American Institute of Mathematical Sciences

Advanced
  • Previous Article
    An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth
  • DCDS Home
  • This Issue
  • Next Article
    Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems
doi: 10.3934/dcds.2020361

Liouville type theorems for fractional and higher-order fractional systems

Daomin Cao 1,2, and  Guolin Qin 3,4,

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

Received  May 2020 Revised  August 2020 Published  October 2020

Fund Project: D. Cao was supported by NNSF of China (No.11831009) and Chinese Academy of Sciences (No.QYZDJ-SSW-SYS021)

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 [16] to derive a Liouville type theorem. We also construct an interesting example on super $ \frac{\alpha}{2} $-harmonic functions (Proposition 1.2).

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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364.   Google Scholar

[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.  Google Scholar

[25]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 465-479.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970.   Google Scholar
[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.  Google Scholar

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.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364.   Google Scholar

[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.  Google Scholar

[25]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 465-479.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970.   Google Scholar
[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.  Google Scholar

Figure 1.  Spherical coordinate system
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034
[2]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255
[3]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291
[4]

Tomasz Szostok. Inequalities of Hermite-Hadamard type for higher order convex functions, revisited. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020296
[5]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129
[6]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011
[7]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268
[8]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354
[9]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001
[10]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
[11]

Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021001
[12]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
[13]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355
[14]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322
[15]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115
[16]

Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376
[17]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379
[18]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
[19]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292
[20]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395

2019 Impact Factor: 1.338

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences