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Liouville's theorem for a fractional elliptic system
Symmetry properties in systems of fractional Laplacian equations
1. | Department of Applied Mathematics, Donghua University, Shanghai 201620, China |
2. | Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80302, USA |
We consider the systems of fractional Laplacian equations in a domain(bounded or unbounded) in $\mathbb{R}^n$. By using a direct method of moving planes, we show that $u_i(x)$ ($i = 1,2,···,m$) are radial symmetric about the same point and strictly decreasing in the radial direction with respect to this point. Comparing with Zhuo-Chen-Cui-Yuan [
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A. D. Aleksandrov,
Uniqueness theorems for surfaces in the large, Amer. Math. Soc. Transl., 21 (1962), 412-416.
|
[2] |
H. Berestycki and L. Nirenberg,
Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, Journal of Geometry and Physics, 5 (1988), 237-275.
doi: 10.1016/0393-0440(88)90006-X. |
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H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Boletim da Sociedade Brasileira de Matemática-Bulletin/Brazilian Mathematical Society, 22 (1991), 1-37.
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C. Brändle, E. Colorado, A. de Pablo and U. Sánchez,
A concaveconvex elliptic problem involving the fractional Laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 143 (2013), 39-71.
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J. Busca and B. Sirakov,
Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.
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[6] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[7] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Mathematics, 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[8] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[9] |
M. Cai and L. Ma,
Moving planes for nonlinear fractional Lapla- cian equation with negative powers, Discrete and Continuous Dynamical Systems-Series A, 38 (2018), 4603-4615.
doi: 10.3934/dcds.2018201. |
[10] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire,
Regularity of radial extremal solutions for some non-local semilinear equations, Communications in Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[11] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Advances in Mathematics, 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[12] |
W. Chen and J. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[13] |
W. Chen and C. Li,
Classifcation of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 29 (2009), 949-960.
doi: 10.1016/S0252-9602(09)60079-5. |
[14] |
W. Chen, C. Li and B. Ou,
Classifcation of solutions for an integral equation, Communications on Pure and Applied Mathematics, 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[15] |
T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications,
Communications in Contemporary Mathematics, 19 (2017), 1750018(12pages).
doi: 10.1142/S0219199717500183. |
[16] |
S. Dipierro, G. Palatucci and E. Valdinoci,
Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216.
|
[17] |
P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian,
Commun. Contemp. Math., 16 (2014), 1350023(24pages).
doi: 10.1142/S0219199713500235. |
[18] |
D. G. de Figueiredo,
Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123.
doi: 10.1007/BF01193947. |
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D. G. de Figueiredo and P. L. Felmer,
A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 21 (1994), 387-397.
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B. Gidas, W. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Communications in Mathematical Physics, 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[21] |
B. Gidas, W. Ni and L. Nirenberg,
Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, Adv. Math. Suppl. Stud. A, 7 (1981), 369-402.
|
[22] |
S. Jarohs and T. Weth,
Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali di Matematica Pura ed Applicata, 195 (2016), 273-291.
doi: 10.1007/s10231-014-0462-y. |
[23] |
C. Li, Z. G. Wu and H. Xu,
Maximum principles and Bôcher type theorems, Pro Natl Acad Sci USA, 115 (2018), 6976-6979.
|
[24] |
C. Li and Z. G. Wu,
Radial symmetry for systems of fractional Laplacian, Acta Mathematica Scientia, 38 (2018), 1567-1582.
doi: 10.1016/S0252-9602(18)30832-4. |
[25] |
C. Li,
Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Communications in Partial Differential Equations, 16 (1991), 491-526.
doi: 10.1080/03605309108820766. |
[26] |
C. Li,
Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Communications in Partial Differential Equations, 16 (1991), 585-615.
doi: 10.1080/03605309108820770. |
[27] |
B. Liu and L. Ma,
Radial symmetry results for fractional Laplacian systems, Nonlinear Analysis: Theory, Methods Applications, 146 (2016), 120-135.
doi: 10.1016/j.na.2016.08.022. |
[28] |
B. Liu,
Direct method of moving planes for logarithmic Laplacian system in bounded domains, Discrete and Continuous Dynamical Systems - Series A, 38 (2018), 5339-5349.
doi: 10.3934/dcds.2018235. |
[29] |
Y. Lü and C. Zhou,
Symmetry for an integral system with general nonlinearity, Discrete Continuous Dynamical Systems -A, 39 (2019), 1533-1543.
doi: 10.3934/dcds.2018121. |
[30] |
L. Ma and B. Liu,
Symmetry results for decay solutions of elliptic systems in the whole space, Advances in Mathematics, 225 (2010), 3052-3063.
doi: 10.1016/j.aim.2010.05.022. |
[31] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, Journal de Mathématiques Pures et Appliquées, 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[32] |
J. Serrin,
A symmetry problem in potential theory, Archive for Rational Mechanics and Analysis, 43 (1971), 304-318.
doi: 10.1007/BF00250468. |
[33] |
R. Servadei and E. Valdinoci,
Mountain Pass solutions for non-local elliptic operators, Journal Mathematical Analysis Applications, 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[34] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications Pure Applied Mathematics, 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[35] |
B. Sirakov,
On symmetry in elliptic systems, Appl. Anal., 41 (1991), 1-9.
doi: 10.1080/00036819108840012. |
[36] |
W. C. Troy,
Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413.
doi: 10.1016/0022-0396(81)90113-3. |
[37] |
L. Zhang, C. Li, W. Chen and T. Cheng,
A Liouville theorem for $α$-harmonic functions in $\mathbb{R}_+^n$, Discrete Continuous Dynamical Systems -A, 36 (2016), 1721-1736.
