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

Let $\mathrm{\Omega }?{\mathbb{R}}^N$ ($N\ge 3$) be a bounded $C^2$ domain and $\delta (x)=\mathrm{dist}(x,?\mathrm{\Omega })$. Put $L_{\mu }=\mathrm{\Delta }+\frac{\mu }{{\delta }^2}$ with $\mu >0$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to

$?L_{\mu }u=u^p+\tau \text{in }\mathrm{\Omega },u=\nu \text{on }?\mathrm{\Omega },$

where $\mu >0$, $p>0$, τ and ν are measures on Ω and ?Ω respectively. We then establish existence results for the system

$\{\begin{array}{cc}\hfill & ?L_{\mu }u=?v^p+\tau \text{in }\mathrm{\Omega },\hfill \\ \hfill & ?L_{\mu }v=?u^{\stackrel{?}{p}}+\stackrel{?}{\tau }\text{in }\mathrm{\Omega },\hfill \\ \hfill & u=\nu ,v=\stackrel{?}{\nu }\text{on }?\mathrm{\Omega },\hfill \end{array}$

where $?=\pm 1$, $p>0$, $\stackrel{?}{p}>0$, τ and $\stackrel{?}{\tau }$ are measures on Ω, ν and $\stackrel{?}{\nu }$ are measures on ?Ω. We also deal with elliptic systems where the nonlinearities are more general.