CLASSIFICATION OF SOLUTIONS FOR A CLASS OF SINGULAR INTEGRAL SYSTEM

doi:10.1016/S0252-9602(11)60330-5

Abstract

Abstract:

In this paper, we consider the following integral system:
???u(x) = v (y) dy,
? n n-α
R |x-y|
(1.1)
??? up(y)
?v(x) = n-μdy,
n
R |x-y|

where 0 < α, μ < n; p,q ≥ 1. Using the method of moving planes in an integral form which
was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions
of (0.1) are radially symmetric and decreasing with respect to some point under some
general conditions of integrability. The results essentially improve and extend previously
known results [4, 8].

Key words: Hardy-Littlewood-Sobolev inquality, integral equation, moving plane, inter-polation inequality, radial symmetry

CLC Number:

45G15

XU Jian-Kai, TAN Zhong. CLASSIFICATION OF SOLUTIONS FOR A CLASS OF SINGULAR INTEGRAL SYSTEM[J].Acta mathematica scientia,Series B, 2011, 31(4): 1449-1456.

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