A direct method of moving planes for a fully nonlinear nonlocal system

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September 2017, 16(5): 1707-1718. doi: 10.3934/cpaa.2017082

A direct method of moving planes for a fully nonlinear nonlocal system

Pengyan Wang and Pengcheng Niu ^,

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaan xi, 710129, China

^*The Corresponding author

Received September 2016 Revised January 2017 Published May 2017

In this paper we consider the system involving fully nonlinear nonlocal operators:

$ \left\{\begin{array}{ll}{\mathcal F}_{α}(u(x)) = C_{n,α} PV ∈t_{\mathbb{R}^n} \frac{F(u(x)-u(y))}{|x-y|^{n+α}} dy=v^p(x)+k_1(x)u^r(x),\\{\mathcal G}_{β}(v(x)) = C_{n,β} PV ∈t_{\mathbb{R}^n} \frac{G(v(x)-v(y))}{|x-y|^{n+β}} dy=u^q(x)+k_2(x)v^s(x),\end{array}\right.$

where

$0<α, β<2, $

$p, q, r, s>1, $

$k_1(x), k_2(x)\geq0.$

A narrow region principle and a decay at infinity are established for carrying on the method of moving planes. Then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space. Furthermore non-existence of positive solutions to the system on a half space is derived.

Keywords: Fully nonlinear nonlocal operator, narrow region principle, decay at infinity, method of moving planes, non-existence.

Mathematics Subject Classification: 35J45, 35J60, 45G05, 45G15.

Citation: Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082

References:

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[2]	K. Bogdan, T. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.
[3]	L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937.
[4]	L. Cao and Z. Dai, A Liouville-type theorem for an integral system on a half-space, J. Inequal. Appl., 1 (2013), 1-9. doi: 10.1186/1029-242X-2013-37.
[5]	L. Caffarelli and L. Silvestre, An extension problem related to the fraction Laplacian, Comm. PDE., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[6]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure. Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.
[7]	W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, accepted, 2016. doi: 10.1007/s00526-017-1110-3.
[8]	W. Chen, C. Li and Y. Li, A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038.
[9]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.
[10]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
[11]	W. Chen, Y. Fang and R. Yang, Loiuville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198. doi: 10.1016/j.aim.2014.12.013.
[12]	W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Dif. Equa., 260 (2016), 4758-4785. doi: 10.1016/j.jde.2015.11.029.
[13]	R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Cal. Var., 39 (2009), 85-99. doi: 10.1007/s00526-009-0302-x.
[14]	F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Mathematical Research Letters, 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.
[15]	F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincaré Nonl. Anal., 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006.
[16]	X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602. doi: 10.1016/j.jde.2011.07.037.
[17]	Y. Li and P. Ma, Symmetry of solutions for a fractional system, arXiv: 1604.01465v2.
[18]	D. Li and R. Zhuo, An integral equation on half space, Proc. Amer. Math. Soc., 138 (2010), 2779-2791. doi: 10.1090/S0002-9939-10-10368-2.
[19]	G. Lu and J. Zhu, An overdetermined problem in Riese-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036.
[20]	G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Cal. Var., 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7.
[21]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Cal. Var., 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7.
[22]	L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.
[23]	L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.
[24]	J. Li, Monotonicity and symmetry of fractional Lane-Emden-type equation in the parabolic domain, Complex Var. Elliptic Equ., 62 (2017), 135-147. doi: 10.1080/17476933.2016.1208185.
[25]	P. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl.(2), 450 (2017), 982-995.
[26]	R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.

show all references

References:

[1]	C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[2]	K. Bogdan, T. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.
[3]	L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937.
[4]	L. Cao and Z. Dai, A Liouville-type theorem for an integral system on a half-space, J. Inequal. Appl., 1 (2013), 1-9. doi: 10.1186/1029-242X-2013-37.
[5]	L. Caffarelli and L. Silvestre, An extension problem related to the fraction Laplacian, Comm. PDE., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[6]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure. Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.
[7]	W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, accepted, 2016. doi: 10.1007/s00526-017-1110-3.
[8]	W. Chen, C. Li and Y. Li, A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038.
[9]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.
[10]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
[11]	W. Chen, Y. Fang and R. Yang, Loiuville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198. doi: 10.1016/j.aim.2014.12.013.
[12]	W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Dif. Equa., 260 (2016), 4758-4785. doi: 10.1016/j.jde.2015.11.029.
[13]	R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Cal. Var., 39 (2009), 85-99. doi: 10.1007/s00526-009-0302-x.
[14]	F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Mathematical Research Letters, 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.
[15]	F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincaré Nonl. Anal., 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006.
[16]	X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602. doi: 10.1016/j.jde.2011.07.037.
[17]	Y. Li and P. Ma, Symmetry of solutions for a fractional system, arXiv: 1604.01465v2.
[18]	D. Li and R. Zhuo, An integral equation on half space, Proc. Amer. Math. Soc., 138 (2010), 2779-2791. doi: 10.1090/S0002-9939-10-10368-2.
[19]	G. Lu and J. Zhu, An overdetermined problem in Riese-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036.
[20]	G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Cal. Var., 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7.
[21]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Cal. Var., 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7.
[22]	L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.
[23]	L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.
[24]	J. Li, Monotonicity and symmetry of fractional Lane-Emden-type equation in the parabolic domain, Complex Var. Elliptic Equ., 62 (2017), 135-147. doi: 10.1080/17476933.2016.1208185.
[25]	P. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl.(2), 450 (2017), 982-995.
[26]	R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.

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