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Article Contents

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

• 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].

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

 Citation:

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