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

Previous Article
Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation
DCDS Home
This Issue
Next Article
Periodic linear motions with multiple collisions in a forced Kepler type problem

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

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

[1]	Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951
[2]	Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[3]	Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791
[4]	Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
[5]	Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164
[6]	Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057
[7]	Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027
[8]	Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
[9]	Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653
[10]	Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[11]	Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure & Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008
[12]	José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[13]	Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235
[14]	Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223
[15]	Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110
[16]	Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329
[17]	Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061
[18]	Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082
[19]	Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143
[20]	Max Fathi, Emanuel Indrei, Michel Ledoux. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6835-6853. doi: 10.3934/dcds.2016097

American Institute of Mathematical Sciences