Existence of solutions for p(x)−Laplacian equations 1

We discuss the problem ( −div � |∇u| p(x)−2 ∇u � = λ(a(x) |u| q(x)−2 u + b(x) |u| h(x)−2 u), for x ∈ , u = 0, for x ∈ ∂. where is a bounded domain with smooth boundary in R N (N ≥ 2) and p is Lipschitz continuous, q and h are continuous functions on such that 1 < q(x) < p(x) < h(x) < p ∗ (x) and p(x) < N. We show the existence of at least one nontrivial weak solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem.


Introduction
The study of partial differential equations and variational problems involving p(x)-growth conditions has captured special attention in the last decades.This is a consequence of the fact that such equations can be used to model phenomena which arise in mathematical physics, for example: • Electrorheological fluids: see Acebri and Mingione [1], Zhikov [25] and Růžička [20], Fan and Zhang [12], Mihȃilescu and Rȃdulescu [16], Chabrowski and Fu [7], Hästö [14], Diening [8].
A typical model of an elliptic equation with p(x)-growth conditions is The operator − div |∇u| p(x)−2 ∇u is called the p(x)-Laplace operator and it is a natural generalization of the p−Laplace operator, in which p(x) ≡ p > 1 is a constant.The p(x)-Laplacian processes have more complicated nonlinearity, for example, it is nonhomogeneous, so in the discussions some special techniques will be needed.Problems like (1.1) with Dirichlet boundary condition have been largely considered in the literature in the recent years.We give in what follows a concise but complete image of the actual stage of research on this topic.We will use the notations such as p 1 and p 2 where In the case f (x, u) = λ |u| p(x)−2 u in [13] the authors established the existence of infinitely many eigenvalues for problem (1.1) by using an argument based on the Ljusternik-Schnirelmann critical point theory.Denoting by Λ the set of all nonnegative eigenvalues, they showed that Λ is discrete, sup Λ = ∞ and pointed out inf Λ = 0 for general p(x), and only under some special conditions inf Λ > 0. In the case f (x, u) = λ |u| q(x)−2 u, there are different papers, for example, in [12] the same authors proved that any λ > 0 is an eigenvalue of problem (1.1) when p 2 < q 1 and also when q 2 < p 1 .In [18] the authors proved the existence of a continuous family of eigenvalues which lies in a neighborhood of the origin when q 1 < p 1 and q(x) has subcritical growth in problem (1.1).
In the case [17] Mihȃilescu show that there exists λ > 0 such that, for any A, B ∈ (0, λ), problem (1.1) has at least two distinct nontrivial weak solutions.
The aim of this paper is to discuss the existence of a weak solution of the p (x)-Laplacian equation where Ω ⊂ R N (N ≥ 2) is a bounded domain with smooth boundary, λ is a positive real number, p is Lipschitz continuous on Ω, and In the present paper, assuming the condition and using the the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem, we show the existence of at least one nontrivial weak solution of problem (P λ ).

2.Preliminaries
We recall in what follows some definitions and basic properties of variable exponent Lebesgue and Sobolev spaces L p(x) (Ω), W 1,p(x) (Ω) and W 1,p(x) 0 (Ω).In that context we refer to [9,10,15] for the fundamental properties of these spaces.Set EJQTDE, 2010 No. 65, p. 2 For p ∈ L ∞ + (Ω) , we define the variable exponent Lebesgue space L p(.) (Ω) to consist of all measurable functions u : Ω → R for which the modular is finite.We define the Luxembourg norm on this space by the formula Equipped with this norm, L p(.) (Ω) is a separable and reflexive Banach space.Define the variable exponent Sobolev space W 1,p(x) (Ω) by and the norm makes W 1,p(x) (Ω) a separable and reflexive Banach space.The space W 1,p(x) 0 (Ω) is a separable and reflexive Banach space.Proposition 2.1 [10,15] The conjugate space of L p(x) (Ω) is L q(x) (Ω), where 1 p(x) + 1 q(x) = 1.For any u ∈ L p(x) (Ω) and υ ∈ L q(x) (Ω), we have The next proposition illuminates the close relation between the • p(x) and the convex modular . Given two Banach spaces X and Y , the symbol X ֒→ Y means that X is continuously imbedded in Y and the symbol X ֒→֒→ Y means that there is a compact embedding of X in Y .
, and also there is a constant Consequently, u := |∇u| p(x) and u 1,p(x) are equivalent norms on W (Ω), with p ∈ C + (Ω), will be considered as endowed with the norm u 1,p(x) .We will (Ω) in the following discussions.
Finally, we introduce Mountain-Pass Theorem which is the main tool of the present paper.
Palais-Smale condition [24] Let E be a Banach space and where M is a positive constant and E * is the dual space of E, then {u n } possesses a convergent subsequence.
Mountain-Pass Theorem [24] Let E be a Banach space, and let I ∈ C 1 (E, R) satisfy the Palais-Smale condition.Assume that I (0) = 0, and there exists a positive real number ρ and u, υ ∈ E such that EJQTDE, 2010 No. 65, p. 4 Then, β ≥ α and β is a critical value of I.

