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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-02-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.534/ |
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| Summary: | 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 $. |
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| ISSN: | 1778-3569 |