A Liouville-type theorem for an integral system on a half-space R+n

• Published in: Journal of inequalities and applications Vol. 2013; no. 1; p. 37
• Main Authors: Cao, Linfen, Dai, Zhaohui
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 04.02.2013; BioMed Central Ltd
• Subjects: Analysis; Applications of Mathematics; Mathematics; Mathematics and Statistics; Liouville-type theorem; systems of integral equations; moving planes method; monotonicity; HLS inequality
• ISSN: 1029-242X; 1029-242X
• DOI: 10.1186/1029-242X-2013-37

Let R + n be an n -dimensional upper half Euclidean space, and let α be any real number satisfying 0 < α < n . In our previous paper (Cao and Dai in J. Math. Anal. Appl. 389:1365-1373, 2012), we considered the single equation 0.1 u ( x ) = ∫ R + n ( 1 | x − y | n − α − 1 | x ∗ − y | n − α ) u r ( y ) d y , where x ∗ = ( x 1 , … , x n − 1 , − x n ) is the reflection of the point x about the ∂ R + n . We obtained the monotonicity and nonexistence of positive solutions to equation ( 0.1 ) under some integrability conditions when r > n n − α . In (Zhuo and Li in J. Math. Anal. Appl. 381:392-401, 2011), the authors discussed the following system of integral equations in R + n : 0.2 { u ( x ) = ∫ R + n ( 1 | x − y | n − α − 1 | x ∗ − y | n − α ) v q ( y ) d y , v ( x ) = ∫ R + n ( 1 | x − y | n − α − 1 | x ∗ − y | n − α ) u p ( y ) d y with 1 q + 1 + 1 p + 1 = n − α n . They obtained rotational symmetry of positive solutions of (0.2) about some line parallel to x n -axis under the assumption u ∈ L p + 1 ( R + n ) and v ∈ L q + 1 ( R + n ) . In this paper, we derive nonexistence results of such positive solutions for (0.2). In particular, we present a simple and more general method for the study of symmetry and monotonicity which has been extensively used in various forms on a half-space. AMS Subject Classification: 35B05, 35B45.