### Multiplicity results for some quasilinear elliptic problems

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.

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

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.