Figure 1.
Spherical coordinate system
-
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
May
2021, 41(5): 2269-2283.
doi: 10.3934/dcds.2020361
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 [
Keywords:
Fractional Laplacians,
Hénon-Lane-Emden systems,
decay estimates,
higher-order fractional Laplacians,
Liouville type theorems.
Mathematics Subject Classification:
Primary: 35J61; Secondary: 35B53, 35C15.
Citation:
Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete and Continuous Dynamical Systems,
2021, 41
(5)
: 2269-2283.
doi: 10.3934/dcds.2020361
![]()
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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] |
Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure and Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807 |
| [2] |
Mostafa Fazly, Nassif Ghoussoub. On the Hénon-Lane-Emden conjecture. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2513-2533. doi: 10.3934/dcds.2014.34.2513 |
| [3] |
Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131 |
| [4] |
Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134 |
| [5] |
Maha Daoud, El Haj Laamri. Fractional Laplacians : A short survey. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 95-116. doi: 10.3934/dcdss.2021027 |
| [6] |
Giovanni Covi, Keijo Mönkkönen, Jesse Railo. Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems. Inverse Problems and Imaging, 2021, 15 (4) : 641-681. doi: 10.3934/ipi.2021009 |
| [7] |
Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915 |
| [8] |
Philip Korman, Junping Shi. On lane-emden type systems. Conference Publications, 2005, 2005 (Special) : 510-517. doi: 10.3934/proc.2005.2005.510 |
| [9] |
Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011 |
| [10] |
Yuxia Guo, Ting Liu. Liouville-type theorem for high order degenerate Lane-Emden system. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2073-2100. doi: 10.3934/dcds.2021184 |
| [11] |
Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034 |
| [12] |
Fausto Ferrari, Michele Miranda Jr, Diego Pallara, Andrea Pinamonti, Yannick Sire. Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 477-491. doi: 10.3934/dcdss.2018026 |
| [13] |
Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136 |
| [14] |
Pradeep Boggarapu, Luz Roncal, Sundaram Thangavelu. On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2575-2605. doi: 10.3934/cpaa.2019116 |
| [15] |
Wei Dai, Jiahui Huang, Yu Qin, Bo Wang, Yanqin Fang. Regularity and classification of solutions to static Hartree equations involving fractional Laplacians. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1389-1403. doi: 10.3934/dcds.2018117 |
| [16] |
Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057 |
| [17] |
Pengyan Wang, Wenxiong Chen. Hopf's lemmas for parabolic fractional p-Laplacians. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022089 |
| [18] |
Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317 |
| [19] |
Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058 |
| [20] |
Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]