Advanced Search
Article Contents
Article Contents

Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$

Abstract Related Papers Cited by
  • Let $n, m$ be a positive integer and let $R_+^n$ be the $n$-dimensional upper half Euclidean space. In this paper, we study the following integral equation \begin{eqnarray} u(x)=\int_{R_+^n}G(x,y)u^pdy, \end{eqnarray} where \begin{eqnarray*} G(x,y)=\frac{C_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_n y_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{n/2}}dz, \end{eqnarray*} $C_{n}$ is a positive constant, $0 <2m 1$. Using the method of moving planes in integral forms, we show that equation (1) has no positive solution.
    Mathematics Subject Classification: Primary: 35J99, 45E10; Secondary: 45G05.


    \begin{equation} \\ \end{equation}
  • [1]

    H. Berestycki and L. Nirenberg, On the method of moving planes and sliding method, Bol. Soc. Brazil. Mat. (N. S.), 22 (1991), 1-37.doi: 10.1007/BF01244896.


    G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ and $R_+ ^N$ through the method of moving plane, Comn. PDE., 22 (1997), 9-10, 1671-1690.doi: 10.1080/03605309708821315.


    L. CaoA Liouville-Type Theorem for an integral equation on a Half-Space $R_+ ^n$, Submitted.


    W. Chen and C. Li, Radial symmetry of solutions for some integral systems of wolff type, Dis. Cont. Dyn. Sys., 30 (2011), 1083-1093.doi: 10.3934/dcds.2011.30.1083.


    W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Drichlet Problems, J. Math. Anal. Appl., 377 (2011), 744-753.doi: 10.1016/j.jmaa.2010.11.035.


    W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, Diff. Equa. Dyn. Syst., 2010.


    W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematics Scientia, 29 (2009), 949-960.doi: 10.1016/S0252-9602(09)60079-5.


    W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330-343.


    W. Chen and C. Li, Regularity of solutions for a system of integral equation, Commun. Pure Appl. Anal., 4 (2005), 1-8.


    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.


    W. Chen and C. Li, Classification of solutions to some nonlinear equations, Duke Math. J., 63 (1991), 615–-622.doi: 10.1215/S0012-7094-91-06325-8.


    D. Li and R. Zhuo, An integral eequation on Half space, Proc. Amer. Math. Soc., 138 (2010), 2779-2791.


    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: 10.5169/seals-55100.


    Y. Ge, Positive solutions in semilinear critical problems for polyharmonic operators, J. Math. Pures Appl., 84 (2005), 199–-245.doi: 10.1016/j.matpur.2004.10.002.


    Y. Guo and J. Liu, Liouville-type theorems for polyharmonic equations in $R^N$ and in $R_+ ^N$, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 339-359.doi: 10.1017/S0308210506000394.


    B. Gidas and J. Spruk, A prior bounds for positive solutions of nonlinear elliptic equations, Commun. PDEs, 6 (1981), 883-901.


    C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.doi: 10.1016/j.aim.2010.07.020.


    L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.


    W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827.doi: 10.1007/s00209-008-0352-3.


    J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.

  • 加载中

Article Metrics

HTML views() PDF downloads(117) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint