Mountain pass and Ekeland's principle for eigenvalue problem with variable exponent†

Abstract

In this article, we study the boundary value problem −div(|∇u| p(x)−2∇u) + |u|α(x)−2 u = λ|u| q(x)−2 u, in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ N and p, q, α are continuous functions on . We show that for any λ > 0 there exists infinitely many weak solutions (respectively, if λ > 0 and small enough, then there exists a non-negative, non-trivial 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 (respectively on simple variational arguments based on Ekeland's variational principle). †This article is devoted to the special issue ‘Växjö Conference 2008’.