Bifurcation and asymptotics for the Lane–Emden–Fowler equation

Ghergu, Marius; Rădulescu, Vicenţiu D.

Comptes Rendus. Mathématique, Tome 337 (2003) no. 4, pp. 259-264.

On considère l'équation de Lane–Emden–Fowler −Δu=λf(u)+a(x)g(u) dans $\Omega$ avec une condition de Dirichlet u=0 sur $\partial \Omega ,$ où $\Omega \subset {ℝ}^N$ est un domaine borné régulier, λ est un paramètre positif, $a:\overline{\Omega }\to [0,\infty )$ est une fonction de Hölder et f est une fonction continue, 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. Dans cette Note nous étudions l'existence et l'unicité d'une solution positive et nous établissons également son taux de décroissance vers 0 autour du bord. La méthode de démonstration repose sur le principe du maximum et sur des estimations elliptiques.

We are concerned with the Lane–Emden–Fowler equation −Δu=λf(u)+a(x)g(u) in $\Omega$, subject to the Dirichlet boundary condition u=0 on $\partial \Omega ,$ where $\Omega \subset {ℝ}^N$ is a smooth bounded domain, λ is a positive parameter, $a:\overline{\Omega }\to [0,\infty )$ is a Hölder function, and f is a positive nondecreasing continuous function such that f(s)/s is nonincreasing in (0,∞). 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.

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