We investigate the existence and bifurcation of solutions of a model nonlinear degenerate elliptic differential equation: $-x^r\Delta u=\lambda u+|u|^{p-1}u$ 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].