A note on radial solutions to the critical Lane-Emden equation with a variable coefficient

TL;DR

This paper investigates the existence and nonexistence of radial solutions to a critical Lane-Emden equation with a variable coefficient in a unit ball, using variational methods, concentration compactness, and Pohozaev identity.

Contribution

It extends the analysis of the critical Lane-Emden equation by considering a variable coefficient, providing new existence and nonexistence results for radial solutions.

Findings

Established conditions for existence of solutions.

Proved nonexistence under certain coefficient conditions.

Applied variational methods and concentration compactness techniques.

Abstract

In this note, we consider the following problem, \begin{equation*} \begin{cases} -\Delta u=(1+g(x))u^{\frac{N+2}{N-2}},\ u>0\text{ in }B,\\ u=0\text{ on }\partial B, \end{cases} \end{equation*} where $N\ge3$ and $B\subset \mathbb{R}^N$ is a unit ball centered at the origin and $g(x)$ is a radial H\"{o}lder continuous function such that $g(0)=0$. We prove the existence and nonexistence of radial solutions by the variational method with the concentration compactness analysis and the Pohozaev identity.