Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data
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 nondecrea...
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| Format: | Article |
| Language: | English |
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AIMS Press
2019-04-01
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| Series: | Mathematics in Engineering |
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| Online Access: | https://www.aimspress.com/article/10.3934/mine.2019.3.391/fulltext.html |
| _version_ | 1819266379380948992 |
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| author | Huyuan Chen Laurent Véron |
| author_facet | Huyuan Chen Laurent Véron |
| author_sort | Huyuan Chen |
| collection | DOAJ |
| description | 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 ∆<sub>2</sub>-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. |
| first_indexed | 2024-12-23T21:00:20Z |
| format | Article |
| id | doaj.art-e864ce4aa9cc4421bad71a9521dd6be5 |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-12-23T21:00:20Z |
| publishDate | 2019-04-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-e864ce4aa9cc4421bad71a9521dd6be52022-12-21T17:31:24ZengAIMS PressMathematics in Engineering2640-35012019-04-011339141810.3934/mine.2019.3.391Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure dataHuyuan Chen0Laurent Véron11 Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China2 Laboratoire de Mathématiques et Physique Théorique, Université de Tours, 37200 Tours, FranceWe 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 ∆<sub>2</sub>-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.https://www.aimspress.com/article/10.3934/mine.2019.3.391/fulltext.htmlLeray-Hardy potentialRadon measurecapacityweak solution |
| spellingShingle | Huyuan Chen Laurent Véron Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data Mathematics in Engineering Leray-Hardy potential Radon measure capacity weak solution |
| title | Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data |
| title_full | Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data |
| title_fullStr | Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data |
| title_full_unstemmed | Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data |
| title_short | Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data |
| title_sort | weak solutions of semilinear elliptic equations with leray hardy potentials and measure data |
| topic | Leray-Hardy potential Radon measure capacity weak solution |
| url | https://www.aimspress.com/article/10.3934/mine.2019.3.391/fulltext.html |
| work_keys_str_mv | AT huyuanchen weaksolutionsofsemilinearellipticequationswithlerayhardypotentialsandmeasuredata AT laurentveron weaksolutionsofsemilinearellipticequationswithlerayhardypotentialsandmeasuredata |