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

Abstract

In this note, we consider the following problem, \begin{equation*} \begin{cases} -Δu=(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 $N\ge 3$ and $B\subset {\mathbb{R}}^N$ is a unit ball centered at the origin and $g(x)$ is a radial Hölder 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.

typos corrected, references added, acknowledgments added