Existence and uniqueness of solutions to Hardy-H\'enon boundary-value problems

Abstract

We consider the elliptic equation $$ -\Delta u=d(x)^\alpha u^p,\quad x\in \Omega\quad (\text{or } x\in\mathbb{R}^N\setminus \overline{\Omega}), $$ where \(\alpha, p \in \mathbb{R}\), \(d(x) = \text{dist} (x,\partial\Omega)\) and \(\Omega\subset\mathbb{R}^N\) \((N\geq 3)\) is a bounded smooth domain. We establish estimates for the positive solutions when \(1< p< \frac{N+2}{N-2}\), and the nonexistence of positive solutions for exterior domains. For the corresponding Dirichlet problem we show the existence and uniqueness of positive solutions.  For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/72/abstr.html