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Ground states of a system of nonlinear Schrödinger equations with periodic potentials. (English) Zbl 1353.35267

Summary: We are concerned with a system of coupled Schrödinger equations \[ -\Delta u_i + V_i(x)u_i = \partial_{u_i} F(x,u) \;\mathrm{on} \;\mathbb R^N, i=1,2,\dots,K, \] where \(F\) and \(V_i\) are periodic in \(x\) and \(0 \notin \sigma(-\Delta+V_i) \) for \(i=1,2,\dots,K\), where \(\sigma(-\Delta+V_i) \) stands for the spectrum of the Schrödinger operator \(-\Delta+V_i\). We impose general assumptions on the nonlinearity \(F\) with the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with the system on a Nehari-Pankov manifold. Our approach is based on a new linking-type result involving the Nehari-Pankov manifold.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q60 PDEs in connection with optics and electromagnetic theory
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J47 Second-order elliptic systems
35C08 Soliton solutions
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