Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities
In this paper, we prove the existence of nontrivial solutions for the following weighted problem without the Ambrosetti-Rabinowitz condition:$- \mathrm{div} (\sigma(x)|\nabla u|^{N-2} \nabla u) = f(x,u)$ and $u >0$ in $B$, $u=0$ on $\partial B$, where $B$ is the unit ball of $\mathbb{R}^N$, $...
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| Datum: | 2023 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2023
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| Online Zugang: | https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1020 |
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| Назва журналу: | Journal of Mathematical Physics, Analysis, Geometry |
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Journal of Mathematical Physics, Analysis, Geometry| Zusammenfassung: | In this paper, we prove the existence of nontrivial solutions for the following weighted problem without the Ambrosetti-Rabinowitz condition:$- \mathrm{div} (\sigma(x)|\nabla u|^{N-2} \nabla u) = f(x,u)$ and $u >0$ in $B$, $u=0$ on $\partial B$, where $B$ is the unit ball of $\mathbb{R}^N$, $ \sigma(x)=\left(\log\left(\frac{e}{|x|}\right)\right)^{N-1}$ is the singular logarithmic weight in the Trudinger-Moser embedding. The nonlinearity is a critical or subcritical growth in view of Trudinger-Moser inequalities. In order to obtain the existence result, we used minimax techniques combined with the Trudinger-Moser inequality. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to avoid compactness level.
Mathematical Subject Classification 2020: 46E35, 35J20, 35J33, 35J60. |
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