On the critical behavior for a Sobolev-type inequality with Hardy potential

Mohamed Jleli; Bessem Samet

doi:10.5802/crmath.534

Research article - Partial differential equations

On the critical behavior for a Sobolev-type inequality with Hardy potential

Mohamed Jleli ¹; Bessem Samet ¹

¹ Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 87-97.

Abstract

We investigate the existence and nonexistence of weak solutions to the Sobolev-type inequality $- \partial_{t} (Δ u) - Δ u + \frac{σ}{{| x |}^{2}} u \geq {| x |}^{μ} {| u |}^{p}$ 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 $ℝ^{N}$ , $N \geq 2$ , $σ > - {(\frac{N - 2}{2})}^{2}$ , $μ \in ℝ$ and $p > 1$ . In particular, when $σ \neq 0$ , we show that the dividing line with respect to existence and nonexistence is given by a critical exponent that depends on $N$ , $σ$ and $μ$ .

Received: 2023-02-16
Revised: 2023-06-29
Accepted: 2023-07-06
Published online: 2024-02-02

DOI: 10.5802/crmath.534

Classification: 35R45, 35A01, 35B33
Mots-clés : Sobolev-type inequality, Hardy potential, bounded domain, existence, nonexistence, critical exponent

Author's affiliations:

Mohamed Jleli ¹; Bessem Samet ¹

¹ Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Mohamed Jleli and Bessem Samet},
     title = {On the critical behavior for a {Sobolev-type} inequality with {Hardy} potential},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {87--97},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.534},
     language = {en},
}

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Mohamed Jleli; Bessem Samet. On the critical behavior for a Sobolev-type inequality with Hardy potential. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 87-97. doi : 10.5802/crmath.534. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.534/

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