We study the bifurcation problem −Δu=g(u)+λ|∇u|2+μ in on , where λ,μ⩾0 and is a smooth bounded domain in . 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 . 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 sur , où λ,μ⩾0 et est un domaine borné régulier de . 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 . 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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Marius Ghergu 1; Vicenţiu Rădulescu 1
@article{CRMATH_2004__338_11_831_0, 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}, }
TY - JOUR AU - Marius Ghergu AU - Vicenţiu Rădulescu TI - Bifurcation for a class of singular elliptic problems with quadratic convection term JO - Comptes Rendus. Mathématique PY - 2004 SP - 831 EP - 836 VL - 338 IS - 11 PB - Elsevier DO - 10.1016/j.crma.2004.03.020 LA - en ID - CRMATH_2004__338_11_831_0 ER -
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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