Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data

Abstract

We study existence and stability of solutions of $$ -\Delta u +\frac{\mu}{|x|^{2 }}u+ g(u)= ν \text{ in }\Omega,\ \ \ u=0\text{ on } ∂Ω,$$ where $\Omega$ is a bounded, smooth domain of $\mathbb{ R}^N$, $N\geq 2$, containing the origin, $\mu\geq-\frac{(N-2)^2}{4}$ is a constant, $g$ is a nondecreasing function satisfying some integral growth assumption and the weak ∆2-condition, and ν is a Radon measure in Ω. We show that the situation differs depending on whether the measure is diffuse or concentrated at the origin. When $g$ is a power function, we introduce a capacity framework to find necessary and sufficient conditions for solvability.

Keywords

• Leray-Hardy potential
• Radon measure
• capacity
• weak solution