Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities

TL;DR

This paper proves the existence of three positive solutions for a nonlinear Dirichlet problem involving the p-Laplacian with competing power nonlinearities, extending previous results on multiplicity of solutions.

Contribution

It introduces a new approach to establish higher order multiplicity of positive solutions for a perturbed nonlinear Dirichlet problem with concave-convex nonlinearities.

Findings

Existence of three positive solutions established.

Properties of the perturbed energy functional analyzed.

Extension of multiplicity results to p-Laplacian problems.

Abstract

This paper studies a nonlinear Dirichlet problem for the $p$-Laplacian operator with nonlinearity consisting of power components. The problem under consideration can be though of as a perturbation of the Ambrosetti-Brezis-Cerami problem with concave-convex nonlinearity. The combined effect of power components in the perturbed nonlinearity allows to establish a higher order multiplicity of positive solutions. We study properties of the perturbed energy functional and prove the existence of three positive solutions to the problem at hand.