Partial Differential Equations
Bifurcation and asymptotics for the Lane–Emden–Fowler equation
[Bifurcation et analyse asymptotique pour l'équation de Lane–Emden–Fowler]
Comptes Rendus. Mathématique, Tome 337 (2003) no. 4, pp. 259-264.

On considère l'équation de Lane–Emden–Fowler −Δu=λf(u)+a(x)g(u) dans Ω avec une condition de Dirichlet u=0 sur Ω,Ω N est un domaine borné régulier, λ est un paramètre positif, a:Ω ¯[0,) est une fonction de Hölder et f est une fonction continue, positive et croissante telle que l'application f(s)/s soit décroissante sur (0,∞). Le caractère singulier de ce problème est donné par la nonlinéarité g, qui est non bornée autour de l'origine. Dans cette Note nous étudions l'existence et l'unicité d'une solution positive et nous établissons également son taux de décroissance vers 0 autour du bord. La méthode de démonstration repose sur le principe du maximum et sur des estimations elliptiques.

We are concerned with the Lane–Emden–Fowler equation −Δu=λf(u)+a(x)g(u) in Ω, subject to the Dirichlet boundary condition u=0 on Ω, where Ω N is a smooth bounded domain, λ is a positive parameter, a:Ω ¯[0,) is a Hölder function, and f is a positive nondecreasing continuous function such that f(s)/s is nonincreasing in (0,∞). The singular character of the problem is given by the nonlinearity g which is assumed to be unbounded around the origin. In this Note we discuss the existence and the uniqueness of a positive solution of this problem and we also describe the precise decay rate of this solution near the boundary. The proofs rely essentially on the maximum principle and on elliptic estimates.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00335-2
Ghergu, Marius 1 ; Rădulescu, Vicenţiu D. 1

1 Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania
@article{CRMATH_2003__337_4_259_0,
     author = {Ghergu, Marius and R\u{a}dulescu, Vicen\c{t}iu D.},
     title = {Bifurcation and asymptotics for the {Lane{\textendash}Emden{\textendash}Fowler} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {259--264},
     publisher = {Elsevier},
     volume = {337},
     number = {4},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00335-2},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00335-2/}
}
TY  - JOUR
AU  - Ghergu, Marius
AU  - Rădulescu, Vicenţiu D.
TI  - Bifurcation and asymptotics for the Lane–Emden–Fowler equation
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 259
EP  - 264
VL  - 337
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00335-2/
DO  - 10.1016/S1631-073X(03)00335-2
LA  - en
ID  - CRMATH_2003__337_4_259_0
ER  - 
%0 Journal Article
%A Ghergu, Marius
%A Rădulescu, Vicenţiu D.
%T Bifurcation and asymptotics for the Lane–Emden–Fowler equation
%J Comptes Rendus. Mathématique
%D 2003
%P 259-264
%V 337
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00335-2/
%R 10.1016/S1631-073X(03)00335-2
%G en
%F CRMATH_2003__337_4_259_0
Ghergu, Marius; Rădulescu, Vicenţiu D. Bifurcation and asymptotics for the Lane–Emden–Fowler equation. Comptes Rendus. Mathématique, Tome 337 (2003) no. 4, pp. 259-264. doi : 10.1016/S1631-073X(03)00335-2. http://www.numdam.org/articles/10.1016/S1631-073X(03)00335-2/

[1] Cîrstea, F.-C.; Rădulescu, V. Existence and uniqueness of blow-up solutions for a class of logistic equations, Comm. Contemp. Math., Volume 4 (2002), pp. 559-586

[2] Cîrstea, F.-C.; Rădulescu, V. Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 447-452

[3] M. Ghergu, V. Rădulescu, Sublinear singular elliptic problems with two parameters, J. Differential Equations, in press

[4] Gui, C.; Lin, F.H. Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, Volume 123 (1993), pp. 1021-1029

[5] Hörmander, L. The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, 1983

[6] Lazer, A.C.; McKenna, P.J. On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., Volume 3 (1991), pp. 720-730

[7] Mironescu, P.; Rădulescu, V. The study of a bifurcation problem associated to an asymptotically linear function, Nonlinear Anal., Volume 26 (1996), pp. 857-875

[8] Shi, J.; Yao, M. On a singular nonlinear semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A, Volume 128 (1998), pp. 1389-1401

Cité par Sources :