论文标题
存在分数耦合的Hartree-fock类型系统的标准化解决方案
Existence of normalized solutions for fractional coupled Hartree-Fock type system
论文作者
论文摘要
在本文中，我们考虑了以下分数耦合的hartree-fock类型系统\ begin {align*} \ left \ left \ { | v |^r）| U |^{r-2} u \\＆（ - δ）^s v+v_2（x）v+λ_2V=μ_2（i_α\ star | v |^q）| x \ in \ mathbb {r}^n，\ end {align*}在约束\ begin {align*} \ int _ {\ mathbb {\ mathbb {r}^n} | u |^2 = a^2 = a^2，〜\ \ int _ { \ end {align*}其中$ s \ in（0,1），〜n \ ge3，〜μ_1> 0，〜μ_2> 0，〜β> 0，〜β> 0，〜α \ in（0，n），〜1+\fracα{n} $i_α（x）= | x |^{α-n} $。在$ n，α，p，q $和$ r $的某些限制下，我们给出了$ p，q，r \ le 1+ \ frac {α+2S} {n} $的归一化解决方案的积极性。
In this paper, we consider the existence of solutions for the following fractional coupled Hartree-Fock type system \begin{align*} \left\{\begin{aligned} &(-Δ)^s u+V_1(x)u+λ_1u=μ_1(I_α\star |u|^p)|u|^{p-2}u+β(I_α\star |v|^r)|u|^{r-2}u\\ &(-Δ)^s v+V_2(x)v+λ_2v=μ_2(I_α\star |v|^q)|v|^{q-2}v+β(I_α\star |u|^r)|v|^{r-2}v \end{aligned} \right.~\quad x\in\mathbb{R}^N, \end{align*} under the constraint \begin{align*} \int_{\mathbb{R}^N}|u|^2=a^2,~\int_{\mathbb{R}^N}|v|^2=b^2. \end{align*} where $s\in(0,1),~N\ge3,~μ_1>0,~μ_2>0,~β>0,~α\in(0,N),~1+\fracα{N}<p,~q,~r<\frac{N+α}{N-2s}$ and $I_α(x)=|x|^{α-N}$. Under some restrictions of $N,α,p,q$ and $r$, we give the positivity of normalized solutions for $p,q,r\le 1+\frac{α+2s}{N}$.
