zbMATH — the first resource for mathematics

A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis. (English) Zbl 0869.35040
Authors’ summary: The aim of this paper is to prove the existence of weak solutions to the equation \(\Delta u+u^p=0\) which are positive in a domain \(\Omega\subset\mathbb{R}^N\), vanish at the boundary, and have prescribed isolated singularities. The exponent \(p\) is required to lie in the interval \((N/(N-2),(N+2)/(N-2))\). We also prove the existence of solutions to the equation \(\Delta u+ u^p=0\) which are positive in a domain \(\Omega\subset\mathbb{R}^n\) and which are singular along arbitrary smooth \(k\)-dimensional submanifolds in the interior of these domains, provided \(p\) lies in the interval \(((n-k)/(n-k-2),(n-k+2)/(n-k-2))\). A particular case is when \(p=(n+2)/(n-2)\), in which case solutions correspond to solutions of the singular Yamabe problem. The method used here is a mixture of different ingredients used by both authors in their separate constructions of solutions to the singular Yamabe problem, along with a new set of scaling techniques.

35J65 Nonlinear boundary value problems for linear elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
Full Text: DOI