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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) be a bounded smooth domain and $\delta (x)=\text{dist}(x,\mathrm{\partial }\mathrm{\Omega })$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$-\mathrm{\Delta }u-\frac{\mu }{{\delta }^2}u=u^p+\tau \text{in }\mathrm{\Omega },u=\nu \text{on }\mathrm{\partial }\mathrm{\Omega },$$ where $\mu \in \mathbb{R}$, $p>0$, $\tau$ and $\nu$ are measures on $\mathrm{\Omega }$ and $\mathrm{\partial }\mathrm{\Omega }$ respectively. We then establish existence results for the system $$\{\begin{array}{rl}{\displaystyle }& {\displaystyle -\mathrm{\Delta }u-\frac{\mu }{{\delta }^2}u=ϵv^p+\tau \text{in }\mathrm{\Omega },}\\ {\displaystyle }& {\displaystyle -\mathrm{\Delta }v-\frac{\mu }{{\delta }^2}v=ϵu^{\tilde{p}}+\tilde{\tau }\text{in }\mathrm{\Omega },}\\ {\displaystyle }& {\displaystyle u=\nu ,v=\tilde{\nu }\text{on }\mathrm{\partial }\mathrm{\Omega },}\end{array}$$ where $ϵ=\pm 1$, $p>0$, $\tilde{p}>0$, $\tau$ and $\tilde{\tau }$ are measures on $\mathrm{\Omega }$, $\nu$ and $\tilde{\nu }$ are measures on $\mathrm{\partial }\mathrm{\Omega }$. We also deal with elliptic systems where the nonlinearities are more general.