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Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials. (English) Zbl 1245.35036
Summary: For singularly perturbed Schrödinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeros of the potentials. The results generalize some recent work of A. Ambrosetti, A. Malchiodi and W.-M. Ni [C. R., Math., Acad. Sci. Paris 335, No. 2, 145–150 (2002; Zbl 1072.35068)] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in the paper cited above.
MSC:
35J60Nonlinear elliptic equations
35B25Singular perturbations (PDE)
47J30Variational methods (nonlinear operator equations)