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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

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.
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:
[1]

C. BrandleE. ColoradoA. 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.  Google Scholar

[2]

K. BogdanT. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fraction Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[8]

W. ChenC. 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.  Google Scholar

[9]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[10]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.   Google Scholar

[11]

W. ChenY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[15]

F. HangX. 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.  Google Scholar

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X. HanG. 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.  Google Scholar

[17]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, arXiv: 1604.01465v2. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. LeiC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

P. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl.(2), 450 (2017), 982-995.   Google Scholar

[26]

R. ZhuoW. ChenX. 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.  Google Scholar

show all references

References:
[1]

C. BrandleE. ColoradoA. 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.  Google Scholar

[2]

K. BogdanT. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. Caffarelli and L. Silvestre, An extension problem related to the fraction Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. ChenC. 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.  Google Scholar

[9]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[10]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.   Google Scholar

[11]

W. ChenY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. HangX. 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.  Google Scholar

[16]

X. HanG. 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.  Google Scholar

[17]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, arXiv: 1604.01465v2. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. LeiC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

P. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl.(2), 450 (2017), 982-995.   Google Scholar

[26]

R. ZhuoW. ChenX. 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.  Google Scholar

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