Criterion toward understanding non-constant solutions to <i>p</i>-Laplace Neumann boundary value problem

We consider a p-Laplace equation ∆pV + h(V) = 0, with an arbitrary C1-nonlinearity h, in a bounded domain and supplemented with the Neumann boundary condition. We prove a necessary condition for zeros of h = h(V) to be touched by non-constant solutions to this problem.