论文标题
具有Sobolev临界非线性的分数质量超临界NLS系统的标准化解决方案
Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities
论文作者
论文摘要
在本文中，我们研究了以下分数Sobolev临界非线性schrödinger（nls）耦合系统：\ begin {equination*} \ left \ left \ {\ begin {array} {array} {lll} {lll} {-Δ） U+| U |^{2^{*} _ {s} -2} u+η_{1} | u |^{p-2} u+γα| v =μ_{2} v+| v |^{2^{*} _ {s} -2} -2} v+η_{2} | \ | U \ |^{2} _ {l^{2}} = m_ {1}^{2}〜\ text {and}〜\ | v \ |^{2} _ {l^{l^{2}} = m_ \ end {qore*}其中$（ - δ）^{s} $是分数laplacian，$ n = {3,4} $，$ s \ in（0,1）$，$μ_{1}，μ__{2} \ $2^{*}_{s}$ is the fractional Sobolev critical index, $η_{1}, η_{2}, γ, m_{1}, m_{2}>0$, $α>1, β>1$, $p, q, α+β\ in（2+4s/n，2^{*} _ {s}] $。首先，如果$ p，q，q，α+β<2^{*} _ {s} $，我们在$γ$很大时获得阳性归一化解决方案的存在，如果$ p = $ p = q = q = q = al+α+α= 2 = 2 = 2^= 2^= 2 = 2^= 2 = 2 = 2^{*阳性归一化解决方案的不存在。
In this paper, we investigate the following fractional Sobolev critical nonlinear Schrödinger (NLS) coupled systems: \begin{equation*} \left\{\begin{array}{lll} (-Δ)^{s} u=μ_{1} u+|u|^{2^{*}_{s}-2}u+η_{1}|u|^{p-2}u+γα|u|^{α-2}u|v|^β ~ \text{in}~ \mathbb{R}^{N},\\ (-Δ)^{s} v=μ_{2} v+|v|^{2^{*}_{s}-2}v+η_{2}|v|^{q-2}v+γβ|u|^α|v|^{β-2}v ~~\text{in}~ \mathbb{R}^{N},\\ \|u\|^{2}_{L^{2}}=m_{1}^{2} ~\text{and}~ \|v\|^{2}_{L^{2}}=m_{2}^{2}, \end{array}\right. \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $N={3,4}$, $s\in(0,1)$, $μ_{1}, μ_{2}\in\mathbb{R}$ are unknown constants, which will appear as Lagrange multipliers, $2^{*}_{s}$ is the fractional Sobolev critical index, $η_{1}, η_{2}, γ, m_{1}, m_{2}>0$, $α>1, β>1$, $p, q, α+β\in(2+4s/N,2^{*}_{s}]$. Firstly, if $p, q, α+β<2^{*}_{s}$, we obtain the existence of positive normalized solution when $γ$ is big enough. Secondly, if $p=q=α+β=2^{*}_{s}$, we show that nonexistence of positive normalized solution. The main ideas and methods of this paper are scaling transformation, classification discussion and concentration-compactness principle.
