Liouville theorems on the upper half space

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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 and 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.
[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.
[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.
[4]	T. Carleman, Zur Theorie de Minimalflächen, Math. Z., 9 (1921), 154-160. doi: 10.1007/BF01378342.
[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.
[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.
[7]	J. Dou, L. Sun, L. Wang and M. Zhu, Divergent operator with degeneracy and related sharp inequalities, preprint, arXiv: 1910.13924.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[4]	T. Carleman, Zur Theorie de Minimalflächen, Math. Z., 9 (1921), 154-160. doi: 10.1007/BF01378342.
[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.
[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.
[7]	J. Dou, L. Sun, L. Wang and M. Zhu, Divergent operator with degeneracy and related sharp inequalities, preprint, arXiv: 1910.13924.
[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.
[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.
[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.
[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.
[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.

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