Partial Differential Equations
Bifurcation for a class of singular elliptic problems with quadratic convection term
Comptes Rendus. Mathématique, Volume 338 (2004) no. 11, pp. 831-836.

We study the bifurcation problem −Δu=g(u)+λ|∇u|2+μ in Ω,u=0 on Ω, where λ,μ⩾0 and Ω is a smooth bounded domain in 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=limt→+∞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.

On étudie le problème elliptique de bifurcation −Δu=g(u)+λ|∇u|2+μ dans Ω,u=0 sur Ω, où λ,μ⩾0 et Ω est un domaine borné régulier de 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=limt→+∞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.

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DOI: 10.1016/j.crma.2004.03.020
Ghergu, Marius 1; Rădulescu, Vicenţiu 1

1 University of Craiova, Department of Mathematics, Street A.I. Cuza No. 13, 200585 Craiova, Romania
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Ghergu, Marius; Rădulescu, Vicenţiu. 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. http://www.numdam.org/articles/10.1016/j.crma.2004.03.020/

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