Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities (2206.13051v2)

Abstract: In this paper, we investigate the following fractional Sobolev critical nonlinear Schr\"{o}dinger (NLS) coupled systems: \begin{equation*} \left{\begin{array}{lll} (-\Delta)^{s} u=\mu_{1} u+|u|^{2^{*}{s}-2}u+\eta{1}|u|^{p-2}u+\gamma\alpha|u|^{\alpha-2}u|v|^{\beta} ~ \text{in}~ \mathbb{R}^{N},\ (-\Delta)^{s} v=\mu_{2} v+|v|^{2^{*}{s}-2}v+\eta{2}|v|^{q-2}v+\gamma\beta|u|^{\alpha}|v|^{\beta-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 $(-\Delta)^{s}$ is the fractional Laplacian, $N={3,4}$, $s\in(0,1)$, $\mu_{1}, \mu_{2}\in\mathbb{R}$ are unknown constants, which will appear as Lagrange multipliers, $2^{*}_{s}$ is the fractional Sobolev critical index, $\eta_{1}, \eta_{2}, \gamma, m_{1}, m_{2}>0$, $\alpha>1, \beta>1$, $p, q, \alpha+\beta\in(2+4s/N,2^{*}_{s}]$. Firstly, if $p, q, \alpha+\beta<2^{*}_{s}$, we obtain the existence of positive normalized solution when $\gamma$ is big enough. Secondly, if $p=q=\alpha+\beta=2^{*}_{s}$, 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.