Uniqueness of bounded solutions to $p$-Laplace problems in strips

Phuong Le

doi:10.5802/crmath.442

Partial differential equations

Uniqueness of bounded solutions to

p

-Laplace problems in strips

Phuong Le ^1,²

¹ Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam
² Vietnam National University, Ho Chi Minh City, Vietnam

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 795-801.

Abstract

We consider a $p$ -Laplace problem in a strip with two-constant boundary Dirichlet conditions. We show that if the width of the strip is smaller than some $d_{0} \in (0, + \infty]$ , then the problem admits a unique bounded solution, which is strictly monotone. Hence this unique solution is one-dimensional symmetric and belongs to the $C^{2}$ class. We also show that the problem has no bounded solution in the case that $d_{0} < + \infty$ and the width of the strip is larger than or equal to $d_{0}$ . An analogous rigidity result in the whole space was obtained recently by Esposito et al. [8]

Received: 2022-09-26
Revised: 2022-10-31
Accepted: 2022-11-09
Published online: 2023-05-11

DOI: 10.5802/crmath.442

Classification: 35J92, 35A01, 35A02, 35B06
Keywords: $p$-Laplace equation, uniqueness, monotonicity, 1D symmetry

Author's affiliations:

Phuong Le ^{1, 2}

¹ Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam
² Vietnam National University, Ho Chi Minh City, Vietnam

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Uniqueness of bounded solutions to $p${-Laplace} problems in strips},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {795--801},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2023},
     doi = {10.5802/crmath.442},
     language = {en},
}

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Phuong Le. Uniqueness of bounded solutions to $p$-Laplace problems in strips. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 795-801. doi : 10.5802/crmath.442. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.442/

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