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

TL;DR

This paper investigates the fractional Laplacian system with critical exponents, establishing symmetry, monotonicity, and bounds of solutions, extending classical results through advanced analytical methods.

Contribution

It introduces a Liouville theorem for the system and characterizes the symmetry and bounds of solutions with critical exponents.

Findings

Solutions are radially symmetric and decreasing

Established bounds for solutions near singularities

Extended classical results to fractional Laplacian systems

Abstract

This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving sphere to derive a Liouville Theorem, and then prove the solutions in R^n\{0} are radially symmetric and monotonically decreasing radially. Together with blow up analysis and the Pohozaev integral, we get the upper and lower bound of the local solutions in B_1\{0}. Our results is an extension of the classical work by Caffarelli et al [6, 7], Chen et al[16]