论文标题
低维度的耦合分数Schrodinger系统的归一化解决方案
Normalized solutions for a coupled fractional schrodinger system in low dimensions
论文作者
论文摘要
我们考虑以下耦合分数schrödinger系统：\ begin {equination*} \ left \ {\ {\ begin {aligned}＆（ - δ）^su+λ_1U=μ_1| u | u |^{2pp-2 {2pp-2} ＆（δ）^sv+λ_2V=μ_2| v |^{2p-2} v+β| $ 2S <n \ le 4s $和$ 1+\ frac {2s} {n} <p <\ frac {n} {n-2s} $在以下约束\ begin {align*} \ ingign*} \ int _ { \ int _ {\ mathbb {r}^n} | v |^2dx = a_2^2。 \ end {align*}假设参数$μ_1，μ_2，a_1，a_2 $是固定数量，我们证明存在归一化解决方案，用于耦合参数$β> 0 $的不同范围。
We consider the following coupled fractional Schrödinger system: \begin{equation*} \left\{ \begin{aligned} &(-Δ)^su+λ_1u=μ_1|u|^{2p-2}u+β|v|^p|u|^{p-2}u\\ &(-Δ)^sv+λ_2v=μ_2|v|^{2p-2}v+β|u|^p|v|^{p-2}v\\ \end{aligned} \right.\quad\text{in}~{\mathbb{R}^N}, \end{equation*} with $0<s<1$, $2s<N\le 4s$ and $1+\frac{2s}{N}<p<\frac{N}{N-2s}$, under the following constraint \begin{align*} \int_{\mathbb{R}^N}|u|^2dx=a_1^2\quad\text{and}\quad \int_{\mathbb{R}^N}|v|^2dx=a_2^2. \end{align*} Assuming that the parameters $μ_1,μ_2,a_1, a_2$ are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter $β>0$ .
