Weak solutions to Dirichlet boundary value problem driven by p ( x )-Laplacian-like operator

We prove the existence of weak solutions to the Dirichlet boundary value problem for equations involving the p(x)-Laplacian-like operator in the principal part, with reaction term satisfying a sub-critical growth condition. We establish the existence of at least one nontrivial weak solution and three weak solutions, by using variational methods and critical point theory.


Introduction
In this article we consider the following Dirichlet boundary value problem: The function g : Ω × R → R is Carathéodory (that is, for all z ∈ R, x → g(x, z) is measurable and for a.a.x ∈ Ω, z → g(x, z) is continuous) and λ is a real positive parameter.In the sequel of this article, we assume that the reaction term g(x, z) satisfies the hypothesis: Email: calogero.vetro@unipa.it(g 1 ) there exist a 1 , a 2 ∈ [0, +∞[ and α ∈ C(Ω) with 1 < α(x) < p * (x) for all x ∈ Ω, such that |g(x, z)| ≤ a 1 + a 2 |z| α(x)−1 for all (x, z) ∈ Ω × R, where p * (x) = np(x) n − p(x) if p(x) < n and p * (x) = +∞ if p(x) ≥ n.
Here, we prove the existence of weak solutions to the Dirichlet boundary value problem (P λ ), by using variational methods and critical point theory.Precisely, we apply a result of Bonanno [2] for functionals satisfying the Palais-Smale condition cut off upper at r (the (PS) [r] -condition for short), to obtain the existence of at least one nontrivial weak solution.Then, we use a result of Bonanno-Marano [4] to obtain the existence of three weak solutions.The motivation of this study comes from the use of such problems to model the behaviour of electrorheological fluids in physics (as discussed in Diening-Harjulehto-Hästö-R ůžička [8]) and, in particular, the phenomenon of capillarity which depends on solid and liquid interfacial properties such as surface tension, contact angle, and solid surface geometry.

Mathematical background
Let X be a real Banach space and X * its topological dual.In developing our study, we consider both the variable exponent Lebesgue space L p(x) (Ω) and the generalized Lebesgue-Sobolev space W 1,p(x) (Ω).Indeed, these spaces, in respect to the norms defined below, are separable, reflexive and uniformly convex Banach spaces (see Fan-Zhang [9]).So, we have the variable exponent Lebesgue space L p(x) (Ω) given as where we consider the following norm dx ≤ 1 (i.e., Luxemburg norm).
On the other hand, the generalized Lebesgue-Sobolev space W 1,p(x) (Ω) is defined by Also, we take the norm which is equivalent to the norm (see D'Aguì-Sciammetta [6]).In the following, we will use the norm u instead of u W 1,p(x) (Ω) on W 1,p(x) 0 (Ω).In the proofs of our theorems, we use a Sobolev embedding result; precisely we refer to the following proposition due to Fan-Zhao [10].
* is a strictly monotone and bounded homeomorphism.
Finally, consider the functional We conclude this section with the following notion.
Definition 2.4.Let X be a real Banach space and X * its topological dual.Then, I λ : X → R satisfies the Palais-Smale condition cut off upper at r, with fixed r ∈ ] − ∞, +∞], if any sequence {u n } such that has a convergent subsequence.

Existence of one weak solution
In this section we establish an existence theorem producing at least one nontrivial weak solution of (P λ ).To this aim, we apply a theorem proved by Bonanno [2, Theorem 2.3], which reads as follows.

Existence of three weak solutions
In this section we prove a theorem producing at least three weak solutions of (P λ ).To this aim, we apply a theorem proved by Bonanno-Marano [4, Theorem 3.6], which run as follows.
The hypotheses on the function G : Ω × R → R are as follows: for all (x, t) ∈ Ω × R; So, we establish the following result.