The precise boundary trace of solutions of a class of supercritical nonlinear equations

TL;DR

This paper investigates the detailed boundary behavior of positive solutions to a supercritical nonlinear PDE, providing a rigorous construction and analysis of their boundary trace in smooth bounded domains.

Contribution

It introduces a precise boundary trace concept for solutions of the supercritical nonlinear PDE and analyzes its properties, extending understanding in this challenging regime.

Findings

Constructed the boundary trace for solutions in the supercritical case

Established properties and uniqueness of the boundary trace

Extended boundary behavior analysis to supercritical nonlinear equations

Abstract

We construct and study the properties of the precise boundary trace of positive solutions of $-\Delta u+u^q=0$ in a smooth bounded domain of $\mathbb R^N$, in the supercritical case $q\geq q_c=(N+1)/(N-1)$