Acta mathematica scientia, Series B >
CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS
Received date: 2009-01-04
Online published: 2009-07-20
Supported by
Chen partially supported by NSF Grant DMS-0604638; Li partially supported by NSF Grant DMS-0401174
We classify all positive solutions for the following integral system:
ui(x) =∫Rn1/ |x − y|n−α fi(u(y))dy, x ∈ Rn, i = 1, · · · , m,
0 < α < n, and u(x) = (u1(x), u2(x), · · · , um(x)).
Here fi(u), 1 ≤ i ≤ m, are real-valued functions of homogeneous degree n+α/ n−α and are monotone nondecreasing with respect to all the independent variables u1, u2, · · ·, um. In the special case n ≥ 3 and α = 2, we show that the above system is equivalent to the following elliptic PDE system:
−△ui(x) = fi(u(x)), x ∈ Rn, i = 1, · · · , m,
and u(x) = (u1(x), u2(x), · · · , um(x)).
This system is closely related to the stationary Schr¨odinger system with critical exponents for Bose-Einstein condensate.
Wenxiong Chen , Congming Li . CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS[J]. Acta mathematica scientia, Series B, 2009 , 29(4) : 949 -960 . DOI: 10.1016/S0252-9602(09)60079-5
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