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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

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 & 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 $\mathbbR^n$ or $\mathbbR^n_+$ through the method of moving planes, Comm. Partial Diff. Eqs., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, in Math. Anal. Appl., Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.  Google Scholar

[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.  Google Scholar

[17]

C. Li, Local asymptotic symnwtry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Diff. Integ. Eqs., 12 (1999), 601-612.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $\mathbbR^n$ or $\mathbbR^n_+$ through the method of moving planes, Comm. Partial Diff. Eqs., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, in Math. Anal. Appl., Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.  Google Scholar

[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.  Google Scholar

[17]

C. Li, Local asymptotic symnwtry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Diff. Integ. Eqs., 12 (1999), 601-612.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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