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November  2020, 19(11): 5253-5268. doi: 10.3934/cpaa.2020236

Liouville type theorem for Fractional Laplacian system

School of Mathematics and Statistics, Huanghuai University, Zhumadian Academy of Industry Innovation and Development, Zhumadian, Henan, 463000, China

Received  March 2020 Revised  July 2020 Published  November 2020 Early access  September 2020

Fund Project: The author is supported by the National Natural Science Foundation of China (Grant No.11771354)

In this paper, using the method of moving planes combined with integral inequality to handle the fractional Laplacian system, we prove Liouville type theorems of nonnegative solution for the nonlinear system.

Citation: Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236
References:
[1]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[2]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure App. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 2 (2007), 1245–1260. doi: 10.1080/03605300600987306.

[4]

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.

[5]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series, 4 2010.

[6]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2016), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[7]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[8]

W. ChenY. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022.

[9]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differ. Equ., 260 (2014), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.

[10]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[11]

D. G. De Figueiredo and P.L. Felmer, A Liouville type theorem for Elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. 

[12]

R. L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[13]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 35 (1982), 528-598.  doi: 10.1002/cpa.3160340406.

[14]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243. 

[15]

Y. X. Guo and J. Q. Liu, type theorems for positive solutions of elliptic system in $\mathbb{R}^n$, Commun. Partial Differ. Equ., 33 (2008), 263-284.  doi: 10.1080/03605300701257476.

[16]

B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differ. Integral Equ., 7 (1994), 301-313. 

[17]

M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[18]

N. S. Landkof, Foundations of modern potential theory, Springer-Verlag Berlin Heidelberg, New York, 1972.

[19]

D. Li and Z. Li, A radial symmetry and Liouville theorem for systems involving fractional Laplacian, Front. Math. China, 12 (2017), 389-402.  doi: 10.1007/s11464-016-0517-z.

[20]

B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition, Differ. Integral Equ., 9 (1996), 1157-1164. 

[21]

A. Quaas and A. Xia, A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian, Mathematics, 29 (2016), 2279-2297.  doi: 10.1088/0951-7715/29/8/2279.

[22]

J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369-380. 

[23]

J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system, Commun. Partial Differ. Equ., 23 (1998), 577-599.  doi: 10.1080/03605309808821356.

[24]

S. Terracini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differ. Integral Equ., 8 (1995), 1911-1922. 

[25]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differ. Equ., 1 (1996), 241-264. 

[26]

Y. Wan and C. L. Xiang, Uniqueness of positive solutions to some Nonlinear Neumann Problems, J. Math. Anal. Appl., 455 (2017), 1835-1847.  doi: 10.1016/j.jmaa.2017.06.006.

[27]

X. WangX. Cui and P. Niu, A Liouville theorem for the semilinear fractional CR covariant equation on the Heisenberg group, Complex Var. Elliptic Equ., 64 (2019), 1325-1344.  doi: 10.1080/17476933.2018.1523898.

[28]

X. WangP. Niu and X. Cui, A Liouville type theorem to an extension problem relating to the Heisenberg group, Commun. Pure Appl. Anal., 17 (2018), 2379-2394.  doi: 10.3934/cpaa.2018113.

[29]

P. Wang and P. Niu, Liouville's theorem for a fractional elliptic system, Discrete Contin. Dyn. Syst., 39 (2019), 1545-1558.  doi: 10.3934/dcds.2019067.

[30]

X. Yu, Liouville type theorems for two mixed boundary value problems with general nonlinearities, J. Math. Anal. Appl., 462 (2018), 305-322.  doi: 10.1016/j.jmaa.2018.02.009.

[31]

L. ZhangM. Yu and J. He, A Liouville theorem for a class of fractional systems in $\mathbb{R}^{n}_{+}$, J. Differ. Equ., 263 (2017), 6025-6065.  doi: 10.1016/j.jde.2017.07.009.

show all references

References:
[1]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[2]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure App. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 2 (2007), 1245–1260. doi: 10.1080/03605300600987306.

[4]

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.

[5]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series, 4 2010.

[6]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2016), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[7]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[8]

W. ChenY. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022.

[9]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differ. Equ., 260 (2014), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.

[10]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[11]

D. G. De Figueiredo and P.L. Felmer, A Liouville type theorem for Elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. 

[12]

R. L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[13]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 35 (1982), 528-598.  doi: 10.1002/cpa.3160340406.

[14]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243. 

[15]

Y. X. Guo and J. Q. Liu, type theorems for positive solutions of elliptic system in $\mathbb{R}^n$, Commun. Partial Differ. Equ., 33 (2008), 263-284.  doi: 10.1080/03605300701257476.

[16]

B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differ. Integral Equ., 7 (1994), 301-313. 

[17]

M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[18]

N. S. Landkof, Foundations of modern potential theory, Springer-Verlag Berlin Heidelberg, New York, 1972.

[19]

D. Li and Z. Li, A radial symmetry and Liouville theorem for systems involving fractional Laplacian, Front. Math. China, 12 (2017), 389-402.  doi: 10.1007/s11464-016-0517-z.

[20]

B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition, Differ. Integral Equ., 9 (1996), 1157-1164. 

[21]

A. Quaas and A. Xia, A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian, Mathematics, 29 (2016), 2279-2297.  doi: 10.1088/0951-7715/29/8/2279.

[22]

J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369-380. 

[23]

J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system, Commun. Partial Differ. Equ., 23 (1998), 577-599.  doi: 10.1080/03605309808821356.

[24]

S. Terracini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differ. Integral Equ., 8 (1995), 1911-1922. 

[25]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differ. Equ., 1 (1996), 241-264. 

[26]

Y. Wan and C. L. Xiang, Uniqueness of positive solutions to some Nonlinear Neumann Problems, J. Math. Anal. Appl., 455 (2017), 1835-1847.  doi: 10.1016/j.jmaa.2017.06.006.

[27]

X. WangX. Cui and P. Niu, A Liouville theorem for the semilinear fractional CR covariant equation on the Heisenberg group, Complex Var. Elliptic Equ., 64 (2019), 1325-1344.  doi: 10.1080/17476933.2018.1523898.

[28]

X. WangP. Niu and X. Cui, A Liouville type theorem to an extension problem relating to the Heisenberg group, Commun. Pure Appl. Anal., 17 (2018), 2379-2394.  doi: 10.3934/cpaa.2018113.

[29]

P. Wang and P. Niu, Liouville's theorem for a fractional elliptic system, Discrete Contin. Dyn. Syst., 39 (2019), 1545-1558.  doi: 10.3934/dcds.2019067.

[30]

X. Yu, Liouville type theorems for two mixed boundary value problems with general nonlinearities, J. Math. Anal. Appl., 462 (2018), 305-322.  doi: 10.1016/j.jmaa.2018.02.009.

[31]

L. ZhangM. Yu and J. He, A Liouville theorem for a class of fractional systems in $\mathbb{R}^{n}_{+}$, J. Differ. Equ., 263 (2017), 6025-6065.  doi: 10.1016/j.jde.2017.07.009.

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