Positive solutions of semilinear elliptic problems with a Hardy potential

TL;DR

This paper investigates positive solutions to a semilinear elliptic equation with Hardy potential, revealing unique boundary behaviors and solution structures in both radial and general domains, especially for nonlinearities with p>1.

Contribution

It provides a complete characterization of radial solutions in balls and extends results to general domains, establishing existence and uniqueness of boundary singular solutions with prescribed boundary behavior.

Findings

Existence of a unique large solution for p>1 with specific boundary behavior.

Extension of boundary singular solution results to general domains.

Complete description of radial solutions in balls.

Abstract

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and $\delta(x)$ be the distance of a point $x\in \Omega$ to the boundary. We study the positive solutions of the problem $\Delta u +\frac{\mu}{\delta(x)^2}u=u^p$ in $\Omega$, where $p>0, \,p\ne 1$ and $\mu \in \mathbb{R},\,\mu\ne 0$ is smaller then the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper we first give the complete picture of the radial solutions in balls. In particular we establish for $p>1$ the existence of a unique large solution behaving like $\delta^{- \frac2{p-1}}$ at the boundary. In general domains we extend results of arXiv:arch-ive/1407.0288 and show that there exists a unique singular solutions $u$ such that $u/\delta^{\beta_-}\to c$ on the boundary for an arbitrary positive function $c \in C^{2+\gamma}(\partial\Omega) \, (\gamma \in (0,1)), c \ge 0$. Here $\beta_-$ is the smaller root of $\beta(\beta-1)+\mu=0$.