Normalized Solutions to Fractional Mass Supercritical Choquard Systems

In this paper, we mainly study the fractional Choquard systems with a local perturbation under the mass constraints: \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^su=\lambda _1u+(I_\alpha *|u|^{2^*_{\alpha ,s}})|u|^{2^*_{\alpha ,s}-2}u+\mu _1|u|^{p-2}u+\beta r_1|u|^{ r_1-2}u|v|^{ r_2} \quad \text {in}~\mathbb {R}^N,\\ (-\Delta )^sv=\lambda _2v+(I_\alpha *|v|^{2^*_{\alpha ,s}})|v|^{2^*_{\alpha ,s}-2}v+\mu _2|v|^{q-2}v+\beta r_2|v|^{ r_2-2}v|u|^{ r_1} \quad \text {in}~\mathbb {R}^N,\\ \Vert u\Vert ^2_{L^2(\mathbb {R}^N)}=a_1^2 ~\text {and} ~\Vert v\Vert ^2_{L^2(\mathbb {R}^N)}=a_2^2, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{(-\mathrm{\Delta })}^{s}u={\lambda }_{1}u+(I_{\alpha }{\ast |u|}^{2_{\alpha ,s}^{\ast }}{)|u|}^{2_{\alpha ,s}^{\ast }-2}u+{\mu }_1{|u|}^{p-2}u+\beta r_1{|u|}^{r_1-2}u{|v|}^{r_2}\text{in}{\mathbb{R}}^N,\hfill \\ {(-\mathrm{\Delta })}^{s}v={\lambda }_{2}v+(I_{\alpha }{\ast |v|}^{2_{\alpha ,s}^{\ast }}{)|v|}^{2_{\alpha ,s}^{\ast }-2}v+{\mu }_2{|v|}^{q-2}v+\beta r_2{|v|}^{r_2-2}v{|u|}^{r_1}\text{in}{\mathbb{R}}^N,\hfill \\ {\Vert u\Vert }_{L^2({\mathbb{R}}^N)}^2=a_1^2\text{and}{\Vert v\Vert }_{L^2({\mathbb{R}}^N)}^2=a_2^2,\hfill \end{array}\right)\end{array}$$ where \((-\Delta )^su\) ${(-\mathrm{\Delta })}^{s}u$ is the fractional Laplacian, \(I_\alpha (x)\) $I_{\alpha }(x)$ is the Riesz potential, \(0<\alpha <\min \{N,4s\},\) $0<\alpha <min\{N,4s\},$ and \(N>2s, s\in (0,1),\) $N>2s,s\in (0,1),$ \(\lambda _1,\lambda _2\in \mathbb {R}^N\) ${\lambda }_1,{\lambda }_2\in {\mathbb{R}}^N$ are unknown constants, which will appear as Lagrange multipliers, \(\mu _1,\mu _2,\beta ,a_1,a_2>0, r_1,r_2>1,\) ${\mu }_1,{\mu }_2,\beta ,a_1,a_2>0,r_1,r_2>1,$ \(p,q~\text {and}~ r_1+r_2\in \big (2+\frac{4\,s}{N},2^*_s\big ].\) $p,q\text{and}{r}_1+r_2\in (2+\frac{4s}{N},2_s^{\ast }].$ \(2^*_s=\frac{2N}{N-2s}\) $2_s^{\ast }=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and \(2^*_{\alpha ,s}=\frac{2N-\alpha }{N-2s}\) $2_{\alpha ,s}^{\ast }=\frac{2N-\alpha }{N-2s}$ is the fractional Hardy–Littlewood–Sobolev critical exponent. Firstly, if \(p,q~\text {and}~r_1+r_2<2^*_s,\) $p,q\text{and}{r}_1+r_2<2_s^{\ast },$ we obtain the existence of positive normalized solution when \(\beta \) $\beta$ is big enough. Then, for the case of \(p=q=r_1+r_2=2^*_s,\) $p=q=r_1+r_2=2_s^{\ast },$ we may obtain the nonexistence of positive normalized solution.