Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities

TL;DR

This paper studies fractional Sobolev critical nonlinear Schrödinger systems with mass constraints, establishing existence of solutions for certain parameters and nonexistence in the critical case, using advanced analytical techniques.

Contribution

It provides new existence results for positive normalized solutions in fractional NLS systems with Sobolev critical nonlinearities, and identifies conditions for nonexistence.

Findings

Existence of positive solutions when nonlinearities are subcritical and coupling is strong.

Nonexistence of solutions in the critical case where nonlinearities reach Sobolev critical exponent.

Application of concentration-compactness and scaling methods to fractional NLS systems.

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.