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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

 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.
Citation: Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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).  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., (1981), 369.   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.  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.  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.  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.  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.  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.   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.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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).  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., (1981), 369.   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.  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.  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.  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.  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.  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.   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.  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.  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.   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.  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.  doi: 10.1007/s00526-011-0474-z.  Google Scholar
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