April  2021, 20(4): 1431-1445. doi: 10.3934/cpaa.2021027

A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations

1. 

School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical, Sciences, Central China Normal University, Wuhan, 430079, China

2. 

Department of Mathematical Sciences, Yeshiva University, New York, NY, USA

Received  September 2020 Revised  December 2020 Published  March 2021

Fund Project: This work was supported by NSFC grant 11771166, Hubei Key Laboratory of Mathematical Sciences, Program for Changjiang Scholars and Innovative Research Team in University #IRT_17R46 and China Scholarship Council

In this paper, we proved a fractional Kirchhoff version of Hopf lemma for anti-symmetry functions and applied it to prove the symmetry and monotonicity of solutions for fractional Kirchhoff equations in the whole space by method of moving planes. We also obtain radially symmetry and monotonicity of solutions for fractional Kirchhoff equations in the unit ball. As far as we know, this is the first time to apply direct method of moving planes to fractional Kirchhoff problems.

Citation: Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1431-1445. doi: 10.3934/cpaa.2021027
References:
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C. O. Alves and F. J. S. A. Correa, On existence of solutions for a class of problem involving a nonlinear operator, Commun. Appl. Nonlinear Anal., 8 (2001), 43-56.   Google Scholar

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G. AutuoriA. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014.  Google Scholar

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W. ChenC. 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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A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

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X. He and W. Zou, Ground state solutions for a class of fractional Kirchhoff equations with critical growth, Sci. China Math., 62 (2019), 853-890.  doi: 10.1007/s11425-017-9399-6.  Google Scholar

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G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

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C. Li and W. Chen, A Hopf type lemma for fractional equations, Proc. Amer. Math. Soc., 147 (2019), 1565-1575.  doi: 10.1090/proc/14342.  Google Scholar

[13]

G. Li and Y. Niu, The existence and local uniqueness of multi-peak positive solutions to a class of Kirchhoff equation, Acta Math. Sci., 40 (2020), 1-23.  doi: 10.1007/s10473-020-0107-y.  Google Scholar

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Y. Li and W. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in $ {\mathbb R}^n$, Commun. Partial Differ. Equ., 18 (1993), 1043-1054.  doi: 10.1080/03605309308820960.  Google Scholar

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L. J. L, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346.  doi: 10.1016/S0304-0208(08)70870-3.  Google Scholar

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S. I. Pohozaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N. S.), 96 (1975), 152–166.  Google Scholar

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P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $ {\mathbb R}^n$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.  doi: 10.4171/RMI/879.  Google Scholar

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B. Zhang and L. Wang, Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 1061-1081.  doi: 10.1017/prm.2018.105.  Google Scholar

show all references

References:
[1]

C. O. Alves and F. J. S. A. Correa, On existence of solutions for a class of problem involving a nonlinear operator, Commun. Appl. Nonlinear Anal., 8 (2001), 43-56.   Google Scholar

[2]

P. and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605.  Google Scholar

[3]

G. AutuoriA. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014.  Google Scholar

[4]

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

[5]

W. Chen and C. Li, Maximum principle 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

[6]

W. ChenC. 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

[7]

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

[8]

W. Chen and S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. S., 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055.  Google Scholar

[9]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[10]

X. He and W. Zou, Ground state solutions for a class of fractional Kirchhoff equations with critical growth, Sci. China Math., 62 (2019), 853-890.  doi: 10.1007/s11425-017-9399-6.  Google Scholar

[11]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[12]

C. Li and W. Chen, A Hopf type lemma for fractional equations, Proc. Amer. Math. Soc., 147 (2019), 1565-1575.  doi: 10.1090/proc/14342.  Google Scholar

[13]

G. Li and Y. Niu, The existence and local uniqueness of multi-peak positive solutions to a class of Kirchhoff equation, Acta Math. Sci., 40 (2020), 1-23.  doi: 10.1007/s10473-020-0107-y.  Google Scholar

[14]

Y. Li and W. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in $ {\mathbb R}^n$, Commun. Partial Differ. Equ., 18 (1993), 1043-1054.  doi: 10.1080/03605309308820960.  Google Scholar

[15]

L. J. L, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346.  doi: 10.1016/S0304-0208(08)70870-3.  Google Scholar

[16]

S. I. Pohozaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N. S.), 96 (1975), 152–166.  Google Scholar

[17]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $ {\mathbb R}^n$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.  doi: 10.4171/RMI/879.  Google Scholar

[18]

B. Zhang and L. Wang, Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 1061-1081.  doi: 10.1017/prm.2018.105.  Google Scholar

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