In this paper, we study the critical fractional Laplacian system with Choquard type nonlinearities
where is the well known fractional Laplacian operator, , , , , , and is the fractional upper critical exponent in the Hardy–Littlewood–Sobolev inequality, is the Hilbert space as the completion of . We show the existence and nonexistence results for the critical nonlocal system. These extend results in the literature for the case and , where 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 , whose key is providing an accurate upper bound of the least energy.
"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