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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| Format: | Article |
| Language: | English |
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AIMS Press
2024-09-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024247 |
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| Summary: | 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. |
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| ISSN: | 2688-1594 |