## Abstract

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

$$\left\{\begin{array}{cc}{\left(-\mathrm{\Delta}\right)}^{s}u={k}_{1}\left({\int}_{{\mathbb{R}}^{N}}\frac{|u\left(y\right){|}^{{2}_{\mu}^{\ast}}}{|x-y{|}^{\mu}}dy\right)|u{|}^{{2}_{\mu}^{\ast}-2}u+\frac{\alpha \gamma}{{2}_{\mu}^{\ast}}\left({\int}_{{\mathbb{R}}^{N}}\frac{|v\left(y\right){|}^{\beta}}{|x-y{|}^{\mu}}dy\right)|u{|}^{\alpha -2}u\phantom{\rule{1em}{0ex}}\hfill & \text{in}{\mathbb{R}}^{N},\hfill \\ {\left(-\mathrm{\Delta}\right)}^{s}v={k}_{2}\left({\int}_{{\mathbb{R}}^{N}}\frac{|v\left(y\right){|}^{{2}_{\mu}^{\ast}}}{|x-y{|}^{\mu}}dy\right)|v{|}^{{2}_{\mu}^{\ast}-2}v+\frac{\beta \gamma}{{2}_{\mu}^{\ast}}\left({\int}_{{\mathbb{R}}^{N}}\frac{|u\left(y\right){|}^{\alpha}}{|x-y{|}^{\mu}}dy\right)|v{|}^{\beta -2}v\phantom{\rule{1em}{0ex}}\hfill & \text{in}{\mathbb{R}}^{N},\hfill \\ u,v\in {D}_{s}\left({\mathbb{R}}^{N}\right),\phantom{\rule{1em}{0ex}}\hfill \end{array}\right.$$

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 ${2}_{\mu}^{\ast}=\left(2N-\mu \right)\u2215\left(N-2s\right)$ is the fractional upper critical exponent in the Hardy–Littlewood–Sobolev inequality, ${D}_{s}\left({\mathbb{R}}^{N}\right)$ is the Hilbert space as the completion of ${C}_{c}^{\infty}\left({\mathbb{R}}^{N}\right)$. 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 ${2}_{\mu}=\left(2N-\mu \right)\u2215\left(N-2\right)$ 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.

## Citation

Qianyu Hong. Yang Yang. "On critical fractional systems with Hardy–Littlewood–Sobolev nonlinearities." Rocky Mountain J. Math. 50 (5) 1661 - 1683, October 2020. https://doi.org/10.1216/rmj.2020.50.1661

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