
-
Previous Article
Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators
- DCDS Home
- This Issue
-
Next Article
Martingale solution for stochastic active liquid crystal system
Liouville type theorems for fractional and higher-order fractional systems
1. | School of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, China |
2. | Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, China |
3. | Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, China |
4. | University of Chinese Academy of sciences, Beijing 100049, China |
In this paper, we first establish decay estimates for the fractional and higher-order fractional Hénon-Lane-Emden systems by using a nonlocal average and integral estimates, which deduce a result of non-existence. Next, we apply the method of scaling spheres introduced in [
References:
[1] |
A. Biswas,
Liouville type results for systems of equations involving fractional Laplacian in exterior domains, Nonlinearity, 32 (2019), 2246-2268.
doi: 10.1088/1361-6544/ab091b. |
[2] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[3] |
D. Cao and W. Dai,
Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994.
doi: 10.1017/prm.2018.67. |
[4] |
D. Cao, W. Dai and G. Qin, Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians, preprint, arXiv: 1905.04300. Google Scholar |
[5] |
W. Chen, W. Dai and G. Qin, Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in $\mathbb{R}^n$, preprint, arXiv: 1808.06609. Google Scholar |
[6] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[7] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[8] |
W. Chen and C. Li,
Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure Appl. Anal., 12 (2013), 2497-2514.
doi: 10.3934/cpaa.2013.12.2497. |
[9] |
W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., 2020,344 pp, https://doi.org/10.1142/10550. Google Scholar |
[10] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[11] |
W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang,
Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst. - A, 39 (2019), 1389-1403.
doi: 10.3934/dcds.2018117. |
[12] |
W. Dai and Z. Liu,
Classification of positive solutions to a system of Hardy-Sobolev type equations, Acta Mathematica Scientia, 37 (2017), 1415-1436.
doi: 10.1016/S0252-9602(17)30082-6. |
[13] |
W. Dai and Z. Liu, Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 156, 24 pp.
doi: 10.1007/s00526-019-1595-z. |
[14] |
W. Dai, Z. Liu and G. Qin, Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations, preprint, arXiv: 1909.00492. Google Scholar |
[15] |
W. Dai and G. Qin,
Classification of nonnegative classical solutions to third-order equations, Adv. Math., 328 (2018), 822-857.
doi: 10.1016/j.aim.2018.02.016. |
[16] |
W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres, preprint, arXiv: 1810.02752. Google Scholar |
[17] |
W. Dai and G. Qin, Liouville type theorem for critical order Hénon-Lane-Emden type equations on a half space and its applications, preprint, arXiv: 1811.00881. Google Scholar |
[18] |
W. Dai and G. Qin,
Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains, Journal of Differential Equations, 269 (2020), 7231-7252.
doi: 10.1016/j.jde.2020.05.026. |
[19] |
W. Dai and G. Qin,
Liouville type theorems for Hardy-Henon equations with concave nonlinearities, Math. Nachr., 293 (2020), 1084-1093.
doi: 10.1002/mana.201800532. |
[20] |
W. Dai, G. Qin and Y. Zhang,
Liouville type theorem for higher order Hénon equations on a half space, Nonlinear Analysis, 183 (2019), 284-302.
doi: 10.1016/j.na.2019.01.033. |
[21] |
M. Fazly and J. Wei, On stable solutions of the fractional Hénon-Lane-Emden equation, Commun. Contemp. Math., 18 (2016), 1650005, 24 pp.
doi: 10.1142/S021919971650005X. |
[22] |
M. Fazly and J. Wei,
On finite Morse index solutions of higher order fractional Lane-Emden equations, Amer. J. Math., 139 (2017), 433-460.
doi: 10.1353/ajm.2017.0011. |
[23] |
T. Kulczycki,
Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364.
|
[24] |
K. Li and Z. Zhang,
Proof of the Hénon-Lane-Emden conjecture in $\mathbb{R}^{3}$, Journal of Differential Equations, 266 (2017), 202-226.
doi: 10.1016/j.jde.2018.07.036. |
[25] |
E. Mitidieri,
Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 465-479.
|
[26] |
S. Peng, Liouville theorems for fractional and higher order Hénon-Hardy systems on $\mathbb{R}^n$, Complex Var. Elliptic Equ., (2020), 25 pp.
doi: 10.1080/17476933.2020.1783661. |
[27] |
P. Poláčik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[28] |
A. Quaas and A. Xia,
A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian, Nonlinerity, 29 (2016), 2279-2297.
doi: 10.1088/0951-7715/29/8/2279. |
[29] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653.
|
[30] |
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. |
[31] |
P. Souplet,
The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[32] |
M. A. S. Souto,
A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258.
|
[33] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970.
