A dynamical system approach to the Chandrasekhar–Hamilton–Jacobi equation

Marie-Françoise Bidaut-Véron; Laurent Véron

doi:10.5802/crmath.793

Article de recherche - Équations aux dérivées partielles

A dynamical system approach to the Chandrasekhar–Hamilton–Jacobi equation
[Une approche systèmes dynamiques de l’équation de Chandrasekhar–Hamilton–Jacobi]

Marie-Françoise Bidaut-Véron ¹ ; Laurent Véron ¹

¹ Institut Denis Poisson, CNRS UMR 7013, Université de Tours, 37200 Tours, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1177-1217

Cet article fait partie du numéro thématique De génération en génération : l'héritage mathématique de Haïm Brezis coordonné par Henri Berestycki et al..

Abstract (VO)
Résumé

We study the local properties of positive solutions of the equation $-\Delta u=e^u-M \vert {\nabla u}\vert ^q$ in a punctured domain $\Omega \setminus \lbrace 0\rbrace $ of $\mathbb{R}^N$ in the range of parameters $q>1$ and $M> 0$. We prove a series of a priori estimates near a singular point. In the case of radial solutions we use various techniques inherited from the dynamical systems theory to obtain the precise behaviour of singular solutions. We prove also the existence of singular solutions with these precise behaviours.

Nous étudions les propriétés locales des solutions de l’équation $-\Delta u= e^u-M \vert {\nabla u}\vert ^q$ dans un domaine épointé $\Omega \setminus \lbrace 0\rbrace $ de $\mathbb{R}^N$ avec des paramètres $q > 1$ et $M > 0$. Nous donnons une série d’estimations a priori près d’un point singulier. Dans le cas de solutions radiales, nous obtenons le comportement précis des solutions avec des méthodes issues de la théorie des systèmes dynamiques. Nous démontrons aussi l’existence de solutions singulières avec les comportements singuliers obtenus.

Reçu le : 2025-06-25
Révisé le : 2025-08-07
Accepté le : 2025-09-08
Publié le : 2025-10-13

DOI : 10.5802/crmath.793

Classification : 35J62, 35B08, 37C25, 37C75
Keywords: Elliptic equations, limit sets, saddle points, stable manifolds, energy functions
Mots-clés : Équations elliptiques, ensembles limites, points selles, variété stable, fonctions d’énergie

Affiliations des auteurs :

Marie-Françoise Bidaut-Véron ¹ ; Laurent Véron ¹

¹ Institut Denis Poisson, CNRS UMR 7013, Université de Tours, 37200 Tours, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2025__363_G11_1177_0,
     author = {Marie-Fran\c{c}oise Bidaut-V\'eron and Laurent V\'eron},
     title = {A dynamical system approach to the {Chandrasekhar{\textendash}Hamilton{\textendash}Jacobi} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1177--1217},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.793},
     language = {en},
}

TY  - JOUR
AU  - Marie-Françoise Bidaut-Véron
AU  - Laurent Véron
TI  - A dynamical system approach to the Chandrasekhar–Hamilton–Jacobi equation
JO  - Comptes Rendus. Mathématique
PY  - 2025
SP  - 1177
EP  - 1217
VL  - 363
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.793
LA  - en
ID  - CRMATH_2025__363_G11_1177_0
ER  -

%0 Journal Article
%A Marie-Françoise Bidaut-Véron
%A Laurent Véron
%T A dynamical system approach to the Chandrasekhar–Hamilton–Jacobi equation
%J Comptes Rendus. Mathématique
%D 2025
%P 1177-1217
%V 363
%I Académie des sciences, Paris
%R 10.5802/crmath.793
%G en
%F CRMATH_2025__363_G11_1177_0

Marie-Françoise Bidaut-Véron; Laurent Véron. A dynamical system approach to the Chandrasekhar–Hamilton–Jacobi equation. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1177-1217. doi: 10.5802/crmath.793

Bibliographie
Cité par

[1] Larry R. Anderson; Walter Leighton Liapunov functions for autonomous systems of second order, J. Math. Anal. Appl., Volume 23 (1968), pp. 645-664 | DOI | MR | Zbl

[2] Marie-Françoise Bidaut-Véron; Marta Garcia-Huidobro; Laurent Véron A priori estimates for elliptic equations with reaction terms involving the function and its gradient, Math. Ann., Volume 378 (2020) no. 1-2, pp. 13-56 | DOI | MR | Zbl

[3] Marie-Françoise Bidaut-Véron; Marta Garcia-Huidobro; Laurent Véron Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms, Discrete Contin. Dyn. Syst., Volume 40 (2020) no. 2, pp. 933-982 | DOI | MR | Zbl

[4] Marie-Françoise Bidaut-Véron; Marta Garcia-Huidobro; Laurent Véron Singular solutions of some elliptic equations involving mixed absorption-reaction, Discrete Contin. Dyn. Syst., Volume 42 (2022) no. 8, pp. 3861-3930 | DOI | Zbl

