A Liouville-type theorem for an integral system on a half-space [InlineEquation not available: see fulltext.]

Let [InlineEquation not available: see fulltext.] be an n-dimensional upper half Euclidean space, and let alpha be any real number satisfying [InlineEquation not available: see fulltext.]. In our previous paper (Cao and Dai in J. Math. Anal. Appl. 389:1365-1373, 2012), we considered the single equation [Equation not available: see fulltext.] where [InlineEquation not available: see fulltext.] is the reflection of the point x about the [InlineEquation not available: see fulltext.]. We obtained the monotonicity and nonexistence of positive solutions to equation (0.1) under some integrability conditions when [InlineEquation not available: see fulltext.]. In (Zhuo and Li in J. Math. Anal. Appl. 381:392-401, 2011), the authors discussed the following system of integral equations in [InlineEquation not available: see fulltext.]: [Equation not available: see fulltext.] with [InlineEquation not available: see fulltext.]. They obtained rotational symmetry of positive solutions of (0.2) about some line parallel to [InlineEquation not available: see fulltext.]-axis under the assumption [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. 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.