On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials

Abstract

Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $δ(x)=\text{dist}(x,\partial Ω)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$ -Δu- \fracμ{δ^2}u= u^p +τ\quad \text{in } Ω, \quad \quad u=ν\quad\text{on } \partial Ω, $$ where $μ\in \mathbb{R}$, $p>0$, $τ$ and $ν$ are measures on $Ω$ and $\partial Ω$ respectively. We then establish existence results for the system $$ \left\{ \begin{aligned} &-Δu- \fracμ{δ^2}u = ε\, v^p +τ\quad \text{in } Ω, \\ &-Δv- \fracμ{δ^2}v = ε\, u^{\tilde p}+\tilde τ\quad \text{in } Ω, \\ &u=ν, \quad v= \tilde ν\quad \text{on } \partial Ω, \end{aligned} \right. $$ where $ε=\pm 1$, $p>0$, $\tilde p>0$, $τ$ and $\tilde τ$ are measures on $Ω$, $ν$ and $\tilde ν$ are measures on $\partial Ω$. We also deal with elliptic systems where the nonlinearities are more general.