\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35B45, 35B53; Secondary: 35J91.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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), 805-827.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), 2835-2867.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), 663-678.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), 206-231.doi: 10.1007/s000140050052.

    [5]

    J. Wei and X. Xu, Classification of solution of higer order conformally invariant equations, Math. Ann., 313 (1999), 207-288.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), 330-343.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), 347-354.

    [8]

    W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65.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), 1-8.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), 1167-1184.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), 1661-1670.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), 447-457.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), 1049-1057.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), 347-354.

    [17]

    C. Liu and S. Qiao, Symmetry and monotonicity for a system of integal equations, Commun. Pure Appl. Anal., 6 (2009), 1925-1932.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), 855-859.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), 943-949.doi: 10.1016/j.jmaa.2007.12.064.

    [20]

    B. Ou, A Remark on a singular integral equation, Houston J. Math., 25 (1999), 181-184.

    [21]

    J. Liu, Y. Guo and Y. Zhang, Liouville type theorems for polyharmonic system in $R^n$, J. Differential Equations, 225 (2006), 685-709.doi: 10.1016/j.jde.2005.10.016.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(73) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return