# American Institute of Mathematical Sciences

October  2018, 38(10): 5339-5349. doi: 10.3934/dcds.2018235

## Direct method of moving planes for logarithmic Laplacian system in bounded domains

 School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District Beijing 100083, China

* Corresponding author: Baiyu Liu

Received  May 2018 Revised  May 2018 Published  July 2018

Fund Project: The author is supported by the National Natural Science Foundation of China (No.11671031) and the Fundamental Research Funds for the Central Universities FRF-BR-17-013A.

Chen, Li and Li [Adv. Math., 308(2017), pp. 404-437] developed a direct method of moving planes for the fractional Laplacian. In this paper, we extend their method to the logarithmic Laplacian. We consider both the logarithmic equation and the system. To carry out the method, we establish two kinds of narrow region principle for the equation and the system separately. Then using these narrow region principles, we give the radial symmetry results for the solutions to semi-linear logarithmic Laplacian equations and systems on the ball.

Citation: Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235
##### References:
 [1] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasi. Mat., 22 (1991), 1-37.  doi: 10.1007/BF01244896. [2] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, P. Roy. Soc. Edinb. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [3] X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025. [4] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [5] W. X. Chen and C. M. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Springfield, United States, 2010. [6] W. X. Chen, C. M. 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. [7] W. X. Chen, C. M. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Part. Diff. Eq., 30 (2005), 59-65.  doi: 10.1081/PDE-200044445. [8] W. X. Chen, C. M. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Cont. Dyn.-A, 12 (2005), 347-354. [9] W. X. Chen, C. M. Li and B. Ou, Classification of solutions for an integral equationr, Commun. Pur. Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [10] H. Y. Chen and T. Weth, Qualitative properties of Logarithmic Laplacian and Dirichlet elliptic equations, preprint, arXiv: 1710.03416v3 [11] W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differ. Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029. [12] R. Cont and P. Tankov,, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2004. [13] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., vol. 219, Springer-Verlag, Berlin-New York, 1976. Translated from French by C. W. John. [14] M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Func. Anal., 263 (2012), 2205-2227.  doi: 10.1016/j.jfa.2012.06.018. [15] P. Felmer, A. Quass and J. G. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, P. Roy. Soc. Edinb. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746. [16] P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commum. Contemp. Math., 16 (2014), 1350023, 24pp. doi: 10.1142/S0219199713500235. [17] B. Gidas, Wei-Ming Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125. [18] S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pur. Appl., 195 (2016), 273-291.  doi: 10.1007/s10231-014-0462-y. [19] C. Jin and C. Li, Symmetry of solutions to some integral equations, P.Am. Math. Soc., 134 (2006), 1661-1670.  doi: 10.1090/S0002-9939-05-08411-X. [20] Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [21] C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.  doi: 10.1137/080712301. [22] B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal.-Theor., 146 (2016), 120-135.  doi: 10.1016/j.na.2016.08.022. [23] L. Ma and D. Z. Chen, A Liouville type theorem for an integral system, Commun. Pur. Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855. [24] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3. [25] E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312.  doi: 10.1016/j.aim.2007.08.009. [26] A. Quaas and A. Xia, A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian, Nonlinearity, 29 (2016), 2279-2297.  doi: 10.1088/0951-7715/29/8/2279. [27] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pur. Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [28] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in Nonlinear Partial Differential Equations, (eds.) 7 Springer, Berlin, Heidelberg, (2012), 271–298. doi: 10.1007/978-3-642-25361-4_15. [29] X. H. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial. Dif., 46 (2013), 75-95.  doi: 10.1007/s00526-011-0474-z. [30] R. Zhuo, W. X. Chen, X. W. Cui and Z. X. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Cont. Dyn.-A, 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

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##### References:
 [1] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasi. Mat., 22 (1991), 1-37.  doi: 10.1007/BF01244896. [2] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, P. Roy. Soc. Edinb. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [3] X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025. [4] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [5] W. X. Chen and C. M. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Springfield, United States, 2010. [6] W. X. Chen, C. M. 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. [7] W. X. Chen, C. M. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Part. Diff. Eq., 30 (2005), 59-65.  doi: 10.1081/PDE-200044445. [8] W. X. Chen, C. M. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Cont. Dyn.-A, 12 (2005), 347-354. [9] W. X. Chen, C. M. Li and B. Ou, Classification of solutions for an integral equationr, Commun. Pur. Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [10] H. Y. Chen and T. Weth, Qualitative properties of Logarithmic Laplacian and Dirichlet elliptic equations, preprint, arXiv: 1710.03416v3 [11] W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differ. Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029. [12] R. Cont and P. Tankov,, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2004. [13] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., vol. 219, Springer-Verlag, Berlin-New York, 1976. Translated from French by C. W. John. [14] M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Func. Anal., 263 (2012), 2205-2227.  doi: 10.1016/j.jfa.2012.06.018. [15] P. Felmer, A. Quass and J. G. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, P. Roy. Soc. Edinb. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746. [16] P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commum. Contemp. Math., 16 (2014), 1350023, 24pp. doi: 10.1142/S0219199713500235. [17] B. Gidas, Wei-Ming Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125. [18] S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pur. Appl., 195 (2016), 273-291.  doi: 10.1007/s10231-014-0462-y. [19] C. Jin and C. Li, Symmetry of solutions to some integral equations, P.Am. Math. Soc., 134 (2006), 1661-1670.  doi: 10.1090/S0002-9939-05-08411-X. [20] Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [21] C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.  doi: 10.1137/080712301. [22] B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal.-Theor., 146 (2016), 120-135.  doi: 10.1016/j.na.2016.08.022. [23] L. Ma and D. Z. Chen, A Liouville type theorem for an integral system, Commun. Pur. Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855. [24] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3. [25] E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312.  doi: 10.1016/j.aim.2007.08.009. [26] A. Quaas and A. Xia, A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian, Nonlinearity, 29 (2016), 2279-2297.  doi: 10.1088/0951-7715/29/8/2279. [27] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pur. Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [28] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in Nonlinear Partial Differential Equations, (eds.) 7 Springer, Berlin, Heidelberg, (2012), 271–298. doi: 10.1007/978-3-642-25361-4_15. [29] X. H. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial. Dif., 46 (2013), 75-95.  doi: 10.1007/s00526-011-0474-z. [30] R. Zhuo, W. X. Chen, X. W. Cui and Z. X. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Cont. Dyn.-A, 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.
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