# American Institute of Mathematical Sciences

February  2021, 41(2): 489-505. doi: 10.3934/dcds.2020265

## Symmetry and nonexistence results for a fractional Choquard equation with weights

 1 Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam 2 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 3 Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

* Corresponding author: Phuong Le (lephuong@tdtu.edu.vn)

Received  August 2019 Published  July 2020

Let
 $u$
be a nonnegative solution to the equation
 $(-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * |x|^a u^p \right) |x|^a u^{p-1} \quad\text{ in } \mathbb{R}^n \setminus \{0\},$
where
 $n \ge 2$
,
 $0 < \alpha < 2$
,
 $0 < \beta < n$
and
 $a>\max\{ -\alpha, -\frac{\alpha+\beta}{2} \}$
. By exploiting the method of scaling spheres and moving planes in integral forms, we show that
 $u$
must be zero if
 $1\le p<\frac{n+\beta+2a}{n-\alpha}$
 $a<0$
and
 $\frac{n+\beta+2a}{n-\alpha} \le p \le \frac{n+\beta+a}{n-\alpha}$
.
Citation: Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265
##### References:

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