Multiple solutions for p-Laplacian type problems with asymptotically p-linear terms via a cohomological index theory

Abstract

The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem \[ \left\{\begin{array}{lr} - \divg (A(x,u)\, |\nabla u|^{p-2}\, \nabla u) + \dfrac1p\, A_t(x,u)\, |\nabla u|^p\ =\ f(x,u) & \hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial \Omega$,} \end{array} \right. \] where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain, $N\ge 2$, $p>1$, $A$ is a given function which admits partial derivative $A_t(x,t)=\frac{\partial A}{\partial t}(x,t)$ and $f$ is asymptotically $p$-linear at infinity. Under suitable hypotheses both at the origin and at infinity, and if $A(x,\cdot )$ is even while $f(x,\cdot )$ is odd, by using variational tools, a cohomological index theory and a related pseudo--index argument, we prove a multiplicity result if $p>N$ in the non--resonant case.