Critical Elliptic Equations on Non-Compact Finsler Manifolds

Abstract

In the present paper we deal with a quasilinear elliptic equation involvinga critical Sobolev exponent on non-compact Finsler manifolds, i.e. on 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 calculus of variations. One of the key step 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. At the end, we prove Hardy-type inequalities on Finsler manifolds as an application of this inequality.