Semilinear elliptic equations involving power nonlinearities and hardy potentials with boundary singularities

TL;DR

This paper investigates boundary value problems for semilinear elliptic equations with Hardy potentials and power nonlinearities, analyzing existence and conditions in supercritical and critical cases with boundary singularities.

Contribution

It introduces new conditions based on capacities for existence of solutions, handling supercritical exponents and critical Hardy parameters, which were not addressed in prior work.

Findings

Established necessary and sufficient capacity conditions for solutions.

Analyzed supercritical nonlinear ranges and critical Hardy potential cases.

Differentiated approaches for absorption and source nonlinearities.

Abstract

Let $\Omega \subset\mathbb{R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \partial\Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma=\partial\Omega$ if $k=N-1$. Denote $d_\Sigma(x)=\mathrm{dist}(x,\Sigma)$ and put $L_\mu=\Delta + \mu d_{\Sigma}^{-2}$ where $\mu$ is a parameter. In this paper, we study boundary value problems for equations $-L_\mu u \pm |u|^{p-1}u = 0$ in $\Omega$ with prescribed condition $u=\nu$ on $\partial \Omega$, where $p>1$ and $\nu$ is a given measure on $\partial \Omega$. The nonlinearity $|u|^{p-1}u$ is referred to as \textit{absorption} or \textit{source} depending whether the plus sign or minus sign appears. The distinctive feature of the problems is characterized by the interplay between the concentration of $\Sigma$, the type of nonlinearity, the exponent $p$ and the parameter $\mu$. The absorption case and the source case are sharply different in several aspects and hence require completely different approaches. In each case, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities. In comparison with related works in the literature, by employing a fine analysis, we are able to treat the supercritical ranges for the exponent $p$, and the critical case for the parameter $\mu$, which justifies the novelty of our paper.