Liouville theorem and isolated singularity of fractional Laplacian system with critical exponents

Abstract This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving spheres to derive a Liouville Theorem with at most three radial solutions, and then prove the solutions in R n ∖ { 0 } are radially symmetric and monotonically decreasing. Together with blow up analysis, we get the upper bound of the local solutions in B 1 ∖ { 0 } . Our results is an extension of the classical works by Caffarelli et al. (1989), Caffarelli et al. (2014), Chen and Lin (2015) and Guo and Liu (2008).