March  2019, 39(3): 1573-1583. doi: 10.3934/dcds.2019069

Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China

* Corresponding author: Pengcheng Niu

Received  January 2018 Revised  April 2018 Published  December 2018

Fund Project: The authors are supported by the National Natural Science Foundation of China (No.11771354) and the first author also supported by Excellent Doctorate Cultivating Foundation of Northwestern Polytechnical University.

In this paper, we consider the fractional p-Laplacian equation
$( - \Delta )_p^su(x) = f(u(x)), $
where the fractional p-Laplacian is of the form
$( - \Delta )_p^su(x) = {C_{n, s, p}}PV\int_{{\mathbb{R}^n}} {\frac{{{{\left| {u(x) - u(y)} \right|}^{p - 2}}(u(x) - u(y))}}{{{{\left| {x - y} \right|}^{n + sp}}}}} dy.$
By proving a narrow region principle to the equation above and extending the direct method of moving planes used in fractional Laplacian equations, we establish the radial symmetry in the unit ball and nonexistence on the half space for the solutions, respectively.
Citation: Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
References:
[1]

C. BjorlandL. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379.  Google Scholar

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C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb., 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

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L. BrascoE. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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L. F. Cao and Z. H. Dai, A Liouville-type theorem for an integral equation on a half-space ${\mathbb{R}}_ + ^n,$, J. Math. Anal. Appl., 389 (2012), 1365-1373.  doi: 10.1016/j.jmaa.2012.01.015.  Google Scholar

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H. Chen and Z. X. Lv, The properties of positive solutions to an integral system involving Wolff potential, Discrete Contin. Dyn. Syst., 34 (2014), 1879-1904.  doi: 10.3934/dcds.2014.34.1879.  Google Scholar

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W. X. Chen and C. M. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.  Google Scholar

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W. X. Chen, C. M. Li and G. F. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.  Google Scholar

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W. X. ChenY. Q. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.  Google Scholar

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W. X. ChenC. M. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

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W. X. ChenC. M. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

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W. X. ChenC. 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.  Google Scholar

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W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.  Google Scholar

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R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood -Sobolev inequality, Calc. Var. Partical Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.  Google Scholar

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X. L. HanG. Z. Lu and J. Y. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.  Google Scholar

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Y. T. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.  Google Scholar

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B. Y. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135.  doi: 10.1016/j.na.2016.08.022.  Google Scholar

[19]

G. Z. Lu and J. Y. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.  Google Scholar

[20]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[21]

P. C. NiuL. Y. Wu and X. X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Frac. Calc. Appl. Anal., 21 (2018), 552-574.  doi: 10.1515/fca-2018-0030.  Google Scholar

[22]

A. Quaas and A. Xia, Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in RN involving fractional Laplacian, Discrete Contin. Dyn. Syst., 37 (2017), 2653-2668.  doi: 10.3934/dcds.2017113.  Google Scholar

[23]

P. Y. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl., 450 (2017), 982-995.  doi: 10.1016/j.jmaa.2017.01.070.  Google Scholar

[24]

P. Y. Wang and P. C. Niu, Liouville's Theorem for a Fractional Elliptic System, to appeared in Discrete Contin. Dyn. Syst., 2018. Google Scholar

show all references

References:
[1]

C. BjorlandL. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379.  Google Scholar

[2]

C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb., 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

[3]

L. BrascoE. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.  Google Scholar

[4]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[5]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.  doi: 10.1002/cpa.20274.  Google Scholar

[6]

L. F. Cao and Z. H. Dai, A Liouville-type theorem for an integral equation on a half-space ${\mathbb{R}}_ + ^n,$, J. Math. Anal. Appl., 389 (2012), 1365-1373.  doi: 10.1016/j.jmaa.2012.01.015.  Google Scholar

[7]

H. Chen and Z. X. Lv, The properties of positive solutions to an integral system involving Wolff potential, Discrete Contin. Dyn. Syst., 34 (2014), 1879-1904.  doi: 10.3934/dcds.2014.34.1879.  Google Scholar

[8]

W. X. Chen and C. M. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.  Google Scholar

[9]

W. X. Chen, C. M. Li and G. F. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.  Google Scholar

[10]

W. X. ChenY. Q. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.  Google Scholar

[11]

W. X. ChenC. M. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

[12]

W. X. ChenC. M. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[13]

W. X. ChenC. 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.  Google Scholar

[14]

W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.  Google Scholar

[15]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood -Sobolev inequality, Calc. Var. Partical Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.  Google Scholar

[16]

X. L. HanG. Z. Lu and J. Y. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.  Google Scholar

[17]

Y. T. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.  Google Scholar

[18]

B. Y. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135.  doi: 10.1016/j.na.2016.08.022.  Google Scholar

[19]

G. Z. Lu and J. Y. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.  Google Scholar

[20]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[21]

P. C. NiuL. Y. Wu and X. X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Frac. Calc. Appl. Anal., 21 (2018), 552-574.  doi: 10.1515/fca-2018-0030.  Google Scholar

[22]

A. Quaas and A. Xia, Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in RN involving fractional Laplacian, Discrete Contin. Dyn. Syst., 37 (2017), 2653-2668.  doi: 10.3934/dcds.2017113.  Google Scholar

[23]

P. Y. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl., 450 (2017), 982-995.  doi: 10.1016/j.jmaa.2017.01.070.  Google Scholar

[24]

P. Y. Wang and P. C. Niu, Liouville's Theorem for a Fractional Elliptic System, to appeared in Discrete Contin. Dyn. Syst., 2018. Google Scholar

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