Bifurcation for a class of singular elliptic problems with quadratic convection term

Marius Ghergu; Vicenţiu Rădulescu

doi:10.1016/j.crma.2004.03.020

Partial Differential Equations

Bifurcation for a class of singular elliptic problems with quadratic convection term
[Bifurcation pour une classe de problèmes elliptiques singuliers à terme quadratique de convection]

Présenté par : Philippe G. Ciarlet

Marius Ghergu ¹ ; Vicenţiu Rădulescu ¹

¹ University of Craiova, Department of Mathematics, Street A.I. Cuza No. 13, 200585 Craiova, Romania

Comptes Rendus. Mathématique, Volume 338 (2004) no. 11, pp. 831-836.

Résumé
Abstract

On étudie le problème elliptique de bifurcation −Δu=g(u)+λ|∇u|²+μ dans $Ω, u = 0$ sur $\partial Ω$ , où λ,μ⩾0 et $Ω$ est un domaine borné régulier de $ℝ^{N}$ . Le caractère singulier de ce problème est donné par la nonlinéarité g, qui est décroissante et non bornée autour de l'origine. Dans cette Note on montre que le problème ci-dessus admet une solution classique positive (qui, de plus, est unique) si et seulement si λ(a+μ)<λ₁, où a=lim_t→+∞g(t) et λ₁ est la première valeur propre de l'opérateur de Laplace dans $H_{0}^{1} (Ω)$ . Nous établissons également le taux de décroissance de cette solution, ainsi qu'un résultat d'explosion autour du paramètre de bifurcation.

We study the bifurcation problem −Δu=g(u)+λ|∇u|²+μ in $Ω, u = 0$ on $\partial Ω$ , where λ,μ⩾0 and $Ω$ is a smooth bounded domain in $ℝ^{N}$ . The singular character of the problem is given by the nonlinearity g which is assumed to be decreasing and unbounded around the origin. In this Note we prove that the above problem has a positive classical solution (which is unique) if and only if λ(a+μ)<λ₁, where a=lim_t→+∞g(t) and λ₁ is the first eigenvalue of the Laplace operator in $H_{0}^{1} (Ω)$ . We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter.

Reçu le : 2004-02-23
Accepté le : 2004-03-01
Publié le : 2004-04-17

DOI : 10.1016/j.crma.2004.03.020

Affiliations des auteurs :

Marius Ghergu ¹ ; Vicenţiu Rădulescu ¹

¹ University of Craiova, Department of Mathematics, Street A.I. Cuza No. 13, 200585 Craiova, Romania

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     author = {Marius Ghergu and Vicen\c{t}iu R\u{a}dulescu},
     title = {Bifurcation for a class of singular elliptic problems with quadratic convection term},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {831--836},
     publisher = {Elsevier},
     volume = {338},
     number = {11},
     year = {2004},
     doi = {10.1016/j.crma.2004.03.020},
     language = {en},
}

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Marius Ghergu; Vicenţiu Rădulescu. Bifurcation for a class of singular elliptic problems with quadratic convection term. Comptes Rendus. Mathématique, Volume 338 (2004) no. 11, pp. 831-836. doi : 10.1016/j.crma.2004.03.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.03.020/

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