Three solutions for a Neumann partial differential inclusion via nonsmooth Morse theory

TL;DR

This paper investigates a Neumann boundary value problem involving a p-Laplacian and a nonsmooth potential, establishing the existence of multiple solutions using nonsmooth Morse theory.

Contribution

It introduces a novel application of nonsmooth Morse theory to find multiple solutions for a p-Laplacian partial differential inclusion with nonsmooth potential.

Findings

Existence of at least two constant sign solutions (positive and negative)

Existence of a third non-zero solution

Application of nonsmooth Morse relation to PDE inclusion

Abstract

We study a partial differential inclusion, driven by the p-Laplacian operator, involving a p-superlinear nonsmooth potential, and subject to Neumann boundary conditions. By means of nonsmooth critical point theory, we prove the existence of at least two constant sign solutions (one positive, the other negative). Then, by applying the nonsmooth Morse relation, we find a third non-zero solution.