Elliptic differential inclusions on non-compact Riemannian manifolds

TL;DR

This paper studies elliptic differential inclusions on non-compact Riemannian manifolds involving the Laplace-Beltrami operator and Hardy-type singularities, establishing conditions for existence, multiplicity, or non-existence of solutions.

Contribution

It introduces new existence and multiplicity results for elliptic differential inclusions on non-compact Riemannian manifolds, utilizing nonsmooth variational analysis and geometric properties.

Findings

Conditions for solution existence depend on nonlinear term behavior and manifold curvature.

Established non-existence results under certain geometric and nonlinear conditions.

Proved multiplicity of solutions using variational methods and eigenvalue analysis.

Abstract

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace-Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions for the studied differential inclusion. The proofs are based on nonsmooth variational analysis as well as isometric actions and fine eigenvalue properties on Riemannian manifolds. The results are also new in the smooth setting.