Isolated Boundary Singularities of Semilinear Elliptic Equations

Abstract

Given a smooth domain $Ω\subset\RR^N$ such that $0\in \partial \Omega$ and given a nonnegative smooth function $\zeta$ on $\partial \Omega$, we study the behavior near 0 of positive solutions of $-\Delta u=u^q$ in $\Omega$ such that $u=\zeta$ on $\partial \Omega \setminus \{0\}$. We prove that if $\frac{N+1}{N-1}<q<\frac{N+2}{N-2}$, then $u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$ and we compute the limit of $\abs{x}^{\frac{2}{q-1}} u(x)$ as $x\to 0$. We also investigate the case $q=\frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.