Existence of Solution for Dirichlet Problem with p ( x ) -Laplacian

: In this paper we study an elliptic equation involving the p ( x ) -laplacian operator, for that equation we prove the existence of a non trivial weak solution. The proof relies on simple variational arguments based on the Mountain-Pass theorem.


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M. Moussaoui and L. Elbouyahyaoui 1.It is known that (H 4 ) is weaker than the condition (AR), moreover we assume the condition on measuring portion of the set Ω.
2. Similar result can be obtained, if we replace in (H 4 ) the assumption s ∈ R by This paper is divided into three sections.In the second section, we introduce some basic properties of the generalized Lebesgue-Sobolev spaces and several important properties of p(x)-Laplace operator.In the third section, we give some existence results of weak solutions of problem (1.1).

Preliminary results
In this section we recall some results on variable exponent Sobolev space, the reader is referred to [2], [6], [3] and the references therein for more details. Set We define on L p(x) the so-called Luxemburg norm by the formula: Variable exponent Lebesgue spaces (L p(x) (Ω), |.| p(x) ) resemble to classical Lebesgue spaces in many respects; they are reflexive and Banach space.On L p(x) (Ω) we also consider the function ϕ p(x) : L p(x) (Ω) → R defined by: We have the equivalence: 5. For a sequence (u n ) ⊂ L p(x) (Ω) and an element u ∈ L p(x) (Ω), the following statements are equivalent: We define the variable Sobolev space and equipp it with the norm Proposition 2.2.(see [2]) 1. W 1,p(x) (Ω) and W 1,p(x) 0 (Ω) are separable reflexive Banach spaces.

Proof:
The embedding E ֒→ L α(x) (Ω) is compact, hence the diagram shows that Nf : E −→ E * is strongly continuous.✷

The main results
Let the functional Φ defined by: Under assymption (H 1 ), the result from proposition 2.4 and proposition 2.6, show that Φ is a C 1 functional on E and It is obvious that u ∈ E is a weak solution for problem (1.1) if and only if Φ ′ (u) = 0.For that we will apply a mountain pass type argument to find nonzero critical point of Φ.Our main result is given by the following theorem.Theorem 3.1.Assume (H 1 ), (H 2 ), (H 3 ) and (H 4 ) hold, then the problem (1.1) has a non trivial weak solution.Definition 3.2.We say that a C 1 functional I : E −→ R satisfies the Palais-Smale condition (P S) if any sequence (u n ) ⊂ E such that (I(u n )) is bounded and and (H 2 ) hold, then the functional Φ : E −→ R satisfies the (PS) condition.
We will show that (u n ) n∈N is bounded in E.
Arguing by contradiction and passing to a subsequence, we have ||u n || → +∞.Using (3.1) it follows that for n large enough, we have So, we obtain The above inequalities combined with (H 2 ) and proposition 2.1, yields: passing to the limit as n → +∞, taking account that, 1 < p − and η < p − , we obtain a contradiction, so (u n ) is bounded, hence, up to a subsequence we may assume that u n ⇀ u.
In other hand Since −∆ p(x) is of type (S + ), we deduce that u n → u, and so Φ satisfies (PS) condition.✷ We will show that Φ satisfies conditions of Mountain Pass lemma.
thus the Carathéodory function f defines an operator N f : M → M , which is called the Nemytskii operator.Proposition 2.5.( [12]) Suppose f : Ω × R → R is a Carathéodory function and satisfies the growth condition |f (x, t)| ≤ c|t| α(x)