Harmonic functions with nonlinear Neumann boundary condition and their Morse indices

We consider the solutions of a nonlinear Neumann elliptic equation Δu=0 in Ω, ∂u/∂ν=f(x,u) on ∂Ω, where Ω is a bounded open smooth domain in RN, N≥2 and f satisfies super-linear and subcritical growth conditions. We prove that L∞-bounds on solutions are equivalent to bounds on their Morse indices.