CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS

doi:10.1016/S0252-9602(09)60079-5

摘要/Abstract

摘要：

We classify all positive solutions for the following integral system:

u_i(x) =∫_Rn1/ |x − y|^n−α f_i(u(y))dy, x ∈ Rⁿ, i = 1, · · · , m,

0 < α < n, and u(x) = (u₁(x), u₂(x), · · · , u_m(x)).

Here f_i(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, · · ·, u_m. In the special case n ≥ 3 and α = 2, we show that the above system is equivalent to the following elliptic PDE system:

−△u_i(x) = f_i(u(x)), x ∈ Rⁿ, i = 1, · · · , m,

and u(x) = (u₁(x), u₂(x), · · · , u_m(x)).

This system is closely related to the stationary Schr¨odinger system with critical exponents for Bose-Einstein condensate.

关键词: integral and PDE systems, positive solutions, method of moving planes, radial symmetry, uniqueness

Abstract:

We classify all positive solutions for the following integral system:

u_i(x) =∫_Rn1/ |x − y|^n−α f_i(u(y))dy, x ∈ Rⁿ, i = 1, · · · , m,

0 < α < n, and u(x) = (u₁(x), u₂(x), · · · , u_m(x)).

−△u_i(x) = f_i(u(x)), x ∈ Rⁿ, i = 1, · · · , m,

and u(x) = (u₁(x), u₂(x), · · · , u_m(x)).

This system is closely related to the stationary Schr¨odinger system with critical exponents for Bose-Einstein condensate.

Key words: integral and PDE systems, positive solutions, method of moving planes, radial symmetry, uniqueness

中图分类号:

35J45

Wenxiong Chen, Congming Li. CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS[J]. 数学物理学报(英文版), 2009, 29(4): 949-960.

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.

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