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: -x2u″ = 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. © 1997 Academic Press.