Zhang, Guoping; Pankov, Alexander Standing waves of the discrete nonlinear Schrödinger equations with growing potentials. (English) Zbl 1168.35437 Commun. Math. Anal. 5, No. 2, 38-49 (2008). 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. Cited in 1 ReviewCited in 34 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations 47J30 Variational methods involving nonlinear operators Keywords:discrete nonlinear Schrödinger equation; standing wave; variation methods; Nehari manifold PDFBibTeX XMLCite \textit{G. Zhang} and \textit{A. Pankov}, Commun. Math. Anal. 5, No. 2, 38--49 (2008; Zbl 1168.35437)