Main Results
The energy functional corresponding to problem (P λ ) is defined as We say that u ∈ W 1,p(x) 0 (Ω) is a weak solution for problem (P λ ) provided for all υ ∈ W 1,p(x) 0 (Ω) .

Standard arguments imply that
(Ω) , R with (Ω).Thus the weak solution of (P λ ) are exactly the critical points of J λ .
The main result of the present paper is the following theorem.
To obtain the proof of Theorem 3.1, we use Mountain-Pass theorem.Therefore, we must show J λ satisfies Palais-Smale condition in the first place.(Ω) is a sequence which satisfies conditions where M is a positive constant, then {u n } possesses a convergent subsequence.
Proof: First, we show that {u n } is bounded in W 1,p(x) 0 (Ω).Assume the contrary.Then, passing to a subsequence if necessary, we may assume that u n → ∞ as n → ∞.Thus, we may consider EJQTDE, 2010 No. 65, p. 5 that u n > 1 for any integer n.By (3.4) we deduce that there exists N 1 > 0 such that for any n > N 1 , we have On the other hand, for any n > N 1 fixed, the application , υ is linear and continuous.The above information implies Setting υ = u n we have Using the assumption u n > 1, relations (3.3) , (3.4), Proposition 2.1, Lemma 2.4 and Proposition 2.6 (ii) we have where C 3 > 0 is a constant independent of u n and x, for n large enough.Dividing (3.5) by u n p1 and passing to the limit as n → ∞ we obtain (Ω).Since {u n } is bounded, up to a subsequence (which we still denote by {u n }), we may assume that there exists u ∈ W 1,p(x) 0 (Ω) such that EJQTDE, 2010 No. 65, p. 6 By Proposition 2.6 (iii) we obtain Furthermore, from [3,23] we have where K is compact subset of Ω.The above information and relation (3.4) imply On the other hand, we have Propositions 2.1, 2.3 and Lemma 2.4 we have where C 4 , C 5 > 0 and 1 β(x) + q(x)−1 p(x) + 1 p(x) = 1.Similarly, Propositions 2.1, 2.3 and Lemma 2.4 we have where C 6 , C 7 > 0 and Therefore, from above inequalities we deduce that and lim respectively.By (3.8) and (3.9) we obtain This result and the following inequality This fact and Proposition 2.3 imply ∇u n − ∇u p(x) → 0 as n → ∞.Relation (3.12) and fact that (Ω) enable us to apply [12] in order to obtain that u n ⇀ u (strongly) in W 1,p(x) 0 (Ω).Thus, Lemma 3.2 is proved.Now, we show that the Mountain-Pass theorem can be applied in this case.Lemma 3.3 Assume p, q, h ∈ C + (Ω) and condition (1.2) is fulfilled.The following assertions hold.
According to Lemma 3.3 (ii) and (iii), we know that υ > ρ, so every path g ∈ G intersects the sphere v = ρ.Then Lemma 3.3 (i) implies with the constant α > 0 in Lemma 3.3 (i), thus β > 0. By the Mountain-Pass theorem J λ admits a critical value β ≥ α.

≤
λa 2 q 1 a 2 h 2 b 1 q 1 q 2 h 1 −q 2 + a 2 h 2 b 1 q 1 q 1 h 2 −q 1 := A (3.17 where A is a positive constant independent of u and x, by using the following elementary inequality (Ω) with u > 1.We infer that J λ (u) → ∞ as u → ∞.Therefore the energy functional I λ is coercive on W 1,p(x) 0 (Ω).Moreover, a similar argument as the one used in the proof of [16,Lemma 3.4] shows that I λ is also weakly lower semi-continuous in W 1,p(x) 0 (Ω).These facts enable us to apply [22,Theorem 1.2] in order to find that there exits u λ ∈ W 1,p(x) 0 (Ω) a global minimizer of I λ and thus, a weak solution of problem (P / λ ).