Harmonic functions with nonlinear Neumann boundary condition and their Morse indices

摘要

We consider the solutions of a nonlinear Neumann elliptic equation Delta u = 0 in Omega, partial derivative u/partial derivative v = f(x, u) on partial derivative Omega, where Omega is a bounded open smooth domain in R-N, N >= 2 and f satisfies super-linear and subcritical growth conditions. We prove that L-infinity-bounds on solutions are equivalent to bounds on their Morse indices. (C) 2017 Elsevier Ltd. All rights reserved.