Morse Index and Symmetry for Elliptic Problems with Nonlinear Mixed   Boundary Conditions
  Under a Morse index condition we prove symmetry results for solutions of a nonlinear mixed boundary condition elliptic problem. As an intermediate step we relate the Morse index of a solution to a mixed boundary condition linear eigenvalue problem for which we construct sequences of eigenvalues and provide variational characterization of them. 
Lucio Damascelli and Filomena Pacella
math.AP