Multiplicity results for some quasilinear elliptic problems

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

DOI: http://dx.doi.org/10.12775/TMNA.2009.030

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 = \dive(|\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.

Keywords


Quasilinear elliptic problems; p-Laplace operator; multiplicity of solutions; critical groups; linking theorems

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