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September  2020, 40(9): 5373-5381. doi: 10.3934/dcds.2020231

Liouville theorems on the upper half space

Lei Wang 1,2,3, and  Meijun Zhu 3,

1. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
2. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
3. 

Department of Mathematics, The University of Oklahoma, Norman, OK 73019, USA

Received  October 2019 Revised  March 2020 Published  June 2020

In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $ a \in (0, 1) $ constants are the only $ C^1 $ up to the boundary positive solutions to $ div(x_n^a \nabla u) = 0 $ on the upper half space.

Keywords: Divergent operator, Liouville theorem, extension problem, positive kernels, modified Kelvin transformation, moving sphere method.
Mathematics Subject Classification: 35A09, 35B06, 35J15, 35J70, 35J75.
Citation: Lei Wang, Meijun Zhu. Liouville theorems on the upper half space. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5373-5381. doi: 10.3934/dcds.2020231
References:
[1]

S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Graduate Texts in Mathematics, 137, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-8137-3.  Google Scholar

[2]

H. P. Boas and R. P. Boas, Short proofs of three theorems on harmonic functions, Proc. Amer. Math. Soc., 102 (1988), 906-908.  doi: 10.1090/S0002-9939-1988-0934865-6.  Google Scholar

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S. Chen, A new family of sharp conformally invariant integral inequalities, Int. Math. Res. Not. IMRN, 2014 (2012), 1205-1220.  doi: 10.1093/imrn/rns248.  Google Scholar

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J. Dou, Q. Guo and M. Zhu, Subcritical approach to sharp Hardy-Littlewood-Sobolev type inequalities on the upper half space, Adv. Math., 312 (2017), 1-45.  doi: 10.1016/j.aim.2017.03.007.  Google Scholar

[7]

J. Dou, L. Sun, L. Wang and M. Zhu, Divergent operator with degeneracy and related sharp inequalities, preprint, arXiv: 1910.13924. Google Scholar

[8]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not. IMRN, 2015 (2013), 651-687.  doi: 10.1093/imrn/rnt213.  Google Scholar

[9]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar

[10]

M. Gluck, Subcritical approach to conformally invariant extension operators on the upper half space, J. Funct. Anal., 278 (2020), 46pp. doi: 10.1016/j.jfa.2018.08.012.  Google Scholar

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F. Hang, X. Wang and X. Yan, Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math., 61 (2008), 54-95.  doi: 10.1002/cpa.20193.  Google Scholar

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Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

show all references

References:
[1]

S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Graduate Texts in Mathematics, 137, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-8137-3.  Google Scholar

[2]

H. P. Boas and R. P. Boas, Short proofs of three theorems on harmonic functions, Proc. Amer. Math. Soc., 102 (1988), 906-908.  doi: 10.1090/S0002-9939-1988-0934865-6.  Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[4]

T. Carleman, Zur Theorie de Minimalflächen, Math. Z., 9 (1921), 154-160.  doi: 10.1007/BF01378342.  Google Scholar

[5]

S. Chen, A new family of sharp conformally invariant integral inequalities, Int. Math. Res. Not. IMRN, 2014 (2012), 1205-1220.  doi: 10.1093/imrn/rns248.  Google Scholar

[6]

J. Dou, Q. Guo and M. Zhu, Subcritical approach to sharp Hardy-Littlewood-Sobolev type inequalities on the upper half space, Adv. Math., 312 (2017), 1-45.  doi: 10.1016/j.aim.2017.03.007.  Google Scholar

[7]

J. Dou, L. Sun, L. Wang and M. Zhu, Divergent operator with degeneracy and related sharp inequalities, preprint, arXiv: 1910.13924. Google Scholar

[8]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not. IMRN, 2015 (2013), 651-687.  doi: 10.1093/imrn/rnt213.  Google Scholar

[9]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar

[10]

M. Gluck, Subcritical approach to conformally invariant extension operators on the upper half space, J. Funct. Anal., 278 (2020), 46pp. doi: 10.1016/j.jfa.2018.08.012.  Google Scholar

[11]

F. Hang, X. Wang and X. Yan, Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math., 61 (2008), 54-95.  doi: 10.1002/cpa.20193.  Google Scholar

[12]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

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