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On nonlinear Schrödinger equations with attractive inverse-power potentials. (English) Zbl 1477.35237

Summary: We study the Cauchy problem for nonlinear Schrödinger equations with attractive inverse-power potentials. By using variational arguments, we first determine a sharp threshold of global well-posedness and blow-up for the equation in the mass-supercritical case. We next study the existence and orbital stability of standing waves for the problem in the mass-subcritical and mass-critical cases. In the mass-critical case, we give a detailed description of the blow-up behavior of standing waves when the mass tends to a critical value.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
35J35 Variational methods for higher-order elliptic equations
35B44 Blow-up in context of PDEs
35B35 Stability in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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