On the critical behavior for a Sobolev-type inequality with Hardy potential
We investigate the existence and nonexistence of weak solutions to the Sobolev-type inequality $ -\partial _t(\Delta u)-\Delta u+ \frac{\sigma }{|x|^2}u \ge |x|^{\mu }|u|^p$ in $(0,\infty )\times B$, under the inhomogeneous Dirichlet-type boundary condition $u(t,x)=f(x)$ on $(0,\infty )\times \parti...
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Académie des sciences
2024-02-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.534/ |
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| author | Jleli, Mohamed Samet, Bessem |
| author_facet | Jleli, Mohamed Samet, Bessem |
| author_sort | Jleli, Mohamed |
| collection | DOAJ |
| description | We investigate the existence and nonexistence of weak solutions to the Sobolev-type inequality $ -\partial _t(\Delta u)-\Delta u+ \frac{\sigma }{|x|^2}u \ge |x|^{\mu }|u|^p$ in $(0,\infty )\times B$, under the inhomogeneous Dirichlet-type boundary condition $u(t,x)=f(x)$ on $(0,\infty )\times \partial B$, where $B$ is the unit open ball of $\mathbb{R}^N$, $N\ge 2$, $\sigma >-\bigl (\frac{N-2}{2}\bigr )^2$, $\mu \in \mathbb{R}$ and $p>1$. In particular, when $\sigma \ne 0$, we show that the dividing line with respect to existence and nonexistence is given by a critical exponent that depends on $N$, $\sigma $ and $\mu $. |
| format | Article |
| id | doaj-art-3a8b318bfbb848148029ea6c177cf97a |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-02-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-3a8b318bfbb848148029ea6c177cf97a2025-02-07T11:12:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-02-01362G1879710.5802/crmath.53410.5802/crmath.534On the critical behavior for a Sobolev-type inequality with Hardy potentialJleli, Mohamed0Samet, Bessem1Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi ArabiaWe investigate the existence and nonexistence of weak solutions to the Sobolev-type inequality $ -\partial _t(\Delta u)-\Delta u+ \frac{\sigma }{|x|^2}u \ge |x|^{\mu }|u|^p$ in $(0,\infty )\times B$, under the inhomogeneous Dirichlet-type boundary condition $u(t,x)=f(x)$ on $(0,\infty )\times \partial B$, where $B$ is the unit open ball of $\mathbb{R}^N$, $N\ge 2$, $\sigma >-\bigl (\frac{N-2}{2}\bigr )^2$, $\mu \in \mathbb{R}$ and $p>1$. In particular, when $\sigma \ne 0$, we show that the dividing line with respect to existence and nonexistence is given by a critical exponent that depends on $N$, $\sigma $ and $\mu $.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.534/Sobolev-type inequalityHardy potentialbounded domainexistencenonexistencecritical exponent |
| spellingShingle | Jleli, Mohamed Samet, Bessem On the critical behavior for a Sobolev-type inequality with Hardy potential Comptes Rendus. Mathématique Sobolev-type inequality Hardy potential bounded domain existence nonexistence critical exponent |
| title | On the critical behavior for a Sobolev-type inequality with Hardy potential |
| title_full | On the critical behavior for a Sobolev-type inequality with Hardy potential |
| title_fullStr | On the critical behavior for a Sobolev-type inequality with Hardy potential |
| title_full_unstemmed | On the critical behavior for a Sobolev-type inequality with Hardy potential |
| title_short | On the critical behavior for a Sobolev-type inequality with Hardy potential |
| title_sort | on the critical behavior for a sobolev type inequality with hardy potential |
| topic | Sobolev-type inequality Hardy potential bounded domain existence nonexistence critical exponent |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.534/ |
| work_keys_str_mv | AT jlelimohamed onthecriticalbehaviorforasobolevtypeinequalitywithhardypotential AT sametbessem onthecriticalbehaviorforasobolevtypeinequalitywithhardypotential |