Ghergu, Marius; Rădulescu, Vicenţiu D. Bifurcation and asymptotics for the Lane–Emden–Fowler equation. (English. Abridged French version) Zbl 1073.35087 C. R., Math., Acad. Sci. Paris 337, No. 4, 259-264 (2003). Summary: We are concerned with the Lane–Emden–Fowler equation \(-{\Delta}u={\lambda}f(u)+a(x)g(u)\) in \(\varOmega\), subject to the Dirichlet boundary condition \(u=0\) on \(\partial \varOmega\), where \(\varOmega \subset \mathbb R^N\) is a smooth bounded domain, \(\lambda\) is a positive parameter, \(a:\overline{\varOmega}\to [o,{\infty})\) is a Hölder function, and \(f\) is a positive nondecreasing continuous function such that \(f(s)/s\) is nonincreasing in \((0,{\infty})\). 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. Cited in 33 Documents MSC: 35J60 Nonlinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 47J15 Abstract bifurcation theory involving nonlinear operators PDF BibTeX XML Cite \textit{M. Ghergu} and \textit{V. D. Rădulescu}, C. R., Math., Acad. Sci. Paris 337, No. 4, 259--264 (2003; Zbl 1073.35087) Full Text: DOI OpenURL References: [1] Cı̂rstea, F.-C.; Rădulescu, V., Existence and uniqueness of blow-up solutions for a class of logistic equations, Comm. contemp. math., 4, 559-586, (2002) [2] Cı̂rstea, F.-C.; Rădulescu, V., Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. acad. sci. Paris, ser. I, 335, 447-452, (2002) [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, 123, 1021-1029, (1993) · Zbl 0805.35032 [5] Hörmander, L., The analysis of linear partial differential operators I, (1983), Springer-Verlag Berlin [6] Lazer, A.C.; McKenna, P.J., On a singular nonlinear elliptic boundary value problem, Proc. amer. math. soc., 3, 720-730, (1991) · Zbl 0727.35057 [7] Mironescu, P.; Rădulescu, V., The study of a bifurcation problem associated to an asymptotically linear function, Nonlinear anal., 26, 857-875, (1996) · Zbl 0842.35008 [8] Shi, J.; Yao, M., On a singular nonlinear semilinear elliptic problem, Proc. roy. soc. Edinburgh sect. A, 128, 1389-1401, (1998) · Zbl 0919.35044 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.