Infinitely many solutions for the discrete Schrödinger equations with a nonlocal term

Abstract

In the present paper, we consider the following discrete Schrödinger equations $$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$ − ( a + b ∑ k ∈ Z | Δ u k − 1 | 2 ) Δ 2 u k − 1 + V k u k = f k ( u k ) k ∈ Z , where a, b are two positive constants and $V=\{V_{k}\}$ V = { V k } is a positive potential. $\Delta u_{k-1}=u_{k}-u_{k-1}$ Δ u k − 1 = u k − u k − 1 and $\Delta ^{2}=\Delta (\Delta )$ Δ 2 = Δ ( Δ ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities $\{f_{k}\}$ { 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.

Infinitely Many Solutions for Partial Discrete Kirchhoff Type Problems Involving p-Laplacian

“…In recent years, an increasing number of researchers have focused on the Schrödinger equations, especially for the DKP. We refer to [47,48] and their references for the relevant studies.…”