We investigate the existence, multiplicity and bifurcation of solutions of a model nonlinear degenerate elliptic differential equation: &x 2 u"=*u+ |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.
1997 Academic Press
Nous e tudions ici l'existence, multiplicite et proprie te s de bifurcation des solutions d'un probleร me elliptique de ge ne re : &x 2 u"=*u+ |u| p&1 u in (0, 1); u(0)=u(1)=0. Ce probleร me modeร le est proche d'une version simplifie e et non-line aire de l'e quation de Wheeler DeWitt, utilise e dans des modeร les de Cosmologie quantique. Nous prouvons l'existence d'une infinite de branches de solutions qui bifurquent aร partir de l'infimum du spectre continu de l'ope rateur line aire correspondant. 1997 Academic Press
where p>1, * is a real parameter and u=u(x) is defined as a continuous function on [0, 1]. We address here the questions of existence, multiplicity and bifurcation of connected components of solutions of (1.1). This equation is related to the Wheeler DeWitt equation as we explain below.