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

Abstract

Abstract In the present paper, we consider the following discrete Schrödinger equations − ( a + b ∑ k ∈ Z | Δ u k − 1 | 2 ) Δ 2 u k − 1 + V k u k = f k ( u k ) k ∈ Z , $$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$ where a, b are two positive constants and V = { V k } $V=\{V_{k}\}$ is a positive potential. Δ u k − 1 = u k − u k − 1 $\Delta u_{k-1}=u_{k}-u_{k-1}$ and Δ 2 = Δ ( Δ ) $\Delta ^{2}=\Delta (\Delta )$ 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 } $\{f_{k}\}$ 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.

Keywords

• Solutions
• Discrete Schrödinger equations
• Kirchhoff type