论文标题
关于施加潜力强的schrödinger方程的批判性耦合（S_1，S_2）
On critically coupled (s_1, s_2)-fractional system of Schrödinger equations with Hardy potential
论文作者
论文摘要
在本文中，我们的主要关注点是研究具有强势的Schrödinger方程的以下分数系统的结合和基态解的存在：\ begin {equation*} \ left \ left \ {\ begin {aligned} {aligned}（-Δ） \ frac {u ~~} {| x |^{2S_ {1}}} - u^{2_ {s_ {s_ {1}}}}^{*} - 1} =ναH（x） （-Δ）^{s_ {2}} v - λ_{2} \ frac {v ~~} {| x |^{2s_ {2}}}}}}}}} - v^{2_ {s_ {s_ {2}}}}}^{*}^{*} - 1} - 1} - 1} - 1} - 1} = c^n（x） \ mbox {in}〜\ Mathbb {r}^{n}，u，v> 0 \ quad \ mbox {in}〜\ mathbb {r}^{n} \ setMinus \ {0 \ {0 \}，\ end eend {aligned}} \ right。 \ end {qore*}其中$ s_ {1}，s_ {2} \ in（0,1）〜\ text {and} 〜λ_ {i} \ in（0，λ_{n，s_ {n，s_ {i}}）$带有$λ_{ \ frac {γ^{2}（\ frac {n+2s_i} {4}）γ（\ frac {n+2s_i} {2} {2} {2}）} {γ^{2}（\ frac}（\ frac {n-2s_i}通过对参数和函数h施加某些假设，我们使用浓度 - 缝合原理和山通定理获得了基态溶液。
In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of Schrödinger equations with Hardy potentials: \begin{equation*} \left\{ \begin{aligned} (-Δ)^{s_{1}} u - λ_{1} \frac{u~~}{|x|^{2s_{1}}} - u^{2_{s_{1}}^{*}-1} = ναh(x) u^{α-1}v^β & \quad \mbox{in} ~ \mathbb{R}^{N}, (-Δ)^{s_{2}} v - λ_{2} \frac{v~~}{|x|^{2s_{2}}} - v^{2_{s_{2}}^{*}-1} = νβh(x) u^αv^{β-1} & \quad \mbox{in} ~ \mathbb{R}^{N}, u,v >0 \quad \mbox{in} ~ \mathbb{R}^{N} \setminus \{0\}, \end{aligned} \right. \end{equation*} where $s_{1},s_{2} \in (0,1)~\text{and}~λ_{i}\in (0, Λ_{N,s_{i}})$ with $Λ_{N,s_{i}} = 2 π^{N/2} \frac{Γ^{2}(\frac{N+2s_i}{4}) Γ(\frac{N+2s_i}{2})}{Γ^{2}(\frac{N-2s_i}{4}) ~|Γ(-s_{i})|}, (i=1,2)$. By imposing certain assumptions on the parameters and on the function h, we obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem.
