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

Abstract

AbstractWe consider an elliptic problem of the type $$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr {\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$ where Ω is a bounded Lipschitz domain in ℝN with a cylindrical symmetry, ν stands for the outer normal and $\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $.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 $$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$ 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 L2 (Ω) × L2(Γ2).