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Standing waves of nonlinear Schrödinger equations with the gauge field. (English) Zbl 1248.35193
Summary: We study standing waves for nonlinear Schrödinger equations with the gauge field. Some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field \(A_{\mu }\) in terms of complex scalar field \(\phi \). We also show that there exists no standing wave for certain range of parameters by establishing a new inequality of Sobolev type.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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