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Zhang, Lihong; Ahmad, Bashir; Wang, Guotao; Ren, Xueyan

Radial symmetry of solution for fractional \(p\)- Laplacian system. (English) Zbl 1437.35015

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 196, Article ID 111801, 16 p. (2020).
Summary: In this paper, we investigate the method of moving planes for fractional \(p\)-Laplacian system. We firstly discuss the key ingredients for the method of moving planes such as maximum principle for anti-symmetric functions, decay at infinity and boundary estimate. Then we apply the method of moving planes to establish the radial symmetry and the monotonicity of the positive solutions for fractional \(p\)-Laplacian system in a unit ball or in the whole space.

Cited in 1 Review
Cited in 28 Documents

MSC:

35B06 Symmetries, invariants, etc. in context of PDEs
35A16 Topological and monotonicity methods applied to PDEs
35R11 Fractional partial differential equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B50 Maximum principles in context of PDEs

Keywords:

maximum principle for anti-symmetric function; decay at infinity; boundary estimate; radial symmetry and monotonicity; fractional \(p\)-Laplacian system
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References:

[1] Brandle, C.; Colorado, E.; de Pablo, A.; Sanchez, U., A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh, 143, 39-71 (2013) · Zbl 1290.35304
[2] Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32, 1245-1260 (2007) · Zbl 1143.26002
[3] Caffarelli, L.; Silvestre, L., Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62, 597-638 (2009) · Zbl 1170.45006
[4] Caffarelli, L.; Silvestre, L., Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62, 597-638 (2009) · Zbl 1170.45006
[5] Cai, M.; Ma, L., Moving planes for nonlinear fractional Laplacian equation with negative powers, Discrete Contin. Dyn. Syst. Ser. A, 38, 4603-4615 (2018) · Zbl 1397.35330
[6] Canino, A.; Montoro, L.; Sciunzi, B., The moving plane method for singular semilinear elliptic problems, Nonlinear Anal., 156, 61-69 (2017) · Zbl 1378.35132
[7] Chen, W.; Fang, Y.; Yang, R., Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274, 167-198 (2015) · Zbl 1372.35332
[8] Chen, W.; Li, C., (Methods on Nonlinear Elliptic Equations. Methods on Nonlinear Elliptic Equations, AIMS Book Series, vol. 4 (2010)) · Zbl 1214.35023
[9] Chen, W.; Li, C., Maximum principle for the fractional \(p -\) Laplacian and symmetry of solutions, Adv. Math., 335, 735-758 (2018) · Zbl 1395.35055
[10] Chen, W.; Li, C.; Li, Y., A direct method of moving planes for fractional Laplacian, Adv. Math., 308, 404-437 (2017) · Zbl 1362.35320
[11] Chen, W.; Li, C.; Li, G., Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations., 56, 29 (2017) · Zbl 1368.35110
[12] W. Chen, Y. Li, P. Ma, The fractional Laplacian, http://dx.doi.org/10.1142/10550. · Zbl 1451.35002
[13] Chen, Y.; Liu, B., Symmetry and non-existence of positive solutions for fractional \(p\)-Laplacian systems, Nonlinear Anal., 183, 303-322 (2019) · Zbl 1415.35276
[14] Chen, W.; Zhu, J., Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260, 4758-4785 (2016) · Zbl 1336.35089
[15] ciunzi, B., On the moving plane method for singular solutions to semilinear elliptic equations, J. Math. Pures Appl. (9), 108, 111-123 (2017) · Zbl 1371.35114
[16] Fang, Y.; Chen, W., A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Adv. Math., 229, 2835-2867 (2012) · Zbl 1250.35051
[17] Frank, R. L.; Lenzmann, E., Uniqueness and nondegeneracy of ground states for \(( - \Delta )^s Q + Q - Q^{\alpha + 1} = 0\), Acta Math., 210, 261-318 (2013) · Zbl 1307.35315
[18] Frank, R. L.; Lenzmann, E.; Silvestre, L., Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69, 1671-1726 (2016) · Zbl 1365.35206
[19] Frank, R. L.; Lieb, E., Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39, 85-99 (2010) · Zbl 1204.39024
[20] Jarohs, S.; Weth, T., Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl., 195, 273-291 (2016) · Zbl 1346.35010
[21] Jin, C.; Li, C., Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134, 1661-1670 (2006) · Zbl 1156.45300
[22] Jin, T.; Li, Y. Y.; Xiong, J., On a fractional Nirenberg problem, part I: blow up analysis and com-pactness of solutions, J. Eur. Math. Soc. (JEMS), 16, 1111-1171 (2014) · Zbl 1300.53041
[23] Jin, T.; Xiong, J., A fractional yemabe flow and some applications, J. Reine Angew. Math., 696, 187-223 (2014) · Zbl 1305.53067
[24] Lei, Y.; Li, C.; Ma, C., Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. Partial Differential Equations, 45, 43-61 (2012) · Zbl 1257.45005
[25] Lei, Y.; Li, C.; Ma, C., Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. Partial Differential Equations, 45, 43-61 (2012) · Zbl 1257.45005
[26] Li, C.; Ma, L., Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40, 1049-1057 (2008) · Zbl 1167.35347
[27] Liu, B.; Ma, L., Radial symmetry results for fractional Laplacian system, Nonlinear Anal., 146, 120-135 (2016) · Zbl 1348.35296
[28] Lu, G.; Zhu, J., Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42, 563-577 (2011) · Zbl 1231.35290
[29] Ma, L.; Chen, D., A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5, 855-859 (2006) · Zbl 1134.45007
[30] Ma, L.; Zhao, L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195, 455-467 (2010) · Zbl 1185.35260
[31] Silvestre, L., Regilarity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60, 1, 67-112 (2007) · Zbl 1141.49035
[32] Wang, G.; Ren, X.; Bai, Z.; Hou, W., Radial symmetry of standing waves for nonlinear fractional Hardy-Schrödinger equation, Appl. Math. Lett., 96, 131-137 (2019) · Zbl 1447.35302
[33] Zhuo, R.; Chen, W.; Cui, X.; Yuan, Z., Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36, 1125-1141 (2016) · Zbl 1322.31007
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