A Dirichlet inclusion problem on Finsler manifolds

TL;DR

This paper investigates a Dirichlet-type differential inclusion involving the Finsler-Laplace operator on Finsler manifolds, establishing existence, non-existence, and multiplicity results using non-smooth variational methods.

Contribution

It introduces new existence and multiplicity results for nonlinear Finslerian differential inclusions using variational techniques in a nonsmooth setting.

Findings

Existence of solutions for certain parameter ranges.

Non-existence results for other parameter values.

Multiplicity of solutions under specific conditions.

Abstract

In this paper we study a Dirichlet-type differential inclusion involving the Finsler-Laplace operator on a complete Finsler manifold. Depending on the positive $\lambda$ parameter of the inclusion, we establish non-existence, as well as existence and multiplicity results by applying non-smooth variational methods. The main difficulties are given by the problem's highly nonlinear nature due to the general Finslerian setting, as well as the nonsmooth context.