An improved existence criterion and an optimal result

TL;DR

This paper establishes an existence criterion for a semi-linear elliptic equation with critical nonlinearity on bounded domains, and proves its optimality in specific cases using Pohozaev identities.

Contribution

It introduces a new existence criterion involving parameters and functions, and demonstrates its optimality for certain domain and function configurations.

Findings

Existence criterion involving $or ixed

Optimality of the criterion proved via Pohozaev identity

Results applicable to specific domain and function cases

Abstract

We are concerned with a semi-linear elliptic equation on a smooth bounded domain $\Omega$ of $\mathbb{R}^n,\,n\geq 5,$ which involves a critical nonlinearity and a linear term of the form $K(x)u^{(n+2)/(n-2)}$ and $\mu u,$ respectively. By using a test function procedure, we give an existence criterion involving the parameter $\mu$ and the function $K(x).$ For a particular case of $\Omega,\,K(x)$ and $n,$ we prove its optimality through a Pohozaev type identity.