Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials

For singularly perturbed Schr ¨ odinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti-Malchiodi-Ni (3) which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in (3).