Nehari method for locally Lipschitz functionals with examples in problems in the space of bounded variation functions

TL;DR

This paper develops abstract results ensuring the existence of minimizers for locally Lipschitz functionals on Nehari-inspired sets, with applications to problems involving the 1-Laplacian and mean-curvature operators in bounded variation spaces.

Contribution

It introduces new existence results for minimizers of non-homogeneous locally Lipschitz functionals on Nehari-type sets, with concrete applications to variational problems in BV spaces.

Findings

Existence of ground state BV solutions for 1-Laplacian problems.

Existence of solutions involving mean-curvature operators.

Applicability to nonlinear problems with mild assumptions.

Abstract

In this work we prove some abstract results about the existence of a minimizer for locally Lipschitz functionals, without any assumption of homogeneity, over a set which has its definition inspired in the Nehari manifold. As applications we present a result of existence of ground state bounded variation solutions of problems involving the 1-Laplacian and the mean-curvature operator, where the nonlinearity satisfies mild assumptions.