Abstract
Using variational methods, we study the existence and multiplicity of homoclinic solutions for a class of discrete Schrödinger equations in infinite m-dimensional lattices with nonlinearities being superlinear at infinity. Our results generalize some existing results in the literature by using some weaker conditions.
Highlights
Introduction and main resultsThe discrete nonlinear Schrödinger equation is a very important discrete model, which has many important applications in many fields, such as nonlinear optics [ ], biomolecular chains [ ], Bose-Einstein condensates [ ], and so on.In general, discrete nonlinear Schrödinger equation can be divided into two different cases, the periodic and nonperiodic cases
Discrete nonlinear Schrödinger equation can be divided into two different cases, the periodic and nonperiodic cases
A few results are about the nonperiodic cases, such as [ – ]; in paticular, the papers [, ] are only about the case of onedimensional lattice (n ∈ Z)
Summary
Introduction and main resultsThe discrete nonlinear Schrödinger equation is a very important discrete model, which has many important applications in many fields, such as nonlinear optics [ ], biomolecular chains [ ], Bose-Einstein condensates [ ], and so on.In general, discrete nonlinear Schrödinger equation can be divided into two different cases, the periodic and nonperiodic cases. Inspired by the papers mentioned, we study homoclinic solutions (lim|n|=|n |+|n |+···+|nm|→∞ un = ) of the following nonperiodic discrete nonlinear equation: ) possesses at least one nontrivial homoclinic solution u if conditions (V ) and (F )-(F ) hold. ) has infinitely many nontrivial homoclinic solutions if conditions (V ) and (F )-(F ) hold and fn(–s) = –fn(s) for all (n, s) ∈ Zm × R.
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