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.
Let Ω be a bounded domain of ℝN(N ≥ 1) whose boundary ∂Ω is a C2 compact manifolds. In the present paper we shall study a variational problem relating the weighted Hardy inequalities established in . As weights we adopt powers of the distance function 𝛿(x) to the boundary ∂Ω.