Parametric nonlinear resonant Robin problems

TL;DR

This paper investigates a nonlinear Robin boundary value problem involving the p-Laplacian with competing nonlinearities, demonstrating the existence of multiple solutions for large parameter values using variational methods.

Contribution

It introduces a novel analysis of a p-Laplacian Robin problem with parametric and resonant nonlinearities, establishing multiple solutions via critical point theory.

Findings

Existence of at least five nontrivial solutions for large parameters

Application of critical groups in nonlinear resonance problems

Analysis of competing sublinear and linear nonlinearities

Abstract

We consider a nonlinear Robin problem driven by the $p$-Laplacian. In the reaction we have the competing effects of two nonlinearities. One term is parametric, strictly $(p-1)$-sublinear and the other one is $(p-1)$-linear and resonant at any nonprincipal variational eigenvalue. Using variational tools from the critical theory (critical groups), we show that for all large enough values of parameter $\lambda$ the problem has at least five nontrivial smooth solutions.