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

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ ($N\ge 3$) 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 $$-\Delta u-\frac{\mu }{{\delta }^2}u=u^p+\tau \text{in }\Omega ,u=\nu \text{on }\partial \Omega ,$$ 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 $$\{\begin{array}{rl}& -\Delta u-\frac{\mu }{{\delta }^2}u=\epsilon v^p+\tau \text{in }\Omega ,\\ & -\Delta v-\frac{\mu }{{\delta }^2}v=\epsilon u^{\tilde{p}}+\tilde{\tau }\text{in }\Omega ,\\ & u=\nu ,v=\tilde{\nu }\text{on }\partial \Omega ,\end{array}$$ 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.

arXiv comment34 pages