Moderate solutions of semilinear elliptic equations with Hardy potential

TL;DR

This paper classifies positive moderate solutions of a semilinear elliptic equation with Hardy potential in bounded domains, introducing a normalized boundary trace to handle boundary behavior in the subcritical case.

Contribution

It introduces a normalized boundary trace for solutions and provides a complete classification of positive moderate solutions in the subcritical regime.

Findings

Complete classification of moderate solutions using normalized boundary trace

Existence of no moderate solutions with isolated boundary singularities for supercritical exponents

Representation of harmonic functions via Martin boundary in Hardy potential setting

Abstract

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$. We study positive solutions of equation (E) $-L_\mu u+ u^q = 0$ in $\Omega$ where $L_\mu=\Delta + \frac{\mu}{\delta^2}$, $0<\mu$, $q>1$ and $\delta(x)=\mathrm{dist}\,(x,\partial\Omega)$. A positive solution of (E) is moderate if it is dominated by an $L_\mu$-harmonic function. If $\mu<C_H(\Omega)$ (the Hardy constant for $\Omega$) every positive $L_\mu$- harmonic functions can be represented in terms of a finite measure on $\partial\Omega$ via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, $1<q<q_{\mu,c}$. (The critical value depends only on $N$ and $\mu$.) For $q\geq q_{\mu,c}$ there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator $L_\mu$. These results form the basis for the study of the nonlinear problem.