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

We consider an elliptic problem of the type-Delta u=f(x,u u=0 in Omega on Gamma(1) partial derivative u/partial derivative v = g(x, u) on Gamma(2)where O is a bounded Lipschitz domain in R-N with a cylindrical symmetry, nu stands for the outer normal and partial derivative Omega = (Gamma(1)) over bar boolean OR (Gamma(2)) over bar.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.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-Delta w(j) + c(x)w(j) = lambda(j)w(j) in Omega on Gamma(1) partial derivative wj/partial derivative v + d(x)w(j) on Gamma(2)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(Omega) x L-2 (Gamma(2)).