Exact singular behavior of positive solutions to nonlinear elliptic equations with a Hardy potential
The authors study the singular behaviour at \(x=0\) of positive solutions to the problem \[ -\Delta u=\frac{\lambda}{|x|^2}u-|x|^\sigma u^p,\;\;x\in \Omega\setminus \{0\}, \] where \(\Omega\) is a bounded smooth domain of \(\mathbb{R}^N\) containing the origin, and \(p>1\), \(\sigma >-2\) are constants. If \(\lambda \leq (N-2)^2/4\), the behaviour at the origin of all positive solutions has recently been classified, and includes subcases where \(u\) converges and blows-up as \(x\to 0\). In the paper under review, it is shown that if \(\lambda > (N-2)^2/4\) then all positive solutions blow-up at the origin with the same rate, which is given explicitly.
scientific article; zbMATH DE number 6678280