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Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation
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 |
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.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
Y. Y. Li,
Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
Y. Y. Li,
Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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