Nonexistence of positive solutions for a system of integral equations on R^n

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November 2013, 12(6): 2601-2613. doi: 10.3934/cpaa.2013.12.2601

Nonexistence of positive solutions for a system of integral equations on $R^n_+$ and applications

1.	Department of Applied Mathematics, Key Laboratory of Space Applied Physics and Chemistry, Ministry of Education, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China, China

Received July 2012 Revised January 2013 Published May 2013

Abstract
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Let $R^n_+$ be the $n$-dimensional upper half Euclidean space, $m$ be a positive integer. In this paper, we consider the following system of integral equations on $R^n_+$: \begin{eqnarray} u(x)=\int_{R^n_+}G(x,y)v^q(y)dy, \\ v(x)=\int_{R^n_+}G(x,y)u^p(y)dy, \end{eqnarray} where \begin{eqnarray} G(x,y)=\frac{c_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_ny_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{\frac{n}{2}}}dz \end{eqnarray} with $0 < 2m < n$ and $p,q>1$. Nonexistence of positive solution is proved by using the method of moving planes in integral forms. We also obtain the equivalence between the system of integral equations and corresponding partial differential equations.

Keywords: System of integral equations, method of moving planes in integral forms, a priori bound., equivalence, nonexistence.

Mathematics Subject Classification: Primary: 35B45, 35B53; Secondary: 35J9.

Citation: Dongyan Li, Yongzhong Wang. Nonexistence of positive solutions for a system of integral equations on $R^n_+$ and applications. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2601-2613. doi: 10.3934/cpaa.2013.12.2601

References:

[1]	W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirchlet problems,, Math.Z., 261 (2009), 805. doi: 10.1007/s00209-008-0352-3.
[2]	Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, Adv. Math., 229 (2012), 2835. doi: 10.1016/j.aim.2012.01.018.
[3]	Y. Fang and J. Zhang, Nonexistence of positive solition for an integral equation on $R^n_+$,, Commun. Pure Appl. Anal., 12 (2013), 663. doi: 10.3934/cpaa.2013.12.663.
[4]	C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comment. Math. Helv, 73 (1998), 206. doi: 10.1007/s000140050052.
[5]	J. Wei and X. Xu, Classification of solution of higer order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258.
[6]	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.20130.
[7]	W. Chen, C. Li and B. Ou, Qualitative problems of solutions for a system of integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347.
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445.
[9]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, appear to in Commun. Pure Appl. Anal., (2012).
[10]	W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Commun. Pure Appl. Anal., 4 (2005), 1. doi: 10.3934/cpaa.2005.4.1.
[11]	W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 4 (2009), 1167. doi: 10.3934/dcds.2009.24.1167.
[12]	C. Jin and C. Li, Symmetry of solutions to some system of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X.
[13]	W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations,", AIMS Book Series on Differ. Equ. Dyn. Syst., 4 (2010). doi: 10.3934/dcds.2009.24.1167.
[14]	C. Jin and C. Li, Quantitative analysis of some system of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447. doi: 10.1007/s00526-006-0013-5.
[15]	C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301.
[16]	W. Chen, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations,, Discrete Contin. Dyn. Syst., 12 (2005), 347.
[17]	C. Liu and S. Qiao, Symmetry and monotonicity for a system of integal equations,, Commun. Pure Appl. Anal., 6 (2009), 1925. doi: 10.3934/cpaa.2009.8.1925.
[18]	L. Ma and D. Z. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855.
[19]	L. Ma and D. Z. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 342 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064.
[20]	B. Ou, A Remark on a singular integral equation,, Houston J. Math., 25 (1999), 181.
[21]	J. Liu, Y. Guo and Y. Zhang, Liouville type theorems for polyharmonic system in $R^n$,, J. Differential Equations, 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016.

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References:

[1]	W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirchlet problems,, Math.Z., 261 (2009), 805. doi: 10.1007/s00209-008-0352-3.
[2]	Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, Adv. Math., 229 (2012), 2835. doi: 10.1016/j.aim.2012.01.018.
[3]	Y. Fang and J. Zhang, Nonexistence of positive solition for an integral equation on $R^n_+$,, Commun. Pure Appl. Anal., 12 (2013), 663. doi: 10.3934/cpaa.2013.12.663.
[4]	C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comment. Math. Helv, 73 (1998), 206. doi: 10.1007/s000140050052.
[5]	J. Wei and X. Xu, Classification of solution of higer order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258.
[6]	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.20130.
[7]	W. Chen, C. Li and B. Ou, Qualitative problems of solutions for a system of integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347.
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445.
[9]	W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, appear to in Commun. Pure Appl. Anal., (2012).
[10]	W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Commun. Pure Appl. Anal., 4 (2005), 1. doi: 10.3934/cpaa.2005.4.1.
[11]	W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 4 (2009), 1167. doi: 10.3934/dcds.2009.24.1167.
[12]	C. Jin and C. Li, Symmetry of solutions to some system of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X.
[13]	W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations,", AIMS Book Series on Differ. Equ. Dyn. Syst., 4 (2010). doi: 10.3934/dcds.2009.24.1167.
[14]	C. Jin and C. Li, Quantitative analysis of some system of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447. doi: 10.1007/s00526-006-0013-5.
[15]	C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301.
[16]	W. Chen, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations,, Discrete Contin. Dyn. Syst., 12 (2005), 347.
[17]	C. Liu and S. Qiao, Symmetry and monotonicity for a system of integal equations,, Commun. Pure Appl. Anal., 6 (2009), 1925. doi: 10.3934/cpaa.2009.8.1925.
[18]	L. Ma and D. Z. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855.
[19]	L. Ma and D. Z. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 342 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064.
[20]	B. Ou, A Remark on a singular integral equation,, Houston J. Math., 25 (1999), 181.
[21]	J. Liu, Y. Guo and Y. Zhang, Liouville type theorems for polyharmonic system in $R^n$,, J. Differential Equations, 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016.

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