Maximal solutions of equation u = uq in arbitrary domains

Abstract

We prove bilateral capacitary estimates for the maximal solution $U_F$ of $-\Delta u+u^q=0$ in the complement of an arbitrary closed set $F\subset {\mathbb{R}}^N$, involving the Bessel capacity $C_{2,q^{\prime }}$, for $q$ in the supercritical range $q\ge q_c:=N/(N-2)$. We derive a pointwise necessary and sufficient condition, via a Wiener type criterion, in order that $U_F(x)\to \infty$ as $x\to y$ for given $y\in\prt F$. Finally we prove a general uniqueness result for large solutions.

References in corpus (1)

• Capacitary representations of positive solutions of semilinear parabolic equations