American Institute of Mathematical Sciences

Advanced
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

[1]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[2]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[3]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[4]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[5]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[6]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[7]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[8]

Xianchao Xiu, Ying Yang, Wanquan Liu, Lingchen Kong, Meijuan Shang. An improved total variation regularized RPCA for moving object detection with dynamic background. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1685-1698. doi: 10.3934/jimo.2019024
[9]

Alexey Yulin, Alan Champneys. Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1341-1357. doi: 10.3934/dcdss.2011.4.1341
[10]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[11]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[12]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
[13]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
[14]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
[15]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
[16]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[17]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006
[18]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[19]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[20]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences