Boundary Trace of Positive Solutions of Semilinear Elliptic Equations in Lipschitz Domains: The Subcritical Case

TL;DR

This paper investigates the boundary behavior of positive solutions to semilinear elliptic equations in Lipschitz domains, establishing existence, uniqueness, and characterization results for solutions with prescribed boundary traces, especially in subcritical cases.

Contribution

It introduces a boundary trace concept using harmonic measure and provides sharp removability and singularity results, extending understanding of solutions in Lipschitz domains with subcritical nonlinearities.

Findings

Defined boundary trace via harmonic measure in Lipschitz domains

Proved sharp removability and singularity characterization for cone domains

Established existence and uniqueness of solutions with arbitrary boundary trace in subcritical case

Abstract

We study the generalized boundary value problem for nonnegative solutions of of $-\Delta u+g(u)=0$ in a bounded Lipschitz domain $\Omega$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Omega$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g(u)=|u|^{q-1}u$, $q>1$. When $\Omega$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $\Omega$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace.