Existence and Bifurcation of Solutions for an Elliptic Degenerate Problem

Abstract We investigate the existence, multiplicity and bifurcation of solutions of a model nonlinear degenerate elliptic differential equation: − x 2 u ″= λu +| u | p −1 u in (0, 1); u (0)= u (1)=0. This model is related to a simplified version of the nonlinear Wheeler–DeWitt equation as it appears in quantum cosmological models. We prove the existence of multiple positive solutions. More precisely, we show that there exists an infinite number of connected branches of solutions which bifurcate from the bottom of the essential spectrum of the corresponding linear operator. Nous etudions ici l'existence, multiplicite et proprietes de bifurcation des solutions d'un probleme elliptique degenere: − x 2 u ″= λu +| u | p −1 u in (0, 1); u (0)= u (1)=0. Ce probleme modele est proche d'une version simplifiee et non-lineaire de l'equation de Wheeler–DeWitt, utilisee dans des modeles de Cosmologie quantique. Nous prouvons l'existence d'une infinite de branches de solutions qui bifurquent a partir de l'infimum du spectre continu de l'operateur lineaire correspondant.