On vector solutions for coupled nonlinear Schrodinger equations with critical exponents

Abstract

In this paper, we study the existence and asymptotic behavior of a solution with positive components (which we call a vector solution) for the coupled system of nonlinear Schrodinger equations with doubly critical exponents ... as the coupling coefficient β ∈? tends to 0 or + ∞ where the domain Ω ⊂ ?<sup>N</sup> (N ≥ 3)is smooth bounded and certain conditions on λ<sub>1</sub>, λ<sub>2</sub> > 0 and μ<sub>1</sub>, μ<sub>2</sub> 0 are imposed. This system naturally arises as a counterpart of the Brezis-Nirenberg problem (Comm. Pure Appl. Math. 36: 437-477, 1983).