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A Liouville type theorem to an extension problem relating to the Heisenberg group
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi, 710129, China |
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.
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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