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March  2013, 12(2): 663-678. doi: 10.3934/cpaa.2013.12.663

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

1. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu

Received  February 2011 Revised  May 2011 Published  September 2011

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.
Citation: Yanqin Fang, Jihui Zhang. Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$. Communications on Pure & Applied Analysis, 2013, 12 (2) : 663-678. doi: 10.3934/cpaa.2013.12.663
References:
[1]

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

[2]

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.  doi: 10.1080/03605309708821315.  Google Scholar

[3]

L. Cao, A Liouville-Type Theorem for an integral equation on a Half-Space $R_+ ^n$, , Submitted., ().   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math. \textbf{59} (2006), 59 (2006), 330.   Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comment. Math. Helv., 73 (1998), 206.  doi: 10.5169/seals-55100.  Google Scholar

[14]

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

[15]

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.  doi: 10.1017/S0308210506000394.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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.  doi: 10.1007/s00209-008-0352-3.  Google Scholar

[20]

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

show all references

References:
[1]

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

[2]

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.  doi: 10.1080/03605309708821315.  Google Scholar

[3]

L. Cao, A Liouville-Type Theorem for an integral equation on a Half-Space $R_+ ^n$, , Submitted., ().   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math. \textbf{59} (2006), 59 (2006), 330.   Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comment. Math. Helv., 73 (1998), 206.  doi: 10.5169/seals-55100.  Google Scholar

[14]

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

[15]

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.  doi: 10.1017/S0308210506000394.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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.  doi: 10.1007/s00209-008-0352-3.  Google Scholar

[20]

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

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