摘要

Let R-+(n) be an n-dimensional upper half Euclidean space, and let alpha be any real number satisfying 0 < alpha < n. In our previous paper (Cao and Dai in J. Math. Anal. Appl. 389:1365-1373, 2012), we considered the single equation u(x) = integral(Rn+) (1/vertical bar x - y vertical bar(n-alpha) - 1/vertical bar x* - y vertical bar(n-alpha))u(r) (y) dy, (0.1) where x* = (x(1), ... , x(n-1), -x(n)) is the reflection of the point x about the partial derivative R-+(n). We obtained the monotonicity and nonexistence of positive solutions to equation (0.1) under some integrability conditions when r > n/n-alpha. 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): { u(x) = integral(R+n) (1 vertical bar x-y vertical bar(n-alpha) -1/vertical bar x*-y vertical bar(n-alpha))V-q(y) dy, (0.2) V(x) = integral(R+n) (1/vertical bar x-y vertical bar(n-alpha) -1 vertical bar x*-y vertical bar(n-alpha))u(p)(y) dy with 1/q+1 + 1/p+1 = n-alpha/n. They obtained rotational symmetry of positive solutions of (0.2) about some line parallel to x(n)-axis under the assumption u is an element of Lp+1(R-+(n)) and v is an element of Lq+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.

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