Minimal potential results for Schrodinger equations with Neumann boundary conditions

We consider the boundary value problem -Delta(p)u = V vertical bar u vertical bar(p-2) u - C, where u is an element of W-1,W- p (D) is assumed to satisfy Neumann boundary conditions, and D is a bounded domain in R-n. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for the product of a sharp Sobolev constant and an L-p norm of V. When p = n, Orlicz norms are used. In many cases, these inequalities are best possible. Applications to linear and non-linear eigenvalue problems are also discussed.