Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth
In this work, we investigated the existence of nontrivial weak solutions for the equation \begin{document}$ -{\rm div}(w(x)\nabla u) \ = \ f(x,u),\qquad x \in \mathbb{R}^2, $\end{document} where $ w(x) $ is a positive radial weight, the nonlinearity $ f(x, s) $ possesses growth at infinity...
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AIMS Press
2024-09-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024247 |
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| author | Yony Raúl Santaria Leuyacc |
| author_facet | Yony Raúl Santaria Leuyacc |
| author_sort | Yony Raúl Santaria Leuyacc |
| collection | DOAJ |
| description | In this work, we investigated the existence of nontrivial weak solutions for the equation \begin{document}$ -{\rm div}(w(x)\nabla u) \ = \ f(x,u),\qquad x \in \mathbb{R}^2, $\end{document} where $ w(x) $ is a positive radial weight, the nonlinearity $ f(x, s) $ possesses growth at infinity of the type $ {\rm \exp}\big((\alpha_0+h(|x|)\big)|s|^{2/(1-\beta)}) $, with $ \alpha_0 > 0 $, $ 0 < \beta < 1 $ and $ h $ is a continuous radial function that may be unbounded at infinity. To show the existence of weak solutions, we used variational methods and a new type of the Trudinger-Moser inequality defined on the whole two-dimensional space. |
| format | Article |
| id | doaj-art-cf0e39da357b41fca1c1c425165294e8 |
| institution | Kabale University |
| issn | 2688-1594 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj-art-cf0e39da357b41fca1c1c425165294e82025-01-23T07:52:42ZengAIMS PressElectronic Research Archive2688-15942024-09-013295341535610.3934/era.2024247Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growthYony Raúl Santaria Leuyacc0Universidad Nacional Mayor de San Marcos, Lima, PerúIn this work, we investigated the existence of nontrivial weak solutions for the equation \begin{document}$ -{\rm div}(w(x)\nabla u) \ = \ f(x,u),\qquad x \in \mathbb{R}^2, $\end{document} where $ w(x) $ is a positive radial weight, the nonlinearity $ f(x, s) $ possesses growth at infinity of the type $ {\rm \exp}\big((\alpha_0+h(|x|)\big)|s|^{2/(1-\beta)}) $, with $ \alpha_0 > 0 $, $ 0 < \beta < 1 $ and $ h $ is a continuous radial function that may be unbounded at infinity. To show the existence of weak solutions, we used variational methods and a new type of the Trudinger-Moser inequality defined on the whole two-dimensional space.https://www.aimspress.com/article/doi/10.3934/era.2024247trudinger-moser inequalitysupercritical exponential growthmountain pass theoremelliptic equationvariational method |
| spellingShingle | Yony Raúl Santaria Leuyacc Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth Electronic Research Archive trudinger-moser inequality supercritical exponential growth mountain pass theorem elliptic equation variational method |
| title | Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth |
| title_full | Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth |
| title_fullStr | Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth |
| title_full_unstemmed | Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth |
| title_short | Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth |
| title_sort | elliptic equations in mathbb r 2 involving supercritical exponential growth |
| topic | trudinger-moser inequality supercritical exponential growth mountain pass theorem elliptic equation variational method |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024247 |
| work_keys_str_mv | AT yonyraulsantarialeuyacc ellipticequationsinmathbbr2involvingsupercriticalexponentialgrowth |