Symmetry and nonexistence results for a fractional Choquard equation with weights

Anh Tuan Duong; Phuong Le; Nhu Thang Nguyen

doi:10.3934/dcds.2020265

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2021, Volume 41, Issue 2: 489-505. Doi: 10.3934/dcds.2020265

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

Received: August 2019

Early access: July 2020

Published: February 2021

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Abstract

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} $ and must be radially symmetric about the origin if $ a<0 $ and $ \frac{n+\beta+2a}{n-\alpha} \le p \le \frac{n+\beta+a}{n-\alpha} $.

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Mathematics Subject Classification: 35R11, 35B06, 35B53, 45G10.

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