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

We study positive bound states for the equation−ε2Δu+V(x)u=K(x)f(u),x∈RN, where ε>0ε>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−11⩽k⩽N−1, as ε→0ε→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.