Elliptic problems with singular nonlinearities of indefinite sign
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1,1}$ boundary. We consider problems of the form $-\Delta u=\chi_{\left\{ u>0\right\}}\left( au^{-\alpha}-g\left( .,u\right) \right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u\geq0$ in $\Omega,$ where $\Omega$ is a bounded dom...
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| Format: | Article |
| Language: | English |
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AIMS Press
2020-02-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/10.3934/math.2020120/fulltext.html |
| _version_ | 1818410024119042048 |
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| author | Tomas Godoy |
| author_facet | Tomas Godoy |
| author_sort | Tomas Godoy |
| collection | DOAJ |
| description | Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1,1}$ boundary. We consider problems of the form $-\Delta u=\chi_{\left\{ u>0\right\}}\left( au^{-\alpha}-g\left( .,u\right) \right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u\geq0$ in $\Omega,$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $0\not \equiv a\in L^{\infty}\left( \Omega\right) ,$ $\alpha\in\left( 0,1\right) ,$ and $g:\Omega\times\left[ 0,\infty\right) \rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. We prove, under suitable assumptions on $a$ and $g,$ the existence of nontrivial and nonnegative weak solutions $u\in H_{0}^{1}\left( \Omega\right) \cap L^{\infty}\left( \Omega\right) $ of the stated problem. Under additional assumptions, the positivity, $a.e.$ in $\Omega,$ of the found solution $u$, is also proved. |
| first_indexed | 2024-12-14T10:08:56Z |
| format | Article |
| id | doaj.art-c92a231cc0b742fb848b80ee9c52405d |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-14T10:08:56Z |
| publishDate | 2020-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-c92a231cc0b742fb848b80ee9c52405d2022-12-21T23:07:04ZengAIMS PressAIMS Mathematics2473-69882020-02-01531779179810.3934/math.2020120Elliptic problems with singular nonlinearities of indefinite signTomas Godoy0Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, ArgentinaLet $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1,1}$ boundary. We consider problems of the form $-\Delta u=\chi_{\left\{ u>0\right\}}\left( au^{-\alpha}-g\left( .,u\right) \right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u\geq0$ in $\Omega,$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $0\not \equiv a\in L^{\infty}\left( \Omega\right) ,$ $\alpha\in\left( 0,1\right) ,$ and $g:\Omega\times\left[ 0,\infty\right) \rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. We prove, under suitable assumptions on $a$ and $g,$ the existence of nontrivial and nonnegative weak solutions $u\in H_{0}^{1}\left( \Omega\right) \cap L^{\infty}\left( \Omega\right) $ of the stated problem. Under additional assumptions, the positivity, $a.e.$ in $\Omega,$ of the found solution $u$, is also proved.https://www.aimspress.com/article/10.3934/math.2020120/fulltext.htmlsingular elliptic problemsnonnegative solutionssub and supersolutions |
| spellingShingle | Tomas Godoy Elliptic problems with singular nonlinearities of indefinite sign AIMS Mathematics singular elliptic problems nonnegative solutions sub and supersolutions |
| title | Elliptic problems with singular nonlinearities of indefinite sign |
| title_full | Elliptic problems with singular nonlinearities of indefinite sign |
| title_fullStr | Elliptic problems with singular nonlinearities of indefinite sign |
| title_full_unstemmed | Elliptic problems with singular nonlinearities of indefinite sign |
| title_short | Elliptic problems with singular nonlinearities of indefinite sign |
| title_sort | elliptic problems with singular nonlinearities of indefinite sign |
| topic | singular elliptic problems nonnegative solutions sub and supersolutions |
| url | https://www.aimspress.com/article/10.3934/math.2020120/fulltext.html |
| work_keys_str_mv | AT tomasgodoy ellipticproblemswithsingularnonlinearitiesofindefinitesign |