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February  2020, 19(2): 1111-1128. doi: 10.3934/cpaa.2020051

## Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres

 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaan xi, 710129, China

* Corresponding author: Pengcheng Niu

Received  April 2019 Revised  June 2019 Published  October 2019

Fund Project: The first author is supported by NSFC grant No. 11771354 and Natural Science Basic Research Plan in Shaanxi Province of China grant No.2017JM5140.

 $(-\Delta)^{\frac{\alpha}{2}}\vec {u}(x) = \vec {h}(x,\vec {u}(x)),$
where
 $0<\alpha <2$
,
 $\vec u$
and
 $\vec h$
stand for
 $k$
-dimentional vector-valued functions, and
 $\vec {h}(x,\vec {u}(x))$
is locally Lipschitz in
 $\vec {u}$
. We first establish two narrow region principles for different cases. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions of the above system in bounded star-shaped domains and the whole space.
Citation: Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051
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