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

Abstract

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 $(-\Delta {)}^s$ 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, $2_s^{\ast }$ is the fractional Sobolev critical index, ${\eta }_1,{\eta }_2,\gamma ,m_1,m_2>0$, $\alpha >1,\beta >1$, $p,q,\alpha +\beta \in (2+4s/N,2_s^{\ast }]$. Firstly, if $p,q,\alpha +\beta <2_s^{\ast }$, we obtain the existence of positive normalized solution when $\gamma$ is big enough. Secondly, if $p=q=\alpha +\beta =2_s^{\ast }$, 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.

References in corpus (3)

• A mass supercritical problem revisited
• Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials
• On a doubly critical system involving fractional Laplacian with partial weight

Cited by in corpus (1)

• Normalized ground state solutions for the fractional Sobolev critical NLSE with an extra mass supercritical nonlinearity