A multiplicity theorem for anisotropic Robin equations

TL;DR

This paper proves the existence of at least five distinct smooth solutions, including positive, negative, and nodal solutions, for an anisotropic Robin problem involving the p(x)-Laplacian with superlinear reaction.

Contribution

It introduces a novel multiplicity theorem for anisotropic Robin problems driven by the p(x)-Laplacian, using variational methods and critical group analysis.

Findings

At least five nontrivial solutions exist for the problem.

Solutions include two positive, two negative, and one nodal solution.

Solutions are ordered and have sign information.

Abstract

In this paper we consider an anisotropic Robin problem driven by the $p(x)$-Laplacian and a superlinear reaction. Applying variational tools along with truncation and comparison techniques as well as critical groups, we prove that the problem has at least five nontrivial smooth solutions to be ordered and with sign information: two positive, two negative and the fifth nodal.