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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.
35J60Nonlinear elliptic equations
35B25Singular perturbations (PDE)
47J30Variational methods (nonlinear operator equations)