Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders

Yunyun Hu

doi:10.3934/cpaa.2020164

Article Contents

2020, Volume 19, Issue 7: 3723-3734. Doi: 10.3934/cpaa.2020164

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Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders

Yunyun Hu

Department of Mathematical Sciences, Yeshiva University, New York, NY, 10033, USA

Received: September 2019

Revised: January 2020

Published: April 2020

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Abstract

The aim of this paper is to study symmetry and monotonicity for positive solutions to fractional equations. We first consider the following problems in bounded domains in the sense of distributions

$ \begin{equation*} \begin{cases} (-\Delta)^su = \frac{g(u)}{|x|^{2s}}+f(x,u)\ \ \ &\mbox{in}\ \Omega,\\ u>0\ \ \ \ \ \ \ \ \ \ \ &\mbox{in}\ \Omega,\\ u = 0\ \ \ \ \ \ \ \ \ \ \ &\mbox{in}\ \mathbb R^n\setminus\Omega, \end{cases} \end{equation*} $

where $ n>2s $, $ 0<s<1 $. We prove that all positive solutions are radically symmetric about the origin. Compare to results in [1], we use a completely different method under the weaker conditions in $ f $. Next we consider a problem in infinite cylinders. We establish the symmetry and monotonicity of positive solutions by using the method of moving planes. This result can be seen as the nonlocal counterparts of [3].

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Mathematics Subject Classification: Primary: 35R11; Secondary: 35B09.

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References

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