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Classification of positive solutions for nonlinear differential and integral systems with critical exponents. (English) Zbl 1212.35103

Summary: We classify all positive solutions for the following integral system:

u i (x)= n 1 |x-y| n-α f i (u(y))dy,

x n , i=1,,m, 0<α<n, and u(x)=(u 1 (x),u 2 (x),,u m (x)). Here f i (u), 1im, are real-valued functions of homogeneous degree n+α n-α and are monotone nondecreasing with respect to all the independent variables u 1 ,u 2 ,,u m . In the special case n3 and α=2, we show that the above system is equivalent to the following elliptic PDE system:

-Δu i (x)=f i (u(x)),x n ,i=1,,m,andu(x)=(u 1 (x),u 2 (x),,u m (x))·

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

MSC:
35J57Second-order elliptic systems, boundary value problems
35B33Critical exponents (PDE)
35J60Nonlinear elliptic equations
35J91Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
45E05Integral equations with kernels of Cauchy type
45G05Singular nonlinear integral equations