Normalized solutions for the discrete Schrödinger equations

Abstract

AbstractIn the present paper, we consider the existence of solutions with a prescribed $l^{2}$ l 2 -norm for the following discrete Schrödinger equations, $$ \textstyle\begin{cases} -\Delta ^{2} u_{k-1}-f(u_{k})= \lambda u_{k} \quad k\in \mathbb{Z}, \\ \sum_{k\in \mathbb{Z}} \vert u_{k} \vert ^{2}=\alpha ^{2}, \end{cases} $$ { − Δ 2 u k − 1 − f ( u k ) = λ u k k ∈ Z , ∑ k ∈ Z | u k | 2 = α 2 , where $\Delta ^{2} u_{k-1}=u_{k+1}+u_{k-1}-2u_{k}$ Δ 2 u k − 1 = u k + 1 + u k − 1 − 2 u k , $f\in C(\mathbb{R}) $ f ∈ C ( R ) , α is a fixed constant, and $\lambda \in \mathbb{R}$ λ ∈ R arises as a Lagrange multiplier. To get the solutions, we investigate the corresponding minimizing problem with the $l^{2}$ l 2 -norm constraint: $$ E_{\alpha}=\inf \biggl\{ \frac{1}{2}\sum \vert \Delta u_{k-1} \vert ^{2}-\sum F(u_{k}): \sum \vert u_{k} \vert ^{2}=\alpha ^{2} \biggr\} . $$ E α = inf { 1 2 ∑ | Δ u k − 1 | 2 − ∑ F ( u k ) : ∑ | u k | 2 = α 2 } . An elaborative analysis on a minimizing sequence with respect to $E_{\alpha}$ E α is obtained. We prove that there is a constant $\alpha _{0}\geq 0$ α 0 ≥ 0 such that there exists a global minimizer if $\alpha >\alpha _{0}$ α > α 0 , and there exists no global minimizer if $\alpha <\alpha _{0}$ α < α 0 . It seems that it is the first time to consider the solution with a prescribed $l^{2}$ l 2 -norm of the discrete Schrödinger equations.