Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case - Chen, Zhijie; Zou, Wenming (Calculus of variations and partial differential equations, 2015)
We study the following nonlinear Schrödinger system which is related to Bose–Einstein condensate: - Δ u + λ 1 u = μ 1 u 2 ∗ - 1 + β u 2 ∗ 2 - 1 v 2 ∗ 2 , x ∈ Ω , - Δ v + λ 2 v = μ 2 v 2 ∗ - 1 + β v 2 ∗ 2 - 1 u 2 ∗ 2 , x ∈ Ω , u ≥ 0 , v ≥ 0 in Ω , u = v = 0 on ∂ Ω . Here Ω ⊂ R N is a smooth bounded domain, 2 ∗ : = 2 N N - 2 is the Sobolev critical exponent, - λ 1 ( Ω ) < λ 1 , λ 2 < 0 , μ 1 , μ 2 > 0 and β ≠ 0 , where λ 1 ( Ω ) is the first eigenvalue of - Δ with the Dirichlet boundary condition. When β = 0 , this is just the well-known Brezis–Nirenberg problem. The special case N = 4 was studied by the authors in (Arch Ration Mech Anal 205:515–551, 2012 ). In this paper we consider the higher dimensional case N ≥ 5 . It is interesting that we can prove the existence of a positive least energy solution ( u β , v β ) for any β ≠ 0 (which can not hold in the special case N = 4 ). We also study the limit behavior of ( u β , v β ) as β → - ∞ and phase separation is expected. In particular, u β - v β will converge to sign-changing solutions of the Brezis–Nirenberg problem, provided N ≥ 6 . In case λ 1 = λ 2 , the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case N = 4 .