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  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 - A, 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
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