On a class of nonlinear problems involving a p(x)-Laplace type operator

Abstract

We study the boundary value problem $- div((\left| {\nabla u} \right|^{p_1 (x) - 2} + \left| {\nabla u} \right|^{p_2 (x) - 2} )\nabla u) = f(x,u)$ in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ^N. Our attention is focused on two cases when $f(x,u) = \pm ( - \lambda \left| u \right|^{m(x) - 2} u + \left| u \right|^{q(x) - 2} u)$, where m(x) = max{p ₁(x), p ₂(x)} for any x ∈ $\bar \Omega$ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ $\bar \Omega$. In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ₂-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.