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

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