Isolated Boundary Singularities of Semilinear Elliptic Equations

TL;DR

This paper analyzes the behavior of positive solutions to a semilinear elliptic equation near boundary singularities, establishing bounds and limits for solutions depending on the exponent q, and exploring critical cases.

Contribution

It provides new results on boundary singularities of solutions to semilinear elliptic equations, including precise asymptotic behavior and limits near the boundary point.

Findings

Solutions are bounded by a specific power law near the boundary singularity.

The limit of the scaled solution as x approaches 0 is computed.

The case q = (N+1)/(N-1) is also thoroughly investigated.

Abstract

Given a smooth domain $\Omega\subset\RR^N$ 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.