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Symmetry properties in systems of fractional Laplacian equations
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Symmetry for an integral system with general nonlinearity
Liouville's theorem for a fractional elliptic system
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China |
$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{\alpha /2}}u(x) = f(x){{v}^{q}}(x),&x\in {{R}^{n}}, \\ {{(-\Delta )}^{\beta /2}}v(x) = h(x){{u}^{p}}(x),&x\in {{R}^{n}}, \\\end{array} \right.$ |
References:
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W. Ao, J. Wei and W. Yang,
Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dyn. Syst., 37 (2017), 5561-5601.
doi: 10.3934/dcds.2017242. |
[2] |
F. Atkinson and L. A. Peletier,
Elliptic equations with nearly critical growth, J. Differential Equations, 70 (1987), 349-365.
doi: 10.1016/0022-0396(87)90156-2. |
[3] |
J. Bertoin,
Lévy Processes, Cambridge Tracts in Mathmatics, 121 Cambridge University Press, Cambridge, 1996. |
[4] |
C. Brandle, E. Colorado, A. de Pablo and U. Sanchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[5] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[6] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[8] |
W. Chen and C. Li,
Super polyharmonic property of solutions for PDE systems and its
applications, Comm. Pure Appl. Anal., 12 (2011), 2497-2514.
doi: 10.3934/cpaa.2013.12.2497. |
[9] |
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. |
[10] |
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. |
[11] |
W. Chen, Y. Li and P. Ma, The fractional Laplacian, in press. |
[12] |
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. |
[13] |
W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, accepted by Discrete Contin. Dyn. Syst. |
[14] |
T. Cheng,
Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.
doi: 10.3934/dcds.2017154. |
[15] |
C. Coffman,
Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.
doi: 10.1007/BF00250684. |
[16] |
P. Constantin,
Euler equations, Navier-Stokes equations and turbulence, in Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., 1871 (2004), 1-43.
doi: 10.1007/11545989_1. |
[17] |
Z. Dai, L. Cao and P. Wang, Liouville type theorems for the system of fractional nonlinear equations in $R^n_+$,
J. Inequal. Appl., (2016), Paper No. 267, 17 pp.
doi: 10.1186/s13660-016-1207-9. |
[18] |
J. Dou and Y. Li,
Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Discrete Contin. Dyn. Syst., 38 (2018), 3939-3953.
doi: 10.3934/dcds.2018171. |
[19] |
D. Figueiredo, P. Lions and R. Nussbaum,
A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.
|
[20] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[21] |
H. Kaper and M. Kwong,
Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States Ⅱ, 13 (1988), 1-17.
doi: 10.1007/978-1-4613-9608-6_1. |
[22] |
E. Leite ang M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, 2015, arXiv:1509.01267. |
[23] |
Y. Li and P. Ma,
Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824.
doi: 10.1007/s11425-016-0231-x. |
[24] |
K. Mcleod and J. Serrin,
Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $ {R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.
doi: 10.1007/BF00275874. |
[25] |
E. D. Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2011), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[26] |
P. Niu, L. Wu and X. Ji,
Positive solutions to nonlinear systems involving fully nonlinear
fractional operators, Fract. Calc. Appl. Anal., 21 (2018), 552-574.
doi: 10.1515/fca-2018-0030. |
[27] |
P. Pucci and V. Radulescu,
The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., 3 (2010), 543-584.
|
[28] |
A. Quaas and A. Xia,
Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659.
doi: 10.1007/s00526-014-0727-8. |
[29] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653.
|
[30] |
R. Servadei and E. Valdinoci,
Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matematiques, 58 (2014), 133-154.
doi: 10.5565/PUBLMAT_58114_06. |
[31] |
R. Servadei and E. Valdinoci,
A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.
doi: 10.3934/cpaa.2013.12.2445. |
[32] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[33] |
V. Tarasov, G. Zaslavsky and M. George,
Fractional dynamics of systems with long-range interaction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-898.
doi: 10.1016/j.cnsns.2006.03.005. |
[34] |
P. Wang and P. Niu,
A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure Appl. Anal., 16 (2017), 1707-1718.
