Normalized solutions for the discrete Schrödinger equations
نویسندگان
چکیده
Abstract In the present paper, we consider existence of solutions with a prescribed $l^{2}$ l 2 -norm for 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 ^{2}=\alpha ^{2}, \end{cases} { − Δ u k 1 f ( ) = λ ∈ Z , ∑ | α where $\Delta u_{k-1}=u_{k+1}+u_{k-1}-2u_{k}$ + , $f\in C(\mathbb{R}) $ C R α is fixed constant, and $\lambda \in \mathbb{R}$ arises as Lagrange multiplier. To get solutions, investigate corresponding minimizing problem constraint: E_{\alpha}=\inf \biggl\{ \frac{1}{2}\sum \Delta u_{k-1} ^{2}-\sum F(u_{k}): \sum \biggr\} . E inf F : } . An elaborative analysis on sequence respect to $E_{\alpha}$ obtained. We prove that there constant $\alpha _{0}\geq 0$ 0 ≥ such exists global minimizer if >\alpha _{0}$ > no <\alpha < It seems it first time solution equations.
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ژورنال
عنوان ژورنال: Boundary Value Problems
سال: 2023
ISSN: ['1687-2770', '1687-2762']
DOI: https://doi.org/10.1186/s13661-023-01754-x