Perturbed Schrödinger lattice systems: existence of homoclinic solutions

Abstract

We study a class of Schrödinger lattice systems with sublinear nonlinearities and perturbed terms. We get an interesting result that the systems do not have nontrivial homoclinic solutions if the perturbed terms are removed, but the systems have ground state homoclinic solutions if the perturbed terms are added. Besides, we also study the continuity of the homoclinic solutions in the perturbation terms at zero. To the best of our knowledge, there is no published result focusing on the perturbed Schrödinger lattice systems.

Infinitely many homoclinic solutions for sublinear and nonperiodic Schrödinger lattice systems

“…However, there are only few results about the nonperiodic Schrödinger lattice systems [5,9,15,16,22,23]. In particular, in [3,6,7] the authors recently obtained the existence and multiplicity of homoclinic solutions for a class of Schrödinger lattice systems with perturbed terms.…”

Multiple solutions for Schrödinger lattice systems with asymptotically linear terms and perturbed terms

“…To the best of our knowledge, there are only four published papers [3,4,5,6] focusing on PSL systems. The authors [3,4] only obtained the existence of one solution of (1.1) with g n (s) being sub-linear and asymptotically-linear as |s| → ∞, respectively, and the authors [5,6] obtained the existence of two solutions of (1.1) with g n (s) being super-linear and sub-linear as |s| → ∞, respectively. However, in this paper, we obtain the existence of two solutions of (1.1) with g n (s) being asymptotically-linear as |s| → ∞.…”