doi: 10.3934/cpaa.2017082. |
[35] |
L. Wu and P. Niu, Symmetry and Nonexistence of Positive Solutions to Fractional p-Laplacian Equations, to appeared in Discrete Contin. Dyn. Syst., 2018. |
[36] |
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] |
W. Ao, J. Wei and W. Yang,
Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dyn. Syst., 37 (2017), 5561-5601.
doi: 10.3934/dcds.2017242. |
[2] |
F. Atkinson and L. A. Peletier,
Elliptic equations with nearly critical growth, J. Differential Equations, 70 (1987), 349-365.
doi: 10.1016/0022-0396(87)90156-2. |
[3] |
J. Bertoin,
Lévy Processes, Cambridge Tracts in Mathmatics, 121 Cambridge University Press, Cambridge, 1996. |
[4] |
C. Brandle, E. Colorado, A. de Pablo and U. Sanchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[5] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[6] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[8] |
W. Chen and C. Li,
Super polyharmonic property of solutions for PDE systems and its
applications, Comm. Pure Appl. Anal., 12 (2011), 2497-2514.
doi: 10.3934/cpaa.2013.12.2497. |
[9] |
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. |
[10] |
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. |
[11] |
W. Chen, Y. Li and P. Ma, The fractional Laplacian, in press. |
[12] |
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. |
[13] |
W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, accepted by Discrete Contin. Dyn. Syst. |
[14] |
T. Cheng,
Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.
doi: 10.3934/dcds.2017154. |
[15] |
C. Coffman,
Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.
doi: 10.1007/BF00250684. |
[16] |
P. Constantin,
Euler equations, Navier-Stokes equations and turbulence, in Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., 1871 (2004), 1-43.
doi: 10.1007/11545989_1. |
[17] |
Z. Dai, L. Cao and P. Wang, Liouville type theorems for the system of fractional nonlinear equations in $R^n_+$,
J. Inequal. Appl., (2016), Paper No. 267, 17 pp.
doi: 10.1186/s13660-016-1207-9. |
[18] |
J. Dou and Y. Li,
Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Discrete Contin. Dyn. Syst., 38 (2018), 3939-3953.
doi: 10.3934/dcds.2018171. |
[19] |
D. Figueiredo, P. Lions and R. Nussbaum,
A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.
|
[20] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[21] |
H. Kaper and M. Kwong,
Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States Ⅱ, 13 (1988), 1-17.
doi: 10.1007/978-1-4613-9608-6_1. |
[22] |
E. Leite ang M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, 2015, arXiv:1509.01267. |
[23] |
Y. Li and P. Ma,
Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824.
doi: 10.1007/s11425-016-0231-x. |
[24] |
K. Mcleod and J. Serrin,
Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $ {R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.
doi: 10.1007/BF00275874. |
[25] |
E. D. Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2011), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[26] |
P. Niu, L. Wu and X. Ji,
Positive solutions to nonlinear systems involving fully nonlinear
fractional operators, Fract. Calc. Appl. Anal., 21 (2018), 552-574.
doi: 10.1515/fca-2018-0030. |
[27] |
P. Pucci and V. Radulescu,
The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., 3 (2010), 543-584.
|
[28] |
A. Quaas and A. Xia,
Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659.
doi: 10.1007/s00526-014-0727-8. |
[29] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653.
|
[30] |
R. Servadei and E. Valdinoci,
Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matematiques, 58 (2014), 133-154.
doi: 10.5565/PUBLMAT_58114_06. |
[31] |
R. Servadei and E. Valdinoci,
A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.
doi: 10.3934/cpaa.2013.12.2445. |
[32] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[33] |
V. Tarasov, G. Zaslavsky and M. George,
Fractional dynamics of systems with long-range interaction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-898.
doi: 10.1016/j.cnsns.2006.03.005. |
[34] |
P. Wang and P. Niu,
A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure Appl. Anal., 16 (2017), 1707-1718.
doi: 10.3934/cpaa.2017082. |
[35] |
L. Wu and P. Niu, Symmetry and Nonexistence of Positive Solutions to Fractional p-Laplacian Equations, to appeared in Discrete Contin. Dyn. Syst., 2018. |
[36] |
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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