## On critical fractional systems with Hardy-Littlewood-Sobolev nonlinearities.(English)Zbl 1471.35005

The authors study the following critical fractional Laplacian system with Choquard type nonlinearities: $\left\{ \begin{array} [c]{lll} (-\Delta)^s u=k_1\displaystyle\left(\int_{\mathbb{R}^N}\frac{|u(y)|^{2_\mu^*}}{|x-y|^\mu}\right)|u|^{2_\mu^*-2}u+\frac{\alpha\gamma}{^{2_\mu^*}}\left(\int_{\mathbb{R}^N}\frac{|v(y)|^{\beta}}{|x-y|^\mu}\right)|u|^{\alpha-2}u & \mathrm{in} & \mathbb{R}^N,\\ (-\Delta)^s v=k_2\displaystyle\left(\int_{\mathbb{R}^N}\frac{|v(y)|^{2_\mu^*}}{|x-y|^\mu}\right)|v|^{2_\mu^*-2}v+\frac{\beta\gamma}{^{2_\mu^*}}\left(\int_{\mathbb{R}^N}\frac{|u(y)|^{\alpha}}{|x-y|^\mu}\right)|v|^{\beta-2}v & \mathrm{in} & \mathbb{R}^N, \end{array} \right.$ where $$s\in(0,1)$$, $$N>2s$$, $$k_1,k_2>0$$, $$\gamma\neq 0$$, $$\mu\in(0,N)$$, $$\alpha,\beta>1$$ and $$2_\mu^*=(2N-\mu)/(N-2s)$$ is the fractional upper critical exponent in the Hardy-Littlewood-Sobolev inequality. By using minimization techniques over the Nehari set they provide existence and non-existence of solutions, for certain regions of parameters $$(\alpha,\beta)$$.

### MSC:

 35A15 Variational methods applied to PDEs 35R09 Integro-partial differential equations 35R11 Fractional partial differential equations 35B33 Critical exponents in context of PDEs 35J47 Second-order elliptic systems 35J62 Quasilinear elliptic equations
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