Topological Methods in Nonlinear Analysis

Multiplicity results for some quasilinear elliptic problems

Francisco Odair de Paiva, João Marcos do Ó, and Everaldo Souto de Medeiros

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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.

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Topol. Methods Nonlinear Anal., Volume 34, Number 1 (2009), 77-89.

First available in Project Euclid: 27 April 2016

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de Paiva, Francisco Odair; do Ó, João Marcos; de Medeiros, Everaldo Souto. Multiplicity results for some quasilinear elliptic problems. Topol. Methods Nonlinear Anal. 34 (2009), no. 1, 77--89.

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