Infinitely many solutions for the discrete Schr\xf6dinger equations with a nonlocal term

Qilin Xie,Huafeng Xiao

doi:10.1186/s13661-022-01583-4

Abstract

In the present paper, we consider the following discrete Schrödinger equations −(a+b∑k∈Z|Δuk−1|2)Δ2uk−1+Vkuk=fk(uk)k∈Z,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ - \\biggl(a+b\\sum_{k\\in \\mathbf{Z}} \\vert \\Delta u_{k-1} \\vert ^{2} \\biggr) \\Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \\quad k\\in \\mathbf{Z}, $$\\end{document} where a, b are two positive constants and V={V_{k}} is a positive potential. Delta u_{k-1}=u_{k}-u_{k-1} and Delta ^{2}=Delta (Delta ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities {f_{k}} satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.

Highlights

In the present paper, we consider the following discrete Schrödinger equations– a + b | uk–1|2 2uk–1 + Vkuk = fk(uk) k ∈ Z, (1) k∈ZThe discrete Schrödinger equations play a significant role in many areas, such as nonlinear optics [7], biomolecular chains [11] and Bose–Einstein condensates [16]
If a = 1 and b = 0, problem (1) reduces to the classical discrete Schrödinger equations, which have been extensively studied by many authors in the past several decades
In Ma and Guo [17] and Zhang and Pankov [32], the authors studied the nontrivial solution of discrete Schrödinger equations with a coercive potential by variation methods

Summary

Introduction

The discrete Schrödinger equations play a significant role in many areas, such as nonlinear optics [7], biomolecular chains [11] and Bose–Einstein condensates [16]. Theorem 1.2 If (V1), (f1), (f2), (f3) and (f4) hold, problem (1) possesses infinitely many nontrivial solutions {un} in E with high energies, i.e., I(un) → +∞ as n → ∞. The above two results obtained infinitely many high-energy solutions are strictly dependent on the 4-superlinear growth assumption. Theorem may be invalid without the superlinear assumption In this case, we try to establish the existence of infinitely many small solutions by the new version of the Symmetric. Theorem 1.3 If (V1) and (f5) hold, problem (1) possesses infinitely many nontrivial solutions {un} in E with small energy, i.e., I(un) → 0– as n → ∞. Combining with (3), we have Fk(t) ≥ –Ct2 for any k ∈ Z and t ∈ R

Note that

Xj and

It follows from the above three inequalities that