Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane–Emden–Fowler type

• Published in: Journal de mathématiques pures et appliquées Vol. 84; no. 4; pp. 493 - 508
• Main Authors: Cîrstea, Florica, Ghergu, Marius, Rădulescu, Vicenţiu
• Format: Journal Article
• Language: English
• Published: Paris Elsevier SAS 01.04.2005; Elsevier
• Subjects: Bifurcation problem; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Maximum principle; Partial differential equations; Sciences and techniques of general use; Singular elliptic equation; Sublinear perturbation; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Sublinear perturbation; 58J55; 35B25; Bifurcation problem; 35J60; Maximum principle; 35A20; 35B50; 58K55; Singular elliptic equation; Solution uniqueness; Singular equation; Bifurcation; Mathematics; Perturbation; Elliptic equation; 35J60: 58J55; Positive solution; 58K55 Singular elliptic equation; Nonlinearity
• ISSN: 0021-7824
• DOI: 10.1016/j.matpur.2004.09.005

We are concerned with the generalized Lane–Emden–Fowler equation − Δ u = λ f ( u ) + a ( x ) g ( u ) in Ω, subject to the Dirichlet boundary condition u | ∂ Ω = 0 , where Ω is a smooth bounded domain in R N , λ ∈ R , a is a nonnegative Hölder function, and f is positive and nondecreasing such that the mapping f ( s ) / s is nonincreasing in ( 0 , ∞ ) . Here, the singular character of the problem is given by the nonlinearity g which is assumed to be unbounded around the origin. We distinguish two different cases which are related to the sublinear (respectively linear) growth of f at infinity. On étudie l'équation de Lane–Emden–Fowler généralisée − Δ u = λ f ( u ) + a ( x ) g ( u ) dans Ω avec une condition de Dirichlet u | ∂ Ω = 0 , où Ω ⊂ R N est un domaine borné régulier, λ ∈ R , a est une fonction de Hölder non-négative et f est 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. Sous des hypothèses différentes concernant f et g, on discute l'existence et l'unicité d'une solution classique positive. On distingue deux cas différents, correspondant aux situations où f a une croissance sous-linéaire ou linéaire à l'infini.