Isolated Boundary Singularities of Semilinear Elliptic Equations

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