Weak solutions of nonlinear elliptic equations with prescribed singular set (Q1914821)

The author studies the equation \[ - \Delta u= u^p\quad \text{in} \quad \Omega,\quad u> 0\quad \text{in} \quad \Omega,\quad u= 0\quad \text{on} \quad \partial\Omega.\tag{1} \] Here \(\Omega\) is a domain in \(\mathbb{R}^n\), \(n> 3\), with smooth boundary \(\partial\Omega\). Given any \(n-m\) submanifold \(\Sigma\) of \(\Omega\) without boundary, with \(n> m> 2\) and \(p>m/(m-2)\), the author shows the existence of at least one positive weak solution to the equation (1) which is singular on \(\Sigma\). The method is a mixture of arguments used by both R. Mazzeo and N. Smale and by F. Pacard and makes use of the Yamabe problem. The 3 steps of the proof are the following: First, the author looks for radial solutions of the problem \(- \Delta u= u^p\) in \(\mathbb{R}^m\). Then he constructs an approximate solution. Finally, he studies the nonlinear problem and by use of the Schauder fixed point theorem he gets a solution to (1) as a perturbation of the approximate solution. This result extends earlier results by Chiun-Chuan Chen and Chang-Shou Lin and F. Pacard.