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

Received  May 2020 Revised  August 2020 Published  May 2021 Early access  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).

Citation: Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete & 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.  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. ChenY. 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. ChenC. 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. DaiY. FangJ. HuangY. 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. DaiG. 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. PoláčikP. 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. ChenY. 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. ChenC. 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. DaiY. FangJ. HuangY. 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. DaiG. 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. PoláčikP. 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
[1]

Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & 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 & 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 & 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 & Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134

[5]

Maha Daoud, El Haj Laamri. Fractional Laplacians : A short survey. Discrete & Continuous Dynamical Systems - S, 2021  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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021184

[11]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057

[17]

Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058

[18]

Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317

[19]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062

[20]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (182)
  • HTML views (182)
  • Cited by (0)

Other articles
by authors

[Back to Top]