-
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
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 |
$ \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.$ |
$0<α, β<2, $ |
$p, q, r, s>1, $ |
$k_1(x), k_2(x)\geq0.$ |
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. |
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. |
[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] |
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 |
[4] |
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 |
[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] |
Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 |
[7] |
Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201 |
[8] |
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 |
[9] |
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 |
[10] |
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 |
[11] |
Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227 |
[12] |
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 |
[13] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
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 |
[17] |
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 |
[18] |
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 |
[19] |
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 |
[20] |
Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059 |
2017 Impact Factor: 0.884
Tools
Metrics
Other articles
by authors
[Back to Top]