The best constant in a weighted Hardy-Littlewood-Sobolev inequality

Chen, Wenxiong; Li, Congming

doi:10.1090/S0002-9939-07-09232-5

The best constant in a weighted Hardy-Littlewood-Sobolev inequality
HTML articles powered by AMS MathViewer

by Wenxiong Chen and Congming Li

Proc. Amer. Math. Soc. 136 (2008), 955-962

DOI: https://doi.org/10.1090/S0002-9939-07-09232-5

Published electronically: November 30, 2007

PDF | Request permission

Abstract:

We prove the uniqueness for the solutions of the singular nonlinear PDE system: \begin{equation}\tag {1} \begin {cases} - \delta ( |x|^{\alpha } u(x) ) = \dfrac {v^q (x)}{|x|^{\beta }} ,\\ - \delta ( |x|^{\beta } v(x) ) = \dfrac {u^p (x)}{|x|^{\alpha }}. \end{cases} \end{equation} In the special case when $\alpha = \beta$ and $p = q$, we classify all the solutions and thus obtain the best constant in the corresponding weighted Hardy-Littlewood-Sobolev inequality.

References

Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
Wen Xiong Chen and Congming Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622. MR 1121147, DOI 10.1215/S0012-7094-91-06325-8
Wenxiong Chen and Congming Li, Regularity of solutions for a system of integral equations, Commun. Pure Appl. Anal. 4 (2005), no. 1, 1–8. MR 2126275, DOI 10.3934/cpaa.2005.4.1
Wenxiong Chen, Congming Li, and Biao Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), no. 3, 330–343. MR 2200258, DOI 10.1002/cpa.20116
Wenxiong Chen, Congming Li, and Biao Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations 30 (2005), no. 1-3, 59–65. MR 2131045, DOI 10.1081/PDE-200044445
Wenxiong Chen, Congming Li, and Biao Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst. 12 (2005), no. 2, 347–354. MR 2122171, DOI 10.3934/dcds.2005.12.347
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\textbf {R}^{n}$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
Chao Jin and Congming Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1661–1670. MR 2204277, DOI 10.1090/S0002-9939-05-08411-X
Chao Jin and Congming Li, Quantitative analysis of some system of integral equations, Calc. Var. Partial Differential Equations 26 (2006), no. 4, 447–457. MR 2235882, DOI 10.1007/s00526-006-0013-5
Congming Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math. 123 (1996), no. 2, 221–231. MR 1374197, DOI 10.1007/s002220050023
Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. MR 717827, DOI 10.2307/2007032
Congming Li and Jisun Lim, The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal. 6 (2007), no. 2, 453–464. MR 2289831, DOI 10.3934/cpaa.2007.6.453
E. M. Stein and Guido Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514. MR 0098285, DOI 10.1512/iumj.1958.7.57030
Juncheng Wei and Xingwang Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (1999), no. 2, 207–228. MR 1679783, DOI 10.1007/s002080050258