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:
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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

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