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A Liouville type theorem to an extension problem relating to the Heisenberg group

  • * Corresponding author

    * Corresponding author 
This work is supported by the Natural Science Basic Research plan in Shaanxi Province of China (Grant No. 2016JM1023). The first author partially supported by NSFC (Grant No. 11471188 & 11771354) and the National Science Foundation for Young Scientists of China (Grant No. 11601427).
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  • We establish a Liouville type theorem for nonnegative cylindrical solutions to the extension problem corresponding to a fractional CR covariant equation on the Heisenberg group by using the generalized CR inversion and the moving plane method.

    Mathematics Subject Classification: 35A01, 35J57, 35D99.


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  • [1] H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a half space, Boundary Value Problems for Partial Differential Equations, ed. by J. L. Lions et al., Masson, Paris (1993), 27–42.
    [2] I. BirindelliI. Capuzzo Dolcetta and A. Cutrí, Liouville theorems for semilinear equations on the Heisenberg group, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 295-308.  doi: 10.1016/S0294-1449(97)80138-2.
    [3] I. Birindelli and A. Cutrí, A semi-linear problem for the Heisenberg Laplacian, Rend. Sem. Mat. Della Univ. Padova, 94 (1995), 137-153. 
    [4] I. Birindelli and J. Prajapat, Nonlinear Liouville theorems in the Heisenberg group via the moving plane method, Comm. Partial Diff. Eqs., 24 (1999), 1875-1890.  doi: 10.1080/03605309908821485.
    [5] I. Birindelli and J. Prajapat, Monotonicity and symmetry results for degenerate elliptic equations on nilpotent Lie groups, Pacific J. Math., 204 (2002), 1-17.  doi: 10.2140/pjm.2002.204.1.
    [6] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monogr. Math., Springer, New York, 2007.
    [7] J. M. Bony, Principe du Maximum, Inegalite de Harnack et unicite du probleme de Cauchy pour les operateurs ellipitiques degeneres, Ann. Inst. Fourier Grenobles, 19 (1969), 277-304. 
    [8] C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.
    [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqs., 2 (2007), 1245-1260.  doi: 10.1080/03605300600987306.
    [10] E. Cinti and J. Tan, A nonlinear Liouville theorem for fractional equations in the Heisenberg group, J. Math. Anal. Appl., 433 (2016), 434-454.  doi: 10.1016/j.jmaa.2015.07.050.
    [11] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.
    [12] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series, vol. 4, 2010.
    [13] 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.
    [14] W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.
    [15] W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Diff. Eqs., 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.
    [16] M. ChipotM. ChlebikM. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in ${\mathbb{H}}_{+}^{n}$ with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471.  doi: 10.1006/jmaa.1998.5958.
    [17] M. ChipotI. Shafrir and M. Fila, On the solutions to some elliptic equations with nonlinear Neumann boundary conditions, Advances in Diff. Equs., 1 (1996), 91-110. 
    [18] F. Ferrari and B. Franchi, Harnack inequality for fractional Laplacians in Carnot groups, Math. Z., 279 (2015), 435-458.  doi: 10.1007/s00209-014-1376-5.
    [19] G. B. Folland, Fundamental solution for subelliptic operators, Bull. Amer. Math. Soc., 79 (1979), 373-376.  doi: 10.1090/S0002-9904-1973-13171-4.
    [20] G. B. Folland and E. M. Stein, Estimates for the b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522.  doi: 10.1002/cpa.3160270403.
    [21] R. FrankM. GonzalezD. Monticelli and J. Tan, An extension problem for the CR fractional Laplacian, Adv. Math., 270 (2015), 97-137.  doi: 10.1016/j.aim.2014.09.026.
    [22] N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J., 41 (1992), 71-98.  doi: 10.1512/iumj.1992.41.41005.
    [23] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 35 (1982), 528-598.  doi: 10.1002/cpa.3160340406.
    [24] B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. 
    [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin: Springer-Verlage, 1983. doi: 10.1007/978-3-642-61798-0.
    [26] L. L. Helms, Introduction to Potential Theory, Pure and Applied Mathematics 22, Wiley-Interscience, New York, London, Sydney, 1969.
    [27] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.
    [28] D. S. Jerison, Boundary regularity in the dirichlet problem for $\Box$b on CR manifolds, Comm. Pure Appl. Math., 36 (1983), 143-181.  doi: 10.1002/cpa.3160360203.
    [29] D. S. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167-197. 
    [30] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag Berlin Heidelberg, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.
    [31] Y. Lou and M. Zhu, Classifications of nonnegative solutions to some elliptic problems, Differ. Integral Eqs., 12 (1999), 601-612. 
    [32] S. Terracini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differ. Integral Eqs., 8 (1995), 1911-1922. 
    [33] X. Wang, X. Cui and P. Niu, A Liouville theorem for the semilinear fractional CR covariant equation on the Heisenberg group, preprint.
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