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

TL;DR

This paper proves the existence of multiple weak solutions for a class of p-Laplacian type problems with asymptotically p-linear terms, using cohomological index theory under certain symmetry and growth conditions.

Contribution

It introduces a novel application of cohomological index theory to establish multiple solutions for quasilinear elliptic problems with asymptotically p-linear nonlinearities.

Findings

Existence of multiple solutions when p > N in the non-resonant case.

Solutions are obtained under symmetry conditions on A and f.

The approach combines variational methods with 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 $\Omega \subset \R^N$ 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.