# American Institute of Mathematical Sciences

July  2020, 19(7): 3723-3734. doi: 10.3934/cpaa.2020164

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

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

Received  September 2019 Revised  January 2020 Published  April 2020

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 . 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]. Citation: Yunyun Hu. Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3723-3734. doi: 10.3934/cpaa.2020164 ##### References:  [1] B. Barrios, L. Montoro and B. Sciunzi, On the moving plane method for nonlocal problems in bounded domains, J. Anal. Math., 135 (2018), 37-57. doi: 10.1007/s11854-018-0031-1. Google Scholar [2] H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a half space, in Boundary Value Problems for Partial Differential Equations and Applications (ed. J. L. Lions et al.), Masson, Paris, (1993), 27–42. Google Scholar [3] H. Berestycki, L. Caffarelli and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains I., Duke Math. J., 81 (1996), 467-494. doi: 10.1215/S0012-7094-96-08117-X. Google Scholar [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in an unbounded Lipschitz domain, Commun. Pure Appl. Math., 50 (1997), 1089-1111. doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6. Google Scholar [5] H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 25 (1997), 69-94. Google Scholar [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar [7] W. Chen and Y. Hu, Monotonicity of positive solutions for nonlocal problems in unbounded domains, J. Func. Anal. (2019), submitted to. Google Scholar [8] W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758. doi: 10.1016/j.aim.2018.07.016. Google Scholar [9] W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ., 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3. Google Scholar [10] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. Google Scholar [11] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347. Google Scholar [12] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar [13] S. Dipierro, L. Montoro, I. Peral and B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 99. doi: 10.1007/s00526-016-1032-5. Google Scholar [14] C. Li, L. Wu and H. Xu, Maximum principle and B$\hat{o}$cher type theorem, Proc. Natl. Acad. Sci. USA, 115 (27), 6976-6979. doi: 10.1073/pnas.1804225115. Google Scholar [15] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. Google Scholar show all references ##### References:  [1] B. Barrios, L. Montoro and B. Sciunzi, On the moving plane method for nonlocal problems in bounded domains, J. Anal. Math., 135 (2018), 37-57. doi: 10.1007/s11854-018-0031-1. Google Scholar [2] H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a half space, in Boundary Value Problems for Partial Differential Equations and Applications (ed. J. L. Lions et al.), Masson, Paris, (1993), 27–42. Google Scholar [3] H. Berestycki, L. Caffarelli and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains I., Duke Math. J., 81 (1996), 467-494. doi: 10.1215/S0012-7094-96-08117-X. Google Scholar [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in an unbounded Lipschitz domain, Commun. Pure Appl. Math., 50 (1997), 1089-1111. doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6. Google Scholar [5] H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 25 (1997), 69-94. Google Scholar [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar [7] W. Chen and Y. Hu, Monotonicity of positive solutions for nonlocal problems in unbounded domains, J. Func. Anal. (2019), submitted to. Google Scholar [8] W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758. doi: 10.1016/j.aim.2018.07.016. Google Scholar [9] W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ., 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3. Google Scholar [10] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. Google Scholar [11] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347. Google Scholar [12] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar [13] S. Dipierro, L. Montoro, I. Peral and B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 99. doi: 10.1007/s00526-016-1032-5. Google Scholar [14] C. Li, L. Wu and H. Xu, Maximum principle and B$\hat{o}$cher type theorem, Proc. Natl. Acad. Sci. USA, 115 (27), 6976-6979. doi: 10.1073/pnas.1804225115. Google Scholar [15] L. 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