• Previous Article
    A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources
  • CPAA Home
  • This Issue
  • Next Article
    Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators
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.

[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.

[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. 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.

[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.

[10]

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

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

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.

[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.

[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. 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.

[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.

[10]

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

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[1]

Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure & Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125

[2]

Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769

[3]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949

[4]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125

[5]

Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015

[6]

Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001

[7]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure & Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048

[8]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041

[9]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335

[10]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209

[11]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237

[12]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549

[13]

Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic & Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205

[14]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395

[15]

Ivan Landjev, Assia Rousseva. The non-existence of $(104,22;3,5)$-arcs. Advances in Mathematics of Communications, 2016, 10 (3) : 601-611. doi: 10.3934/amc.2016029

[16]

J. F. Toland. Non-existence of global energy minimisers in Stokes waves problems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3211-3217. doi: 10.3934/dcds.2014.34.3211

[17]

Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919

[18]

Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707

[19]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897

[20]

Chuanqiang Chen. On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4761-4811. doi: 10.3934/dcds.2016007

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (5)
  • HTML views (12)
  • Cited by (0)

Other articles
by authors

[Back to Top]