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August  2018, 38(8): 3939-3953. doi: 10.3934/dcds.2018171

Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space

Jingbo Dou 1,, and  Ye Li 2,

1. 

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China
2. 

Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA

* Corresponding author: Jingbo Dou

Received  September 2017 Revised  January 2018 Published  May 2018

In this paper we mainly classify the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space $\mathbb{R}_+^{n}$, and also present some remarks on the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the whole space $\mathbb{R}^{n}$. Our main techniques are Kelvin transformation and the method of moving spheres in integral forms.

Keywords: Logarithmic Hardy-Littlewood-Sobolev inequality, extremal functions, integral system, Kelvin transformation, method of moving spheres.
Mathematics Subject Classification: Primary: 45G15; Secondary: 31B10, 34A05.
Citation: Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
References:
[1]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math., 138 (1993), 213-242.  doi: 10.2307/2946638.  Google Scholar

[2]

T. P. Branson, L. Fontana and C. Morpurgo, Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Ann. of Math., 177 (2013), 1-52.  doi: 10.4007/annals.2013.177.1.1.  Google Scholar

[3]

E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $\mathbb{S}^n$, Geom. Funct. Anal., 2 (1992), 90-104.  doi: 10.1007/BF01895706.  Google Scholar

[4]

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.  Google Scholar

[5]

K. S. Chou and T. Y. H. Wan, Asymptotic radial symmetry for solutions of $Δ u + e^u= 0$ in a punctured disc, Pacific J. Math., 163 (1994), 269-276.  doi: 10.2140/pjm.1994.163.269.  Google Scholar

[6]

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

[7]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar

[8]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. D'Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551.  Google Scholar

[9]

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

[10]

C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^n$, Comment. Math. Helv., 73 (1998), 206-231.  doi: 10.1007/s000140050052.  Google Scholar

[11]

C. Morpurgo, The logarithmic Hardy-Littlewood-Sobolev inequality and extremals of zeta functions on $\mathbb{S}^n$, Geom. Funct. Anal., 6 (1996), 146-171.  doi: 10.1007/BF02246771.  Google Scholar

[12]

Q. A. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality on the half space $\mathbb{R}_+^n$, Int. Math. Res. Not. IMRN, 2017 (2017), 6187-6230.  doi: 10.1093/imrn/rnw108.  Google Scholar

[13]

W. M. Ni, On the elliptic equation $Δ u + Ke^\frac{n+2}{n-2} = 0$, its generalizations and applications in geometry, Indiana Univ. Math. J., 31 (1982), 493-529.  doi: 10.1512/iumj.1982.31.31040.  Google Scholar

[14]

E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys., 86 (1982), 321-326.  doi: 10.1007/BF01212171.  Google Scholar

[15]

J. Wei and X. Xu, Prescribing $Q$-curvature problem on $\mathbb{S}^n$, J. Funct. Anal., 257 (2009), 1995-2023.  doi: 10.1016/j.jfa.2009.06.024.  Google Scholar

[16]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.  Google Scholar

[17]

X. Xu, Uniqueness and non-existence theorems for conformally invariant equations, J. Funct. Anal., 222 (2005), 1-28.  doi: 10.1016/j.jfa.2004.07.003.  Google Scholar

show all references

References:
[1]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math., 138 (1993), 213-242.  doi: 10.2307/2946638.  Google Scholar

[2]

T. P. Branson, L. Fontana and C. Morpurgo, Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Ann. of Math., 177 (2013), 1-52.  doi: 10.4007/annals.2013.177.1.1.  Google Scholar

[3]

E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $\mathbb{S}^n$, Geom. Funct. Anal., 2 (1992), 90-104.  doi: 10.1007/BF01895706.  Google Scholar

[4]

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.  Google Scholar

[5]

K. S. Chou and T. Y. H. Wan, Asymptotic radial symmetry for solutions of $Δ u + e^u= 0$ in a punctured disc, Pacific J. Math., 163 (1994), 269-276.  doi: 10.2140/pjm.1994.163.269.  Google Scholar

[6]

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

[7]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar

[8]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. D'Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551.  Google Scholar

[9]

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

[10]

C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^n$, Comment. Math. Helv., 73 (1998), 206-231.  doi: 10.1007/s000140050052.  Google Scholar

[11]

C. Morpurgo, The logarithmic Hardy-Littlewood-Sobolev inequality and extremals of zeta functions on $\mathbb{S}^n$, Geom. Funct. Anal., 6 (1996), 146-171.  doi: 10.1007/BF02246771.  Google Scholar

[12]

Q. A. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality on the half space $\mathbb{R}_+^n$, Int. Math. Res. Not. IMRN, 2017 (2017), 6187-6230.  doi: 10.1093/imrn/rnw108.  Google Scholar

[13]

W. M. Ni, On the elliptic equation $Δ u + Ke^\frac{n+2}{n-2} = 0$, its generalizations and applications in geometry, Indiana Univ. Math. J., 31 (1982), 493-529.  doi: 10.1512/iumj.1982.31.31040.  Google Scholar

[14]

E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys., 86 (1982), 321-326.  doi: 10.1007/BF01212171.  Google Scholar

[15]

J. Wei and X. Xu, Prescribing $Q$-curvature problem on $\mathbb{S}^n$, J. Funct. Anal., 257 (2009), 1995-2023.  doi: 10.1016/j.jfa.2009.06.024.  Google Scholar

[16]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.  Google Scholar

[17]

X. Xu, Uniqueness and non-existence theorems for conformally invariant equations, J. Funct. Anal., 222 (2005), 1-28.  doi: 10.1016/j.jfa.2004.07.003.  Google Scholar

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