Critical elliptic equations on non-compact Finsler manifolds

TL;DR

This paper investigates critical elliptic equations on non-compact Finsler manifolds, establishing the existence of solutions using variational methods and new inequalities, advancing understanding of nonlinear PDEs in geometric analysis.

Contribution

It introduces a novel approach to prove solution existence for critical elliptic equations on non-compact Finsler manifolds under general perturbations.

Findings

Existence of non-trivial solutions established

Energy functional shown to be weakly lower semicontinuous

New inequalities facilitate analysis on Randers spaces

Abstract

In the present paper, we deal with a quasilinear elliptic equation involving a critical Sobolev exponent on non-compact Randers spaces. Under very general assumptions on the perturbation, we prove the existence of a non-trivial solution. The approach is based on the direct methods of the calculus of variations. One of the key steps is to prove that the energy functional associated with the problem is weakly lower semicontinuous on small balls of the Sobolev space, which is provided by a general inequality.