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 which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in .
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Jaeyoung Byeon, Zhi-Qiang Wang, Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials. J. Eur. Math. Soc. 8 (2006), no. 2, pp. 217–228DOI 10.4171/JEMS/48