Normalized solutions for the discrete Schrödinger equations

Abstract

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

Keywords

• Normalized solutions
• Lagrange multiplier
• Non-vanishing
• Minimizer