REGULARITY AND SYMMETRY OF SOLUTIONS OF AN INTEGRAL SYSTEM

doi:10.1016/S0252-9602(12)60139-8

Abstract

Abstract:

In this paper, we are concerned with the regularity and symmetry of positive solutions of the following nonlinear integral system
u(x) = ∫_RⁿG (x − y)v(y)^q/|y| dy, v(x) = ∫_RⁿG (x − y)u(y)^p/|y| dy
for x ∈ Rⁿ, where G (x) is the kernel of Bessel potential of order α, 0 ≤β < α< n, 1 < p, q < n− β/β and 1 /p + 1+1/q + 1>n −α + β/n.
We show that positive solution pairs (u, v) ∈ L^p⁺¹(Rⁿ)×L^q⁺¹(Rⁿ) are H¨older continuous, radially symmetric and strictly decreasing about the origin.

Key words: regularity, radially symmetry, Bessel kernel, nonlinear integral system

CLC Number:

35J25

CHEN Xiao-Li, YANG Jian-Fu. REGULARITY AND SYMMETRY OF SOLUTIONS OF AN INTEGRAL SYSTEM[J].Acta mathematica scientia,Series B, 2012, 32(5): 1759-1780.

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