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

Diem Hang T. Le; Phuong Le

doi:10.5802/crmath.710

Research article - Partial differential equations

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

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 (Original language)
Résumé

This paper is concerned with possibly sign-changing solutions to the critical $p$-Laplace equation $-\Delta _p u = \vert {x}\vert ^\alpha \vert {u}\vert ^{p^*_\alpha -2}u$ in $\Omega $, $u=0$ on $\partial \Omega $, where $1<p<N$, $\alpha >-p$, $p^*_\alpha =\frac{(N+\alpha )p}{N-p}$, and $\Omega $ is a bounded or unbounded domain of $\smash{\mathbb{R}^N}$. 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. $-\Delta _p u = \vert {x}\vert ^\alpha \vert {u}\vert ^{p^*_\alpha -2}u$ dans $\Omega $, $u=0$ sur $\partial \Omega $, où $1<p<N$, $\alpha >-p$, $p^*_\alpha =\frac{(N+\alpha )p}{N-p}$, et $\Omega $ est un domaine borné ou non de $\smash{\mathbb{R}^N}$. 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.

Received: 2024-07-17
Accepted: 2024-12-06
Published online: 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

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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},
}

TY  - JOUR
AU  - Diem Hang T. Le
AU  - Phuong Le
TI  - Liouville theorems for $p$-Laplace equations with critical Hardy–Hénon exponents
JO  - Comptes Rendus. Mathématique
PY  - 2025
SP  - 571
EP  - 582
VL  - 363
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.710
LA  - en
ID  - CRMATH_2025__363_G6_571_0
ER  -

%0 Journal Article
%A Diem Hang T. Le
%A Phuong Le
%T Liouville theorems for $p$-Laplace equations with critical Hardy–Hénon exponents
%J Comptes Rendus. Mathématique
%D 2025
%P 571-582
%V 363
%I Académie des sciences, Paris
%R 10.5802/crmath.710
%G en
%F CRMATH_2025__363_G6_571_0

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/

References
Cited by

[1] Luis Ángel Caffarelli; Robert V. Kohn; Louis Nirenberg First order interpolation inequalities with weights, Compos. Math., Volume 53 (1984) no. 3, pp. 259-275 | Numdam | MR | Zbl

[2] Kai Seng Chou; Chiu Wing Chu On the best constant for a weighted Sobolev–Hardy inequality, J. Lond. Math. Soc. (2), Volume 48 (1993) no. 1, pp. 137-151 | DOI | MR | Zbl

[3] Giulio Ciraolo; Rosario Corso Symmetry for positive critical points of Caffarelli–Kohn–Nirenberg inequalities, Nonlinear Anal., Theory Methods Appl., Volume 216 (2022), 112683, 23 pages | DOI | MR | Zbl

[4] Mónica Clapp Entire nodal solutions to the pure critical exponent problem arising from concentration, J. Differ. Equations, Volume 261 (2016) no. 6, pp. 3042-3060 | DOI | MR | Zbl

[5] Mónica Clapp; Luis Lopez Rios Entire nodal solutions to the pure critical exponent problem for the $p$ -Laplacian, J. Differ. Equations, Volume 265 (2018) no. 3, pp. 891-905 | DOI | MR | Zbl

[6] Jean-Michel Coron Topologie et cas limite des injections de Sobolev, C. R. Math., Volume 299 (1984) no. 7, pp. 209-212 | MR | Zbl

[7] Lucio Damascelli; Alberto Farina; Berardino Sciunzi; Enrico Valdinoci Liouville results for $m$ -Laplace equations of Lane–Emden–Fowler type, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009) no. 4, pp. 1099-1119 | DOI | MR | Zbl

[8] Emmanuele DiBenedetto $C^{1 + α}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl., Volume 7 (1983) no. 8, pp. 827-850 | DOI | MR

[9] Wei Yue Ding On a conformally invariant elliptic equation on $ℝ^{n}$ , Commun. Math. Phys., Volume 107 (1986) no. 2, pp. 331-335 | MR | Zbl

[10] Maria J. Esteban; Pierre-Louis Lions Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. R. Soc. Edinb., Sect. A, Math., Volume 93 (1982/83) no. 1-2, pp. 1-14 | DOI | MR | Zbl

[11] Alberto Farina On the classification of solutions of the Lane–Emden equation on unbounded domains of $ℝ^{N}$ , J. Math. Pures Appl. (9), Volume 87 (2007) no. 5, pp. 537-561 | DOI | MR | Zbl

[12] Alberto Farina; Carlo Mercuri; Michel Willem A Liouville theorem for the $p$ -Laplacian and related questions, Calc. Var. Partial Differ. Equ., Volume 58 (2019) no. 4, 153, 13 pages | DOI | MR | Zbl

[13] Nassif A. Ghoussoub; Chaoqui Yuan Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Am. Math. Soc., Volume 352 (2000) no. 12, pp. 5703-5743 | DOI | MR | Zbl

[14] Mohammed Guedda; Laurent Véron Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., Theory Methods Appl., Volume 13 (1989) no. 8, pp. 879-902 | DOI | MR | Zbl

[15] Phuong Le; Diem Hang T. Le Classification of positive solutions to $p$ -Laplace equations with critical Hardy–Sobolev exponent, Nonlinear Anal., Real World Appl., Volume 74 (2023), 103949, 13 pages | DOI | MR | Zbl

[16] Yuanyuan Li; Qianqiao Guo; Pengcheng Niu Global compactness results for quasilinear elliptic problems with combined critical Sobolev–Hardy terms, Nonlinear Anal., Theory Methods Appl., Volume 74 (2011) no. 4, pp. 1445-1464 | DOI | MR | Zbl

[17] Gary M. Lieberman Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl., Volume 12 (1988) no. 11, pp. 1203-1219 | DOI | MR | Zbl

[18] Carlo Mercuri; Berardino Sciunzi; Marco Squassina On Coron’s problem for the $p$ -Laplacian, J. Math. Anal. Appl., Volume 421 (2015) no. 1, pp. 362-369 | DOI | MR | Zbl

[19] Carlo Mercuri; Michel Willem A global compactness result for the $p$ -Laplacian involving critical nonlinearities, Discrete Contin. Dyn. Syst., Volume 28 (2010) no. 2, pp. 469-493 | DOI | MR | Zbl

[20] Wei Ming Ni A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., Volume 31 (1982) no. 6, pp. 801-807 | DOI | MR | Zbl

[21] Manuel del Pino; Monica Musso; Frank Pacard; Angela Pistoia Large energy entire solutions for the Yamabe equation, J. Differ. Equations, Volume 251 (2011) no. 9, pp. 2568-2597 | DOI | MR | Zbl

[22] Patrizia Pucci; Raffaella Servadei Existence, non-existence and regularity of radial ground states for $p$ -Laplacian equations with singular weights, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 25 (2008) no. 3, pp. 505-537 | DOI | Numdam | MR | Zbl

[23] Shaya Shakerian; Jérôme Vétois Sharp pointwise estimates for weighted critical $p$ -Laplace equations, Nonlinear Anal., Theory Methods Appl., Volume 206 (2021), 112236, 18 pages | DOI | MR | Zbl

[24] Giorgio Talenti Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), Volume 110 (1976), pp. 353-372 | DOI | MR | Zbl

[25] Peter Tolksdorf Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equations, Volume 51 (1984) no. 1, pp. 126-150 | DOI | MR | Zbl

Cited by Sources:

Comments - Policy