Morse index and symmetry for elliptic problems with nonlinear mixed boundary conditions

We consider an elliptic problem of the type where Ω is a bounded Lipschitz domain in ℝ N with a cylindrical symmetry, ν stands for the outer normal and. Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem. As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem For this one, we construct sequences of eigenvalues and provide variational characterization of them, following the usual approach for the Dirichlet case, but working in the product Hilbert space L 2 (Ω) × L 2 (Γ2).