Nonlinear Schrodinger equation with unbounded or vanishing potentials: Solutions concentrating on lower dimensional spheres

We study positive bound states for the equation -epsilon(2) Delta u + V(x)u = K(x)f(u), x is an element of R-N. where epsilon > 0 is a real parameter and V and K are radial positive potentials. We are especially interested in solutions which concentrate on a k-dimensional sphere, 1 <= k <= N - 1, as epsilon -> 0. We adopt a purely variational approach which allows us to consider broader classes of potentials than those treated in previous works. For example, V and K might be singular at the origin or vanish superquadratically at infinity. (C) 2011 Elsevier Inc. All rights reserved.