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Standing waves of the discrete nonlinear Schrödinger equations with growing potentials. (English) Zbl 1168.35437

Summary: We investigate the existence of nontrivial standing wave solution of the discrete nonlinear Schrödinger equation with the growing potential at infinity. Firstly we derive a discrete version of compact embedding theorem. Then combining the Nehari manifold approach and the compact embedding theorem we show the existence of nontrivial standing wave solution without Palais-Smale condition. We also prove the exponential decay of the standing wave solutions.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
47J30 Variational methods involving nonlinear operators
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