American Institute of Mathematical Sciences

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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

Dongyan Li 1, and  Yongzhong Wang 1,

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

J. Wei and X. Xu, Classification of solution of higer order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

B. Ou, A Remark on a singular integral equation,, Houston J. Math., 25 (1999), 181. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

J. Wei and X. Xu, Classification of solution of higer order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

B. Ou, A Remark on a singular integral equation,, Houston J. Math., 25 (1999), 181. Google Scholar

[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. Google Scholar

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