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

Huyuan Chen; Laurent Véron; Huyuan Chen; Laurent Véron

doi:10.3934/mine.2019.3.391

Mathematics in Engineering

2019, Volume 1, Issue 3: 391-418. doi: 10.3934/mine.2019.3.391

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Research article

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

Huyuan Chen ¹,
Laurent Véron ^{2
,
,}

1.
Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China
2.
Laboratoire de Mathématiques et Physique Théorique, Université de Tours, 37200 Tours, France

Received: 10 March 2019 Accepted: 04 April 2019 Published: 28 April 2019

We study existence and stability of solutions of $ -\Delta u +\frac{\mu}{|x|^{2 }}u+ g(u) = \nu\text{ in }\Omega,\ \ \ u = 0\text{ on } \partial\Omega, $ 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 $\Delta_2$-condition, and $\nu$ is a Radon measure in $\Omega$. 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.
- Leray-Hardy potential,
- Radon measure,
- capacity,
- weak solution
Citation: Huyuan Chen, Laurent Véron. Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data[J]. Mathematics in Engineering, 2019, 1(3): 391-418. doi: 10.3934/mine.2019.3.391

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Abstract

We study existence and stability of solutions of $ -\Delta u +\frac{\mu}{|x|^{2 }}u+ g(u) = \nu\text{ in }\Omega,\ \ \ u = 0\text{ on } \partial\Omega, $ 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 $\Delta_2$-condition, and $\nu$ is a Radon measure in $\Omega$. 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.

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