Multiplicity results for some quasilinear elliptic problems

Abstract

In this paper, we study multiplicity of weak solutions for the following class of quasilinear elliptic problems of the form $$ -\Delta_p u -\Delta u = g(u)-\lambda |u|^{q-2}u \quad \text{in } \Omega \text{ with } u=0 \text{ on } \partial\Omega, $$ where $ \Omega $ is a bounded domain in ${\mathbb{R}}^n $ with smooth boundary $\partial\Omega$, $ 1< q< 2< p\leq n$, $\lambda$ is a real parameter, $\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian and the nonlinearity $g(u)$ has subcritical growth. The proofs of our results rely on some linking theorems and critical groups estimates.