Bifurcation of solutions for an elliptic degenerate problem (Q1431396)

The authors establish existence and bifurcation results for the degenerate nonlinear elliptic equation \(-x^r\Delta u=\lambda u+| u| ^{p-1}u\) in \((0,1)\) under the Dirichlet condition \(u(0)=u(1)=0\), where \(r>0\), \(p>1\), and \(\lambda\) is a real parameter. Using the Lyapunov-Schmidt reduction method combined with the Banach contraction principle and standard tools related to the Brouwer degree, the authors prove the existence of a sequence of real eigenvalues. The model studied in this paper is related to a simplified version of the nonlinear Wheeler-De Witt equation that describes some phenomena arising in quantum cosmology.