2) in the case p = 2 and developed by G. Li et al. [23] and C. Ji [19]: 4 ) There exists t 0 > 0 such that f (x,t) is increasing in t ≥ t 0 and decreasing in t ≤ −t 0 for any x ∈ Ω.
The readers may consult the proof and comments on this assertion in the papers by G. Li et al. [23] or by O.H. Miyagaki et al. [28] and the references cited there.
Our main result of this paper is given by the following theorem.
In the rest of this paper we will use the letter X to denote the Sobolev space W 1,p(x) 0 (Ω).
By the continuous embeddings obtained from the hypotheses (F 0 ) and (1.4), some standard arguments assure that the functional J is well defined on X and J ∈ C 1 (X) with the derivative given by for all u, ϕ ∈ X.Thus, non-trivial weak solutions of problem (1.5) are exactly the non-trivial critical points of the functional J.
Lemma 3.3.The functional J satisfies the (C c ) condition for any c > 0.
Proof.Let {u m } ⊂ X be a (C c ) sequence of the functional J, that is, where o(1) → 0 as m → ∞.
We will prove that the sequence {u m } is bounded in X.Indeed, if {u m } is unbounded in X, we may assume that u m → +∞ as m → ∞.We define the sequence {w m } by w m = um um , m = 1, 2, ... It is clear that {w m } ⊂ X and w m = 1 for any m.Therefore, up to a subsequence, still denoted by {w m }, we have that {w m } converges weakly to some function w ∈ X and Let Using the condition (F 3 ), there exists t 0 > 0 such that for all x ∈ Ω and |t| > t 0 > 0. Since F (x, t) is continuous on Ω×[−t 0 , t 0 ], there exists a positive constant C 1 such that for all (x, t) ∈ Ω × R. From (3.9), for all x ∈ Ω and m, we have We also have and by (3.11), and the Fatou lemma, we have . Since the function t → J(tu m ) is continuous in t ∈ [0, 1], for each m there exists t m ∈ [0, 1] such that Now, we fix a big integer k ≥ 1 so that u k > 1 and define the sequence {v m } by From (F 0 ) and (F 2 ), for any ǫ > 0, there exists a positive constant C(ǫ) such that EJQTDE, 2013 No. 26, p. 9 Fix k, since w m → 0 strongly in the spaces L q(x) (Ω), L r(x) (Ω) and L p + (Ω) as m → ∞, using (3.17), we deduce that there exists a constant C 4 > 0 such that We also have Hence, using relations (3.14), (3.18)-(3.20), it follows that for any m > m k > k large enough.
EJQTDE, 2013 No. 26, p. 10 On the other hand, using the conditions (F 4 ) and relation (3.15), for all m > m k > k large enough, we have where C 3 is given by (3.11).
From (3.21) and (3.22), we deduce that for all m > m k > k large enough, or Now, since the Banach space X is reflexive, there exists u ∈ X such that passing to a subsequence, still denoted by {u m }, it converges weakly to u in X and converges strongly to u in the spaces L q(x) (Ω) and L r(x) (Ω).Using the condition (F 0 ) and the Hölder inequality, we deduce that We also have → 0 as m → ∞. (i) There exists λ * > 0 such that for any λ ∈ (0, λ * ), we can choose R > 0 and ρ > 0 so that J(u) ≥ R > 0 for all u ∈ X with u = ρ; (ii) There exists φ ∈ X, φ > 0 such that J(tφ) → −∞ as t → +∞; (iii) There exists ψ ∈ X, ψ > 0 such that J(tψ) < 0 for all t > 0 small enough.
Proof Theorem 3.2.By Lemmas 3.3 and 3.4, there exists λ * > 0 such that for any λ ∈ (0, λ * ), the functional J satisfies all the assumptions of the mountain pass theorem, see Proposition 2.4.Then we deduce u 1 as a non-trivial critical point of the functional J with J(u 1 ) = c > 0 and thus a non-trivial weak solution of problem (1.5).
We now prove that there exists a second weak solution u 2 ∈ X such that u 2 = u 1 .Indeed, by (3.28), the functional J is bounded from below on the ball B ρ (0).

. 23 )
Recall that k ≥ 1 is an arbitrarily big integer and m > m k > k.In(3.23), let k → ∞ we have m → ∞ and the left hand side of (3.23) tends to +∞ since r + < p − .In the right hand side of (3.23), J(u m ) → c and 1 p + J ′ (u m )(u m ) → 0 as m → ∞.Thus, we have a contradiction.This proves that the sequence {u m } is bounded in X.EJQTDE, 2013 No.26, p. 11

(3. 25 )From ( 3 .
24) and(3.25)and the fact thatlim m→∞ J ′ (u m )(u m − u) = 0 we get lim m→∞ Ω |∇u m | p(x)−2 ∇u m • (∇u m − ∇u) dx = 0. (3.26)Now, using standard arguments we can show that the sequence {u m } converges strongly to u in X and the functional J satisfies the (C c ) condition for any c > 0. The proof of Lemma 3