We investigate the existence and bifurcation of solutions of a model nonlinear degenerate elliptic differential equation: $-x^r\mathrm{\Delta }u=\lambda u+|u{|}^{p-1}u$ in $(0;1)$; $u(0)=u(1)=0$. This model is related to a simplified version of the nonlinear Wheeler-De Witt equation as it appears in quantum cosmological models, see [8, 9, 10].