Moderate solutions of semilinear elliptic equations with Hardy potential under minimal restrictions on the potential

TL;DR

This paper investigates semilinear elliptic equations with Hardy potential in arbitrary domains, extending boundary trace concepts and analyzing existence, uniqueness, and properties of solutions for a broad range of parameters.

Contribution

It extends the analysis of boundary value problems for semilinear elliptic equations with Hardy potential to arbitrary domains under minimal restrictions on the potential.

Findings

Existence of solutions depends on two critical exponents.

Normalized boundary trace can be extended to arbitrary domains when <.

Properties of local superharmonic functions are characterized.

Abstract

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and $\delta_\Omega(x)=\mathrm{dist}(x,\partial\Omega)$. Assuming that $0\leq \mu<C_H(\Omega)$, boundary value problems with measure data and discrete boundary singularities for positive solutions of $\mathrm{(E)}$ have been studied earlier. In the present paper we study these problems, in arbitrary domains, assuming only $-\infty<\mu<1/4$, even if $C_H(\Omega)<1/4$. We recall that $C_H(\Omega)\leq 1/4$ and, in general, strict inequality holds. The key to our study is the fact that, if $\mu<1/4$ then in smooth domains there exist local $L_\mu$-superharmonic functions in a neighborhood of $\partial\Omega$ (even if $C_H(\Omega)<1/4$). Using this fact we extend the notion of normalized boundary trace to arbitrary domains, provided that $\mu<1/4$. Further we study the b.v.p. with normalized boundary trace $\nu$ in the space of positive finite measures on $\partial\Omega$. We show that existence depends on two critical values of the exponent $q$ and discuss the question of uniqueness. Part of the paper is devoted to the study of the linear operator: properties of local $L_\mu$ subharmonic and superharmonic functions and the related notion of moderate solutions.