Multiplicity results for some quasilinear elliptic problems. (English) Zbl 1183.35107

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

Topol. Methods Nonlinear Anal. 34, No. 1, 77-89 (2009).

Summary: We study multiplicity of weak solutions for the following class of quasilinear elliptic problems of the form
\[ -\Delta_p u -\Delta u= g(u)-\lambda|u|^{q-2}u \quad \text{in } \Omega \quad\text{with } u=0 \quad\text{on } \partial\Omega, \]
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, \(\Delta_pu= \text{div}(|\nabla u|^{p-2}\nabla u)\) 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.

MSC:

35J25	Boundary value problems for second-order elliptic equations
35J92	Quasilinear elliptic equations with \(p\)-Laplacian
35J20	Variational methods for second-order elliptic equations

Keywords:

quasilinear elliptic problems; \(p\)-Laplace operator; multiplicity of solutions; critical groups; linking theorems