A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. (English) Zbl 1466.35168
Summary: 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.
Keywords:
fractional Kirchhoff equations; Hopf type lemma; anti-symmetric functions; moving plane; symmetry and monotonicityReferences:
| [1] | C. O. Alves; F. J. S. A. Correa, On existence of solutions for a class of problem involving a nonlinear operator, Commun. Appl. Nonlinear Anal., 8, 43-56 (2001) · Zbl 1011.35058 |
| [2] | P.; S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108, 247-262 (1992) · Zbl 0785.35067 · doi:10.1007/BF02100605 |
| [3] | G. Autuori; A. Fiscella; P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125, 699-714 (2015) · Zbl 1323.35015 · doi:10.1016/j.na.2015.06.014 |
| [4] | L. Caffarelli; L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32, 1245-1260 (2007) · Zbl 1143.26002 · doi:10.1080/03605300600987306 |
| [5] | W. Chen; C. Li, Maximum principle for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335, 735-758 (2018) · Zbl 1395.35055 · doi:10.1016/j.aim.2018.07.016 |
| [6] | W. Chen; C. Li; Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308, 404-437 (2017) · Zbl 1362.35320 · doi:10.1016/j.aim.2016.11.038 |
| [7] | W. Chen; C. Li; B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59, 330-343 (2006) · Zbl 1093.45001 · doi:10.1002/cpa.20116 |
| [8] | W. Chen; S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. S., 39, 1269-1310 (2019) · Zbl 1408.35038 · doi:10.3934/dcds.2019055 |
| [9] | A. Fiscella; E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94, 156-170 (2014) · Zbl 1283.35156 · doi:10.1016/j.na.2013.08.011 |
| [10] | X. He; W. Zou, Ground state solutions for a class of fractional Kirchhoff equations with critical growth, Sci. China Math., 62, 853-890 (2019) · Zbl 1415.35121 · doi:10.1007/s11425-017-9399-6 |
| [11] | G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. |
| [12] | C. Li; W. Chen, A Hopf type lemma for fractional equations, Proc. Amer. Math. Soc., 147, 1565-1575 (2019) · Zbl 1417.35034 · doi:10.1090/proc/14342 |
| [13] | G. Li; Y. Niu, The existence and local uniqueness of multi-peak positive solutions to a class of Kirchhoff equation, Acta Math. Sci., 40, 1-23 (2020) · Zbl 1499.35271 · doi:10.1007/s10473-020-0107-y |
| [14] | Y. Li; W. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb R}^n\), Commun. Partial Differ. Equ., 18, 1043-1054 (1993) · Zbl 0788.35042 · doi:10.1080/03605309308820960 |
| [15] | L. J. L., On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30, 284-346 (1978) · Zbl 0404.35002 · 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. · Zbl 0309.35051 |
| [17] | P. Pucci; S. Saldi, Critical stationary Kirchhoff equations in \({\mathbb R}^n\) involving nonlocal operators, Rev. Mat. Iberoam., 32, 1-22 (2016) · Zbl 1405.35045 · doi:10.4171/RMI/879 |
| [18] | B. Zhang; L. Wang, Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 149, 1061-1081 (2019) · Zbl 1442.35501 · doi:10.1017/prm.2018.105 |
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