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Standing waves for periodic discrete nonlinear Schrödinger equations with asymptotically linear terms. (English) Zbl 1359.35175

Summary: In this paper we study the existence of nontrivial solutions for the periodic discrete nonlinear equation \[ Lu_n-\omega u_n=f_n(u_n), \] where \[ Lu_n=a_{n+1}u_{n+1}+a_{n-1}u_{n-1}+b_nu_n \] is the discrete Laplacian in one spatial dimension. The given real-valued sequences \(a_n\), \(b_n\) are assumed to be \(N\)-periodic in \(n\), i.e., \(a_{n+N}=a_n\), \(b_{n+N}=b_n\). The nonlinearity \(f_n(t)\) is \(N\)-periodic in \(n\) and asymptotically linear at infinity. We show that, if \(\omega\) is in the spectrum gap of \(L\), there is a nontrivial solution. The proof is based on the strongly indefinite functional critical points theorem developed recently.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
39A12 Discrete version of topics in analysis
39A70 Difference operators
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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