The solvability of Brezis-Nirenberg type problems of singular quasilinear elliptic equation

TL;DR

This paper investigates the existence and non-existence of solutions for a Brezis-Nirenberg type singular quasilinear elliptic problem, extending classical theorems and establishing new conditions for solution existence.

Contribution

It introduces a generalized compact embedding theorem, derives a Pohozaev identity, and provides new existence and non-existence results for the problem.

Findings

Extended Rellich-Kondrachov compact embedding theorem

Non-existence results via Pohozaev identity

Existence results based on extremal functions

Abstract

In this paper, we consider the existence and non-existence of non-trivial solution to a Brezis-Nirenberg type problem with singular weights. First, we obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a non-existence result. Thirdly, based on a generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)$_c$ condition. Finally, based on the explicit form of the extremal function, we will obtain some existence results to the problem.