Liouville theorem for an integral system on the upper half space

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January 2015, 35(1): 155-171. doi: 10.3934/dcds.2015.35.155

Liouville theorem for an integral system on the upper half space

Jingbo Dou ^1, and Ye Li ^2,

1.	School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi, 710100, China
2.	Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, United States

Received January 2014 Revised June 2014 Published August 2014

Abstract
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In this paper we establish a Liouville type theorem for an integral system on the upper half space $\mathbb{R}_+^{n}$ \begin{equation*} \begin{cases} u(y)=\int_{\mathbb{R}^{n}_+}\frac{f(v(x))}{|x-y|^{n-\alpha}}dx,&\quad y\in\partial\mathbb{R}^{n}_+,\\ v(x)=\int_{\partial\mathbb{R}^{n}_+}\frac{g(u(y))}{|x-y|^{n-\alpha}}dy,&\quad x\in\mathbb{R}_+^{n}. \end{cases} \end{equation*} This integral system arises from the Euler-Lagrange equation corresponding to Hardy-Littlewood-Sobolev inequality on the upper half space. Under natural structure conditions on $f$ and $g$, we classify positive solutions to the above system basing on the method of moving sphere in integral forms and the Hardy-Littlewood-Sobolev inequality on the upper half space.

Keywords: Liouville type theorem, method of moving sphere., Hardy-Littlewood-Sobolev inequality, integral system.

Mathematics Subject Classification: Primary: 35B53; Secondary: 35B6.

Citation: Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155

References:

[1]	G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $\mathbb{R}^{N}$ or $\mathbbR^n_+$ through the method of moving planes, Comm. Partial Diff. Eqs., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.
[2]	L. Cao and Z. Dai, A Liouville-type theorem for an integral equation on a half-space $\mathbbR^n_+$, J. Math. Anal. Appl., 389 (2012), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015.
[3]	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.
[4]	W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167.
[5]	W. Chen and C. Li, Super Polyharmonic Property of Solutions for PDE Systems and Its Applications, Comm. Pure and Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.
[6]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.
[7]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.
[8]	C. Chen and C. S. Lin, Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent, Duke Math. J., 78 (1995), 315-334. doi: 10.1215/S0012-7094-95-07814-4.
[9]	L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86.
[10]	J. Dou, C. Qu and Y. Han, Symmetry and nonexistence of positive solutions to an integral system with weighted functions, Sci. China Math., 54 (2011), 753-768. doi: 10.1007/s11425-011-4177-x.
[11]	J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Notices, 2014 (2014), 37 pp. doi: 10.1093/imrn/rnt213.
[12]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Math. Anal. Appl., Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.
[13]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.
[14]	B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Eqs., 6 (1981), 883-901. doi: 10.1080/03605308108820196.
[15]	Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^{N}$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.
[16]	F. B. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.
[17]	C. Li, Local asymptotic symnwtry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.
[18]	Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
[19]	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.
[20]	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.
[21]	Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Diff. Integ. Eqs., 12 (1999), 601-612.
[22]	W. Reichel and T. Weth, A prior bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827. doi: 10.1007/s00209-008-0352-3.
[23]	X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. PDE, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.

show all references

References:

[1]	G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $\mathbb{R}^{N}$ or $\mathbbR^n_+$ through the method of moving planes, Comm. Partial Diff. Eqs., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.
[2]	L. Cao and Z. Dai, A Liouville-type theorem for an integral equation on a half-space $\mathbbR^n_+$, J. Math. Anal. Appl., 389 (2012), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015.
[3]	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.
[4]	W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167.
[5]	W. Chen and C. Li, Super Polyharmonic Property of Solutions for PDE Systems and Its Applications, Comm. Pure and Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.
[6]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.
[7]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.
[8]	C. Chen and C. S. Lin, Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent, Duke Math. J., 78 (1995), 315-334. doi: 10.1215/S0012-7094-95-07814-4.
[9]	L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86.
[10]	J. Dou, C. Qu and Y. Han, Symmetry and nonexistence of positive solutions to an integral system with weighted functions, Sci. China Math., 54 (2011), 753-768. doi: 10.1007/s11425-011-4177-x.
[11]	J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Notices, 2014 (2014), 37 pp. doi: 10.1093/imrn/rnt213.
[12]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Math. Anal. Appl., Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.
[13]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.
[14]	B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Eqs., 6 (1981), 883-901. doi: 10.1080/03605308108820196.
[15]	Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^{N}$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.
[16]	F. B. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.
[17]	C. Li, Local asymptotic symnwtry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.
[18]	Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
[19]	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.
[20]	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.
[21]	Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Diff. Integ. Eqs., 12 (1999), 601-612.
[22]	W. Reichel and T. Weth, A prior bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827. doi: 10.1007/s00209-008-0352-3.
[23]	X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. PDE, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.

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