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

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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

Abstract
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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 and 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), 1-37. doi: 10.1007/BF01244896.
[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), 9-10, 1671-1690. doi: 10.1080/03605309708821315.
[3]	L. Cao, A Liouville-Type Theorem for an integral equation on a Half-Space $R_+ ^n$, , Submitted., ().
[4]	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.
[5]	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.
[6]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, Diff. Equa. Dyn. Syst., 2010.
[7]	W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematics Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5.
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330-343.
[9]	W. Chen and C. Li, Regularity of solutions for a system of integral equation, Commun. Pure Appl. Anal., 4 (2005), 1-8.
[10]	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.
[11]	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.
[12]	D. Li and R. Zhuo, An integral eequation on Half space, Proc. Amer. Math. Soc., 138 (2010), 2779-2791.
[13]	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.
[14]	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.
[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), 339-359. doi: 10.1017/S0308210506000394.
[16]	B. Gidas and J. Spruk, A prior bounds for positive solutions of nonlinear elliptic equations, Commun. PDEs, 6 (1981), 883-901.
[17]	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.
[18]	L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.
[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), 805-827. doi: 10.1007/s00209-008-0352-3.
[20]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.

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), 1-37. doi: 10.1007/BF01244896.
[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), 9-10, 1671-1690. doi: 10.1080/03605309708821315.
[3]	L. Cao, A Liouville-Type Theorem for an integral equation on a Half-Space $R_+ ^n$, , Submitted., ().
[4]	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.
[5]	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.
[6]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, Diff. Equa. Dyn. Syst., 2010.
[7]	W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematics Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5.
[8]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330-343.
[9]	W. Chen and C. Li, Regularity of solutions for a system of integral equation, Commun. Pure Appl. Anal., 4 (2005), 1-8.
[10]	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.
[11]	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.
[12]	D. Li and R. Zhuo, An integral eequation on Half space, Proc. Amer. Math. Soc., 138 (2010), 2779-2791.
[13]	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.
[14]	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.
[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), 339-359. doi: 10.1017/S0308210506000394.
[16]	B. Gidas and J. Spruk, A prior bounds for positive solutions of nonlinear elliptic equations, Commun. PDEs, 6 (1981), 883-901.
[17]	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.
[18]	L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.
[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), 805-827. doi: 10.1007/s00209-008-0352-3.
[20]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.

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