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Semiclassical solutions for a class of Schrödinger system with magnetic potentials. (English) Zbl 1311.35302

Summary: This paper is concerned with the following nonlinear Schrödinger system with magnetic potentials \[ \begin{cases} (-i\varepsilon\nabla+A(x))^2u+V(x)u=H_u(x,u,v),\quad x\in\mathbb R^N,\\(-i\varepsilon\nabla+A(x))^2v+V(x)v=-H_v(x,u,v),\quad x\in\mathbb R^N,\end{cases} \] where \(N\geqslant 3,\varepsilon\) is a small parameter, \(A:\mathbb R^N\to\mathbb R^N\) is the magnetic vector potential and \(V:\mathbb R^N\to\mathbb R\) is the electric potential. By applying generalized linking theorems for strongly indefinite functionals, we establish the existence and multiplicity of semiclassical solutions for superquadratic and subcritical nonlinearity.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
78A02 Foundations in optics and electromagnetic theory
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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