Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition

Abstract

We discuss the problem – div(a(x, ∇u)) = m(x)|u| r(x)−2 u + n(x)|u| s(x)−2 u in Ω, where Ω is a bounded domain with smooth boundary in ℝ N (N ≥ 2), and div(a(x, ∇u)) is a p(x)-Laplace type operator with 1 < r(x) < p(x) < s(x). We show the existence of infinitely many nontrivial weak solutions in . Our approach relies on the theory of the variable exponent Lebesgue and Sobolev spaces combined with adequate variational methods and a variation of the Mountain Pass lemma and critical point theory.