Radial solutions concentrating on spheres of nonlinear Schrodinger equations with vanishing potentials

We prove the existence of radial solutions of -epsilon(2) Delta u + V(vertical bar x vertical bar)u = u(p), x is an element of R-n, u is an element of W-1,W-2(R-n), u > 0, concentrating on a sphere for potentials which might be zero and might decay to zero at infinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov-Schmidt reduction.