Quasilinear Problems with the Competition Between Convex and Concave Nonlinearities and Variable Potentials

TL;DR

This paper investigates the existence and non-existence of solutions for nonlinear elliptic equations with competing convex and concave nonlinearities and variable potentials, using variational and sub-super solution methods.

Contribution

It extends previous results by analyzing the effects of variable potentials on solutions to quasilinear elliptic problems with mixed nonlinearities.

Findings

Existence of solutions under certain conditions on parameters.

Non-existence results when parameters do not satisfy specific criteria.

Behavior of solutions influenced by the interaction between nonlinearities and potentials.

Abstract

The purpose of this paper is to prove some existence and non-existence theorems for the nonlinear elliptic problems of the form -{\Delta}_{p}u={\lambda}k(x)u^{q}\pmh(x)u^{{\sigma}} if x\in{\Omega}, subject to the Dirichlet conditions u_{1}=u_{2}=0 on \partial{\Omega}. In the proofs of our results we use the sub-super solutions method and variational arguments. Related results as obtained here have been established in [Z. Guo and Z. Zhang, W^{1,p} versus C^{1} local minimizers and multiplicity results for quasilinear elliptic equations, Journal of Mathematical Analysis and Applications, Volume 286, Issue 1, Pages 32-50, 1 October 2003.] for the case k(x)=h(x)=1. Our results reveal some interesting behavior of the solutions due to the interaction between convex-concave nonlinearities and variable potentials.