Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian

Phuong Le

doi:10.3934/cpaa.2020026

Article Contents

2020, Volume 19, Issue 1: 527-539. Doi: 10.3934/cpaa.2020026

This issue Previous Article Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator Next Article Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method

Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian

Phuong Le^1,2,

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

Received: February 2019

Revised: April 2019

Published online: January 2020

This work was done while the author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) in 2019. He wish to thank the institute for their hospitality and support.

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Abstract

Let $ u \in L_{sp} \cap C^{1, 1}_{\rm loc}(\mathbb{R}^n\setminus\{0\}) $ be a positive solution, which may blow up at zero, of the equation

$ (-\Delta)^s_p u = \left(\frac{1}{|x|^{n-\beta }} * \frac{u^q}{|x|^\alpha}\right) \frac{u^{q-1 }}{|x|^\alpha} \quad\text{ in } \mathbb{R}^n \setminus \{0\}, $

where $ 0 < s < 1 $, $ 0 < \beta < n $, $ p>2 $, $ q\ge 1 $ and $ \alpha>0 $. We prove that if $ u $ satisfies some suitable asymptotic properties, then $ u $ must be radially symmetric and monotone decreasing about the origin. In stead of using equivalent fractional systems, we exploit a direct method of moving planes for the weighted Choquard nonlinearity. This method allows us to cover the full range $ 0 < \beta < n $ in our results.

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Mathematics Subject Classification: 35R11, 35J92, 35J75, 35B06.

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