On critical fractional systems with Hardy–Littlewood–Sobolev nonlinearities

Abstract

In this paper, we study the critical fractional Laplacian system with Choquard type nonlinearities

$$\left\{\begin{array}{cc}{\left(-\mathrm{\Delta }\right)}^{s}u=k_1\left({\int }_{{ℝ}^N}\frac{|u\left(y\right){|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\right)|u{|}^{2_{\mu }^{\ast }-2}u+\frac{\alpha \gamma }{2_{\mu }^{\ast }}\left({\int }_{{ℝ}^N}\frac{|v\left(y\right){|}^{\beta }}{|x-y{|}^{\mu }}dy\right)|u{|}^{\alpha -2}u\hfill & \text{ in }{ℝ}^N,\hfill \\ {\left(-\mathrm{\Delta }\right)}^{s}v=k_2\left({\int }_{{ℝ}^N}\frac{|v\left(y\right){|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\right)|v{|}^{2_{\mu }^{\ast }-2}v+\frac{\beta \gamma }{2_{\mu }^{\ast }}\left({\int }_{{ℝ}^N}\frac{|u\left(y\right){|}^{\alpha }}{|x-y{|}^{\mu }}dy\right)|v{|}^{\beta -2}v\hfill & \text{ in }{ℝ}^N,\hfill \\ u,v\in D_s\left({ℝ}^N\right),\hfill \end{array}\right.$$

where ${\left(-\mathrm{\Delta }\right)}^s$ is the well known fractional Laplacian operator, $s\in \left(0,1\right)$, $N>2s$, $k_1,k_2>0$, $\gamma \ne 0$, $\mu \in \left(0,N\right)$, $\alpha ,\beta >1$ and $2_{\mu }^{\ast }=\left(2N-\mu \right)∕\left(N-2s\right)$ is the fractional upper critical exponent in the Hardy–Littlewood–Sobolev inequality, $D_s\left({ℝ}^N\right)$ is the Hilbert space as the completion of $C_c^{\infty }\left({ℝ}^N\right)$. We show the existence and nonexistence results for the critical nonlocal system. These extend results in the literature for the case $s=1$ and $\alpha =\beta =2_{\mu }$, where $2_{\mu }=\left(2N-\mu \right)∕\left(N-2\right)$ is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Particularly, we establish new existence and multiplicity of positive solutions for the system when $\alpha +\beta =2\cdot 2_{\mu }^{\ast }$, whose key is providing an accurate upper bound of the least energy.