Maximal solutions of equation u = uq in arbitrary domains

TL;DR

This paper establishes capacitary estimates and a Wiener criterion for the behavior of maximal solutions to a nonlinear PDE in arbitrary domains, and proves a uniqueness result for large solutions.

Contribution

It introduces bilateral capacitary estimates and a Wiener criterion for maximal solutions of a nonlinear PDE in arbitrary domains, extending understanding of their boundary behavior.

Findings

Capacitary estimates involving Bessel capacity for solutions

A Wiener type criterion for boundary blow-up behavior

A general uniqueness theorem for large solutions

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 $C_{2,q'}$, for $q$ in the supercritical range $q\geq q_{c}:=N/(N-2)$. 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.