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November  2018, 17(6): 2379-2394. doi: 10.3934/cpaa.2018113

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

Xinjing Wang , Pengcheng Niu , and  Xuewei Cui

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi, 710129, China

* Corresponding author

Received  June 2017 Revised  January 2018 Published  June 2018

Fund Project: 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).

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.

Keywords: Heisenberg group, extension problem, Liouville type theorem, generalized CR inversion, moving plane method.
Mathematics Subject Classification: 35A01, 35J57, 35D99.
Citation: Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113
References:
[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. Birindelli, I. 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. Brändle, E. Colorado, A. 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. 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.

[14]

W. Chen, C. 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. Chipot, M. Chlebik, M. 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. Chipot, I. 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. Frank, M. Gonzalez, D. 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. Gidas, W. 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.

show all references

References:
[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. Birindelli, I. 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. Brändle, E. Colorado, A. 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. 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.

[14]

W. Chen, C. 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. Chipot, M. Chlebik, M. 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. Chipot, I. 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. Frank, M. Gonzalez, D. 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. Gidas, W. 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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