Nonexistence of variational minimizers related to a quasilinear singular problem in metric measure spaces

TL;DR

This paper proves the nonexistence of variational minimizers for a quasilinear singular problem in metric measure spaces, using a purely variational approach, marking a novel contribution in the general metric setting.

Contribution

It introduces the first nonexistence result for singular problems in a broad metric measure space framework using variational methods.

Findings

No variational minimizers exist for the problem in the specified setting.

The approach is purely variational, without relying on additional assumptions.

This work extends the understanding of singular problems to general metric measure spaces.

Abstract

In this article we consider a variational problem related to a quasilinear singular problem and obtain a nonexistence result in a metric measure space with a doubling measure and a Poincar\'e inequality. Our method is purely variational and to the best of our knowledge, this is the first work concerning singular problems in a general metric setting.