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 $\mathbb{R}^N$, $...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2023
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oai:jmag.ilt.kharkiv.ua:article-10202023-11-29T18:04:11Z Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities Abid, Imed Jaidane, Rached нерiвнiсть Трудiнґера–Мозера нелiнiйнiсть подвiй- ного експоненцiального зростання критичнi експоненти рiвень компактностi Trudinger-Moser inequality nonlinearity of double exponential growth critical exponents compactness level In this paper, we prove the existence of nontrivial solutions for the following weighted problem without the Ambrosetti-Rabinowitz condition:$- \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. У роботi доведено iснування нетривiального розв’язку для такої ва-гової задачi без умови Амбросеттi–Рабiновiца: $- \mathrm{div} (\sigma(x)|\nabla u|^{N-2} \nabla u) = f(x,u)$ i $u >0$ в $B$, $u=0$ на $\partial B$, де $B$ є одиничною кулею в $\mathbb{R}^N$, $\sigma(x)=\left(\log\left(\frac{e}{|x|}\right)\right)^{N-1}$ є сингулярною логарифмiчною вагою у вкладеннi Трудiнґера–Мозера. Нелiнiйнiсть дає критичне або субкритичне зростання вiдносно нерiвностi Трудiнґера–Мозера. Ми скористалися мiнiмакс технiкою в комбiнацiї з нерiвнiстю Трудiнґера–Мозера, щоб довести iснування розв’язку. Ми запровадили нову умову для зростання та наполягаємо на її важливостi для позбавлення рiвня компактностi. Mathematical Subject Classification 2020: 46E35, 35J20, 35J33, 35J60. Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України 2023-11-29 Article Article application/pdf https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1020 10.15407/mag19.03.527 Journal of Mathematical Physics, Analysis, Geometry; Vol. 19 No. 3 (2023); 527-555 Журнал математической физики, анализа, геометрии; Том 19 № 3 (2023); 527-555 Журнал математичної фізики, аналізу, геометрії; Том 19 № 3 (2023); 527-555 1817-5805 1812-9471 en https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1020/jm19-0527e |
| institution |
Journal of Mathematical Physics, Analysis, Geometry |
| baseUrl_str |
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| datestamp_date |
2023-11-29T18:04:11Z |
| collection |
OJS |
| language |
English |
| topic |
нерiвнiсть Трудiнґера–Мозера нелiнiйнiсть подвiй- ного експоненцiального зростання критичнi експоненти рiвень компактностi |
| spellingShingle |
нерiвнiсть Трудiнґера–Мозера нелiнiйнiсть подвiй- ного експоненцiального зростання критичнi експоненти рiвень компактностi Abid, Imed Jaidane, Rached Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities |
| topic_facet |
нерiвнiсть Трудiнґера–Мозера нелiнiйнiсть подвiй- ного експоненцiального зростання критичнi експоненти рiвень компактностi Trudinger-Moser inequality nonlinearity of double exponential growth critical exponents compactness level |
| format |
Article |
| author |
Abid, Imed Jaidane, Rached |
| author_facet |
Abid, Imed Jaidane, Rached |
| author_sort |
Abid, Imed |
| title |
Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities |
| title_short |
Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities |
| title_full |
Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities |
| title_fullStr |
Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities |
| title_full_unstemmed |
Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities |
| title_sort |
weighted elliptic equations in dimension n with subcritical and critical double exponential nonlinearities |
| title_alt |
Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities |
| description |
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. |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України |
| publishDate |
2023 |
| url |
https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1020 |
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AT abidimed weightedellipticequationsindimensionnwithsubcriticalandcriticaldoubleexponentialnonlinearities AT jaidanerached weightedellipticequationsindimensionnwithsubcriticalandcriticaldoubleexponentialnonlinearities |
| first_indexed |
2025-09-26T01:40:37Z |
| last_indexed |
2025-09-26T01:40:37Z |
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1844288776151498752 |