On a singular elliptic equation with a sign-changing nonlinearity

Abstract

Let $\Omega$ be an open subset of ${ℝ}^n$ with $n\ge 3$ such that $0\in \Omega$. For $s\in \left(0,2\right)$ and $q>1$ fixed, we consider a positive function $u\in C^{\infty }\left(\Omega \backslash \left\{0\right\}\right)$ such that

where $2^{\star }\left(s\right):=\frac{2\left(n-s\right)}{n-2}$ is critical from the viewpoint of the Hardy–Sobolev embeddings. In this talk, I will present recent results on the classification of the behaviour near zero for the positive solutions of (1). This is joint work with Frédéric Robert (University of Lorraine – Nancy).