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).