![]() |
[34] |
R. Zhuo and Y. Li, Liouville theorem for the higher-order fractional Laplacian, Commun. Contemp. Math., 21 (2019), 1850005, 19 pp.
doi: 10.1142/S0219199718500050. |
show all references
References:
[1] |
A. Biswas,
Liouville type results for systems of equations involving fractional Laplacian in exterior domains, Nonlinearity, 32 (2019), 2246-2268.
doi: 10.1088/1361-6544/ab091b. |
[2] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[3] |
D. Cao and W. Dai,
Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994.
doi: 10.1017/prm.2018.67. |
[4] |
D. Cao, W. Dai and G. Qin, Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians, preprint, arXiv: 1905.04300. Google Scholar |
[5] |
W. Chen, W. Dai and G. Qin, Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in $\mathbb{R}^n$, preprint, arXiv: 1808.06609. Google Scholar |
[6] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[7] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[8] |
W. Chen and C. Li,
Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure Appl. Anal., 12 (2013), 2497-2514.
doi: 10.3934/cpaa.2013.12.2497. |
[9] |
W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., 2020,344 pp, https://doi.org/10.1142/10550. Google Scholar |
[10] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[11] |
W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang,
Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst. - A, 39 (2019), 1389-1403.
doi: 10.3934/dcds.2018117. |
[12] |
W. Dai and Z. Liu,
Classification of positive solutions to a system of Hardy-Sobolev type equations, Acta Mathematica Scientia, 37 (2017), 1415-1436.
doi: 10.1016/S0252-9602(17)30082-6. |
[13] |
W. Dai and Z. Liu, Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 156, 24 pp.
doi: 10.1007/s00526-019-1595-z. |
[14] |
W. Dai, Z. Liu and G. Qin, Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations, preprint, arXiv: 1909.00492. Google Scholar |
[15] |
W. Dai and G. Qin,
Classification of nonnegative classical solutions to third-order equations, Adv. Math., 328 (2018), 822-857.
doi: 10.1016/j.aim.2018.02.016. |
[16] |
W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres, preprint, arXiv: 1810.02752. Google Scholar |
[17] |
W. Dai and G. Qin, Liouville type theorem for critical order Hénon-Lane-Emden type equations on a half space and its applications, preprint, arXiv: 1811.00881. Google Scholar |
[18] |
W. Dai and G. Qin,
Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains, Journal of Differential Equations, 269 (2020), 7231-7252.
doi: 10.1016/j.jde.2020.05.026. |
[19] |
W. Dai and G. Qin,
Liouville type theorems for Hardy-Henon equations with concave nonlinearities, Math. Nachr., 293 (2020), 1084-1093.
doi: 10.1002/mana.201800532. |
[20] |
W. Dai, G. Qin and Y. Zhang,
Liouville type theorem for higher order Hénon equations on a half space, Nonlinear Analysis, 183 (2019), 284-302.
doi: 10.1016/j.na.2019.01.033. |
[21] |
M. Fazly and J. Wei, On stable solutions of the fractional Hénon-Lane-Emden equation, Commun. Contemp. Math., 18 (2016), 1650005, 24 pp.
doi: 10.1142/S021919971650005X. |
[22] |
M. Fazly and J. Wei,
On finite Morse index solutions of higher order fractional Lane-Emden equations, Amer. J. Math., 139 (2017), 433-460.
doi: 10.1353/ajm.2017.0011. |
[23] |
T. Kulczycki,
Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364.
|
[24] |
K. Li and Z. Zhang,
Proof of the Hénon-Lane-Emden conjecture in $\mathbb{R}^{3}$, Journal of Differential Equations, 266 (2017), 202-226.
doi: 10.1016/j.jde.2018.07.036. |
[25] |
E. Mitidieri,
Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 465-479.
|
[26] |
S. Peng, Liouville theorems for fractional and higher order Hénon-Hardy systems on $\mathbb{R}^n$, Complex Var. Elliptic Equ., (2020), 25 pp.
doi: 10.1080/17476933.2020.1783661. |
[27] |
P. Poláčik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[28] |
A. Quaas and A. Xia,
A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian, Nonlinerity, 29 (2016), 2279-2297.
doi: 10.1088/0951-7715/29/8/2279. |
[29] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653.
|
[30] |
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. |
[31] |
P. Souplet,
The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[32] |
M. A. S. Souto,
A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258.
|
[33] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970.
![]() |
[34] |
R. Zhuo and Y. Li, Liouville theorem for the higher-order fractional Laplacian, Commun. Contemp. Math., 21 (2019), 1850005, 19 pp.
doi: 10.1142/S0219199718500050. |

[1] |
Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028 |
[2] |
Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021022 |
[3] |
Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021017 |
[4] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[5] |
María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 |
[6] |
Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 |
[7] |
Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021021 |
[8] |
Zhimin Chen, Kaihui Liu, Xiuxiang Liu. Evaluating vaccination effectiveness of group-specific fractional-dose strategies. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021062 |
[9] |
Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021004 |
[10] |
Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021068 |
[11] |
Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 |
[12] |
Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021015 |
[13] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[14] |
Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 |
[15] |
Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021019 |
[16] |
Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021020 |
[17] |
Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021023 |
[18] |
Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 |
[19] |
John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021004 |
[20] |
Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 |
2019 Impact Factor: 1.338
Tools
Article outline
Figures and Tables
[Back to Top]