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

TL;DR

This paper establishes conditions for the existence of weak solutions to semilinear elliptic equations and systems with Hardy potentials, considering measure data and boundary conditions, advancing the understanding of such nonlinear PDEs.

Contribution

It provides new necessary and sufficient conditions for solutions and extends existence results to systems with general nonlinearities involving Hardy potentials.

Findings

Derived criteria for solution existence with measure data.

Proved existence of solutions for coupled systems with Hardy potentials.

Extended results to more general nonlinear elliptic systems.

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.