[5] Marie-Françoise Bidaut-Véron; Laurent Véron Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., Volume 106 (1991) no. 3, pp. 489-539 | DOI | MR | Zbl

[6] Marie-Françoise Bidaut-Véron; Laurent Véron Local behaviour of the solutions of the Chipot–Weissler equation, Calc. Var. Partial Differ. Equ., Volume 62 (2023) no. 9, 241, 59 pages | DOI | MR | Zbl

[7] Marie-Françoise Bidaut-Véron; Laurent Véron Singularities and asymptotics of solutions of the Emden–Chandrasekhar–Riccati equation (2025) | arXiv | Zbl

[8] Haïm Brezis; Pierre-Louis Lions A note on isolated singularities for linear elliptic equations, Mathematical analysis and applications, Part A (Advances in Mathematics. Supplementary Studies), Academic Press Inc., 1981 no. 7a, pp. 263-266 | MR | Zbl

[9] Haïm Brezis; Frank Merle Uniform estimates and blow-up behavior for solutions of $- Δ u = V (x) e^{u}$ in two dimensions, Commun. Partial Differ. Equations, Volume 16 (1991) no. 8-9, pp. 1223-1253 | DOI | MR | Zbl

[10] Haïm Brezis; Louis Nirenberg Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal., Volume 9 (1997) no. 2, pp. 201-219 | DOI | MR | Zbl

[11] Luis A. Caffarelli; Michael G. Crandall Distance functions and almost global solutions of eikonal equations, Commun. Partial Differ. Equations, Volume 35 (2010) no. 3, pp. 391-414 | DOI | MR | Zbl

[12] Subrahmanyan Chandrasekhar An introduction to the study of stellar structure, University of Chicago Press, 1939, ix+509 pages | MR | Zbl

[13] Michel Marie Chipot; Fred B. Weissler On the elliptic problem $Δ u - {| \nabla u |}^{q} + λ u^{p} = 0$ , Nonlinear diffusion equations and their equilibrium states, I (Berkeley, CA, 1986) (W.-M. Ni; L. A. Peletier; James Serrin, eds.) (Mathematical Sciences Research Institute Publications), Springer, 1988 no. 12, pp. 237-243 | DOI | MR | Zbl

[14] Michel Marie Chipot; Fred B. Weissler Some blowup results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., Volume 20 (1989) no. 4, pp. 886-907 | DOI | MR | Zbl

[15] Paul Concus; Robert Finn A singular solution of the capillary equation. I. Existence, Invent. Math., Volume 29 (1975) no. 2, pp. 143-148 | DOI | MR | Zbl

[16] Paul Concus; Robert Finn A singular solution of the capillary equation. II. Uniqueness, Invent. Math., Volume 29 (1975) no. 2, pp. 149-159 | DOI | MR | Zbl

[17] Jean-Michel Lasry; Pierre-Louis Lions Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem, Math. Ann., Volume 283 (1989) no. 4, pp. 583-630 | DOI | MR | Zbl

[18] Yimei Li; Laurent Véron Isolated singularities of solutions of a 2-D diffusion equation with mixed reaction, Calc. Var. Partial Differ. Equ., Volume 64 (2025) no. 7, 235, 32 pages | DOI | MR | Zbl

[19] Hartmut Logemann; Eugene P. Ryan Non-autonomous systems: asymptotic behaviour and weak invariance principles, J. Differ. Equations, Volume 189 (2003) no. 2, pp. 440-460 | DOI | MR | Zbl

[20] Phuoc-Tai Nguyen Isolated singularities of positive solutions of elliptic equations with weighted gradient term, Anal. PDE, Volume 9 (2016) no. 7, pp. 1671-1692 | DOI | MR | Zbl

[21] Peter Poláčik; Pavol Quittner; Philippe Souplet Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., Volume 139 (2007) no. 3, pp. 555-579 | DOI | MR | Zbl

[22] Yves Richard; Laurent Véron Isotropic singularities of solutions of nonlinear elliptic inequalities, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 6 (1989) no. 1, pp. 37-72 | MR | DOI | Numdam | Zbl

[23] James Serrin; Henghui Zou Existence and non-existence results for ground states of quasilinear elliptic equations, Arch. Ration. Mech. Anal., Volume 121 (1992) no. 2, pp. 101-130 | DOI | Zbl | MR

[24] James Serrin; Henghui Zou Classification of positive solutions of quasilinear elliptic equations, Topol. Methods Nonlinear Anal., Volume 3 (1994) no. 1, pp. 1-25 | DOI | MR | Zbl

[25] F. X. Voirol Étude de quelques équations elliptiques fortement non-linéaires, Ph. D. Thesis, Université de Metz (France) (1994)

Cité par Sources :

Commentaires - Politique