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 coupled systems: $$\begin{aligned} \left\{ \begin{array}{lll} (-\Delta )^{s} u=\mu _{1} u+|u|^{2^{*}_{s}-2}u+\eta _{1}|u|^{p-2}u+\gamma \alpha |u|^{\alpha -2}u|v|^{\beta } ~ \text {in}~ {\mathbb {R}}^{N},\\ (-\Delta )^{s} v=\mu _{2} v+|v|^{2^{*}_{s}-2}v+\eta _{2}|v|^{q-2}v+\gamma \beta |u|^{\alpha }|v|^{\beta -2}v ~~\text {in}~ {\mathbb {R}}^{N},\\ \Vert u\Vert ^{2}_{L^{2}}=m_{1}^{2} ~\text {and}~ \Vert v\Vert ^{2}_{L^{2}}=m_{2}^{2}, \end{array}\right. \end{aligned}$$ ( - Δ ) s u = μ 1 u + | u | 2 s ∗ - 2 u + η 1 | u | p - 2 u + γ α | u | α - 2 u | v | β in R N , ( - Δ ) s v = μ 2 v + | v | 2 s ∗ - 2 v + η 2 | v | q - 2 v + γ β | u | α | v | β - 2 v in R N , ‖ u ‖ L 2 2 = m 1 2 and ‖ v ‖ L 2 2 = m 2 2 , where $$(-\Delta )^{s}$$ ( - Δ ) s is the fractional Laplacian, $$N>2s$$ N > 2 s , $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$\mu _{1}, \mu _{2}\in {\mathbb {R}}$$ μ 1 , μ 2 ∈ R are unknown constants, which will appear as Lagrange multipliers, $$2^{*}_{s}$$ 2 s ∗ is the fractional Sobolev critical index, $$\eta _{1}, \eta _{2}, \gamma , m_{1}, m_{2}>0$$ η 1 , η 2 , γ , m 1 , m 2 > 0 , $$\alpha>1, \beta >1$$ α > 1 , β > 1 , $$p, q, \alpha +\beta \in (2+4s/N,2^{*}_{s}]$$ p , q , α + β ∈ ( 2 + 4 s / N , 2 s ∗ ] . Firstly, if $$p, q, \alpha +\beta <2^{*}_{s}$$ p , q , α + β < 2 s ∗ , we obtain the existence of positive normalized solution when $$\gamma $$ γ is big enough. Secondly, if $$p=q=\alpha +\beta =2^{*}_{s}$$ 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.