A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis

Abstract

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 $\Omega \subset {\Bbb R}^N$, 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 $\Omega \subset {\Bbb R}^n$ and which are singular along arbitrary smooth $k$-dimensional submanifolds in the interior of these domains provided $p$ lie 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 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.