Nonhomogeneous Hemivariational Inequalities with Indefinite Potential and Robin Boundary Condition

TL;DR

This paper studies a complex nonlinear Robin boundary problem with an indefinite potential and nonsmooth reaction term, proving the existence of multiple solutions using nonsmooth critical point theory and optimization techniques.

Contribution

It introduces a novel approach to nonhomogeneous hemivariational inequalities with indefinite potentials and nonsmooth reactions, establishing multiple solution existence results.

Findings

Existence of three nontrivial solutions, including a nodal one.

Identification of extremal constant sign solutions.

Application of nonsmooth critical point theory and global optimization methods.

Abstract

We consider a nonlinear, nonhomogeneous Robin problem with an indefinite potential and a nonsmooth primitive in the reaction term. In fact, the right-hand side of the problem (reaction term) is the Clarke subdifferential of a locally Lipschitz integrand. We assume that asymptotically this term is resonant with respect the principal eigenvalue (from the left). We prove the existence of three nontrivial smooth solutions, two of constant sign and the third nodal. We also show the existence of extremal constant sign solutions. The tools come from nonsmooth critical point theory and from global optimization (direct method).