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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 $R_+^n$

Yanqin Fang 1, and  Jihui Zhang 1,

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.
Keywords: monotonicity., moving planes in integral forms, Nonexistence.
Mathematics Subject Classification: Primary: 35J99, 45E10; Secondary: 45G0.
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

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

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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

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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

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

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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

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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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