ABSTRACT
In this paper, we are concerned with the anti-symmetric solutions to the following elliptic system involving fractional Laplacian
where 0<s<1,
. We first show that the solutions only depend on
variable by the method of moving planes. Moreover, we can obtain the monotonicity of solutions with respect to
variable (for the critical and subcritical cases
in the
space). Furthermore, when
, in the cases
, we obtain a Liouville theorem for the cases
in the
space. Then, through the doubling lemma, we obtain the singularity estimates of the positive solutions on a bounded domain Ω. Using the anti-symmetric property of the solutions, one can extend the space from
to
, we can still prove the Liouville theorem in the extended space. With the extension, we prove the existence of nontrivial solutions.
Disclosure statement
No potential conflict of interest was reported by the author(s).