Nodal solutions for the Robin $p$-Laplacian plus an indefinite potential and a general reaction term

TL;DR

This paper proves the existence of multiple nodal solutions for a nonlinear Robin p-Laplacian problem with an indefinite potential and arbitrary growth reaction term, using advanced critical point and perturbation methods.

Contribution

It introduces new techniques to establish multiple nodal solutions for p-Laplacian problems with indefinite potentials and minimal growth conditions on the reaction term.

Findings

Existence of infinitely many nodal solutions converging to zero.

Solutions are smooth and obtained via critical point theory.

Applicable to problems with arbitrary reaction growth and indefinite potentials.

Abstract

We consider a nonlinear Robin problem driven by the $p$-Laplacian plus an indefinite potential. The reaction term is of arbitrary growth and only conditions near zero are imposed. Using critical point theory together with suitable truncation and perturbation techniques and comparison principles, we show that the problem admits a sequence of distinct smooth nodal solutions converging to zero in $C^1(\overline{\Omega})$.