American Institute of Mathematical Sciences

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  April 2021 Early access  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 and Applied Analysis, 2021, 20 (4) : 1431-1445. doi: 10.3934/cpaa.2021027
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. [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. [3] G. Autuori, A. 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. [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. [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. [6] W. Chen, C. 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. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [8] W. Chen and S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. S., 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055. [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. [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. [11] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [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. [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. [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. [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. [16] S. I. Pohozaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N. S.), 96 (1975), 152–166. [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. [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.

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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. [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. [3] G. Autuori, A. 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. [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. [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. [6] W. Chen, C. 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. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [8] W. Chen and S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. S., 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055. [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. [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. [11] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [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. [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. [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. [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. [16] S. I. Pohozaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N. S.), 96 (1975), 152–166. [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. [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.
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