Nodal and multiple solutions for a nonhomogeneous Neumann boundary problem

TL;DR

This paper proves the existence of multiple solutions, including extremal and nodal solutions, for a nonlinear Neumann boundary problem involving a p-Laplacian-type operator using advanced variational and Morse theory techniques.

Contribution

It introduces new existence results for multiple solutions of a nonhomogeneous Neumann problem with a p-Laplacian-type operator, employing variational, truncation, and Morse theory methods.

Findings

Existence of two extremal constant sign solutions

Existence of a nontrivial nodal solution

Application of variational and Morse theory techniques

Abstract

We consider a nonlinear Neumann problem driven by a $p$-Laplacian-type, nonhomogeneous elliptic differential operator and a Carath\'eodory reaction term. In this paper we prove the existence of two extremal constant sign smooth solutions and a nontrivial nodal smooth solution. In the proof we use variational methods with truncation techniques, critical point theory and Morse theory (critical groups).