Symmetry properties in systems of fractional Laplacian equations

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March 2019, 39(3): 1559-1571. doi: 10.3934/dcds.2019068

Symmetry properties in systems of fractional Laplacian equations

Zhigang Wu ^1, and Hao Xu ^2,,

1.	Department of Applied Mathematics, Donghua University, Shanghai 201620, China
2.	Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80302, USA

* Corresponding author: Hao Xu

Received April 2018 Revised May 2018 Published December 2018

Fund Project: The first author is supported by Natural Science Foundation of Shanghai grant 16ZR1402100.

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 [38], our results not only include subcritical case and critical case but also include supercritical case, and we need not the nonlinear terms to be homogenous. In addition, we completely remove the nonnegativity of $\frac{\partial f_i}{\partial u_i}$. Above all, to the best of our knowledge, it is the first result on the symmetric property of the system containing the gradient of the solution in the nonlinear terms.

Keywords: System of fractional Laplacian, method of moving planes, maximum principles.

Mathematics Subject Classification: Primary: 35B50; Secondary: 35R11.

Citation: Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068

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.

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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