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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[24]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[24]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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