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Bifurcation and asymptotics for the Lane–Emden–Fowler equation. (English. Abridged French version) Zbl 1073.35087

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.

MSC:

35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
47J15 Abstract bifurcation theory involving nonlinear operators
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