Liouville theorems for $p$-Laplace equations with critical Hardy–Hénon exponents

Diem Hang T. Le; Phuong Le

doi:10.5802/crmath.710

Article de recherche - Équations aux dérivées partielles

Liouville theorems for

p

-Laplace equations with critical Hardy–Hénon exponents
[Théorèmes de Liouville pour les équations de

p

-Laplace avec des exposants critiques de Hardy–Hénon]

Diem Hang T. Le ^1,² ; Phuong Le ^3,⁴

¹ Faculty of Mathematics and Applications, Saigon University, 273 An Duong Vuong St., Ward 3, District 5, Ho Chi Minh City, Vietnam
² Faculty of Data Science in Business, Ho Chi Minh University of Banking, Ho Chi Minh City, Vietnam
³ Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam
⁴ Vietnam National University, Ho Chi Minh City, Vietnam

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 571-582.

Abstract (VO)
Résumé

This paper is concerned with possibly sign-changing solutions to the critical $p$ -Laplace equation $- Δ_{p} u = | x |^{α} | u |^{p_{α}^{*} - 2} u$ in $Ω$ , $u = 0$ on $\partial Ω$ , where $1 < p < N$ , $α > - p$ , $p_{α}^{*} = \frac{(N + α) p}{N - p}$ , and $Ω$ is a bounded or unbounded domain of $R^{N}$ . We first derive a Liouville type theorem on half-spaces. Then we classify solutions via the radial symmetry and Morse index. Moreover, we characterize the compactness of radial Palais–Smale sequences on radial domains.

Cet article s’intéresse aux solutions de l’équation critique de p-Laplace qui peuvent changer de signe. $- Δ_{p} u = | x |^{α} | u |^{p_{α}^{*} - 2} u$ dans $Ω$ , $u = 0$ sur $\partial Ω$ , où $1 < p < N$ , $α > - p$ , $p_{α}^{*} = \frac{(N + α) p}{N - p}$ , et $Ω$ est un domaine borné ou non de $R^{N}$ . Nous dérivons d’abord un théorème de type Liouville sur les demi-espaces. Ensuite, nous classons les solutions en fonction de la symétrie radiale et de l’indice de Morse. De plus, nous caractérisons la compacité des suites de Palais–Smale radiales sur les domaines radiaux.

Reçu le : 2024-07-17
Accepté le : 2024-12-06
Publié le : 2025-06-05

DOI : 10.5802/crmath.710

Classification : 35J92, 35B53, 35B33, 35B38
Keywords:

p

-Laplace equations, Hardy–Hénon exponents, Liouville theorems, Palais–Smale sequences
Mots-clés : Équations de

p

-Laplace, exposants de Hardy–Hénon, théorèmes de Liouville, suites de Palais–Smale

Affiliations des auteurs :

Diem Hang T. Le ^{1, 2} ; Phuong Le ^{3, 4}

¹ Faculty of Mathematics and Applications, Saigon University, 273 An Duong Vuong St., Ward 3, District 5, Ho Chi Minh City, Vietnam
² Faculty of Data Science in Business, Ho Chi Minh University of Banking, Ho Chi Minh City, Vietnam
³ Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam
⁴ Vietnam National University, Ho Chi Minh City, Vietnam

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2025__363_G6_571_0,
     author = {Diem Hang T. Le and Phuong Le},
     title = {Liouville theorems for $p${-Laplace} equations with critical {Hardy{\textendash}H\'enon} exponents},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {571--582},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.710},
     language = {en},
}

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Diem Hang T. Le; Phuong Le. Liouville theorems for $p$-Laplace equations with critical Hardy–Hénon exponents. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 571-582. doi : 10.5802/crmath.710. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.710/

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