doi: 10.3934/dcds.2016.36.1721. |
[38] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
A Liouville theorem for the fractional Laplacian, Discrete Continuous Dynamical Systems -A, 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
show all references
References:
[1] |
A. D. Aleksandrov,
Uniqueness theorems for surfaces in the large, Amer. Math. Soc. Transl., 21 (1962), 412-416.
|
[2] |
H. Berestycki and L. Nirenberg,
Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, Journal of Geometry and Physics, 5 (1988), 237-275.
doi: 10.1016/0393-0440(88)90006-X. |
[3] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Boletim da Sociedade Brasileira de Matemática-Bulletin/Brazilian Mathematical Society, 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[4] |
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez,
A concaveconvex elliptic problem involving the fractional Laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[5] |
J. Busca and B. Sirakov,
Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.
doi: 10.1006/jdeq.1999.3701. |
[6] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[7] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Mathematics, 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[8] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[9] |
M. Cai and L. Ma,
Moving planes for nonlinear fractional Lapla- cian equation with negative powers, Discrete and Continuous Dynamical Systems-Series A, 38 (2018), 4603-4615.
doi: 10.3934/dcds.2018201. |
[10] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire,
Regularity of radial extremal solutions for some non-local semilinear equations, Communications in Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[11] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Advances in Mathematics, 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[12] |
W. Chen and J. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[13] |
W. Chen and C. Li,
Classifcation of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 29 (2009), 949-960.
doi: 10.1016/S0252-9602(09)60079-5. |
[14] |
W. Chen, C. Li and B. Ou,
Classifcation of solutions for an integral equation, Communications on Pure and Applied Mathematics, 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[15] |
T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications,
Communications in Contemporary Mathematics, 19 (2017), 1750018(12pages).
doi: 10.1142/S0219199717500183. |
[16] |
S. Dipierro, G. Palatucci and E. Valdinoci,
Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216.
|
[17] |
P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian,
Commun. Contemp. Math., 16 (2014), 1350023(24pages).
doi: 10.1142/S0219199713500235. |
[18] |
D. G. de Figueiredo,
Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123.
doi: 10.1007/BF01193947. |
[19] |
D. G. de Figueiredo and P. L. Felmer,
A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 21 (1994), 387-397.
|
[20] |
B. Gidas, W. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Communications in Mathematical Physics, 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[21] |
B. Gidas, W. Ni and L. Nirenberg,
Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, Adv. Math. Suppl. Stud. A, 7 (1981), 369-402.
|
[22] |
S. Jarohs and T. Weth,
Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali di Matematica Pura ed Applicata, 195 (2016), 273-291.
doi: 10.1007/s10231-014-0462-y. |
[23] |
C. Li, Z. G. Wu and H. Xu,
Maximum principles and Bôcher type theorems, Pro Natl Acad Sci USA, 115 (2018), 6976-6979.
|
[24] |
C. Li and Z. G. Wu,
Radial symmetry for systems of fractional Laplacian, Acta Mathematica Scientia, 38 (2018), 1567-1582.
doi: 10.1016/S0252-9602(18)30832-4. |
[25] |
C. Li,
Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Communications in Partial Differential Equations, 16 (1991), 491-526.
doi: 10.1080/03605309108820766. |
[26] |
C. Li,
Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Communications in Partial Differential Equations, 16 (1991), 585-615.
doi: 10.1080/03605309108820770. |
[27] |
B. Liu and L. Ma,
Radial symmetry results for fractional Laplacian systems, Nonlinear Analysis: Theory, Methods Applications, 146 (2016), 120-135.
doi: 10.1016/j.na.2016.08.022. |
[28] |
B. Liu,
Direct method of moving planes for logarithmic Laplacian system in bounded domains, Discrete and Continuous Dynamical Systems - Series A, 38 (2018), 5339-5349.
doi: 10.3934/dcds.2018235. |
[29] |
Y. Lü and C. Zhou,
Symmetry for an integral system with general nonlinearity, Discrete Continuous Dynamical Systems -A, 39 (2019), 1533-1543.
doi: 10.3934/dcds.2018121. |
[30] |
L. Ma and B. Liu,
Symmetry results for decay solutions of elliptic systems in the whole space, Advances in Mathematics, 225 (2010), 3052-3063.
doi: 10.1016/j.aim.2010.05.022. |
[31] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, Journal de Mathématiques Pures et Appliquées, 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[32] |
J. Serrin,
A symmetry problem in potential theory, Archive for Rational Mechanics and Analysis, 43 (1971), 304-318.
doi: 10.1007/BF00250468. |
[33] |
R. Servadei and E. Valdinoci,
Mountain Pass solutions for non-local elliptic operators, Journal Mathematical Analysis Applications, 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[34] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications Pure Applied Mathematics, 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[35] |
B. Sirakov,
On symmetry in elliptic systems, Appl. Anal., 41 (1991), 1-9.
doi: 10.1080/00036819108840012. |
[36] |
W. C. Troy,
Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413.
doi: 10.1016/0022-0396(81)90113-3. |
[37] |
L. Zhang, C. Li, W. Chen and T. Cheng,
A Liouville theorem for $α$-harmonic functions in $\mathbb{R}_+^n$, Discrete Continuous Dynamical Systems -A, 36 (2016), 1721-1736.
doi: 10.3934/dcds.2016.36.1721. |
[38] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
A Liouville theorem for the fractional Laplacian, Discrete Continuous Dynamical Systems -A, 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
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