Existence and symmetry of positive ground states for a doubly critical Schrödinger system

Abstract

We study the following doubly critical Schrödinger system: \[ { − Δ u − λ 1 | x | 2 u = u 2 ∗ − 1 + ν α u α − 1 v β , x ∈ R N , − Δ v − λ 2 | x | 2 v = v 2 ∗ − 1 + ν β u α v β − 1 , x ∈ R N , u , v ∈ D 1 , 2 ( R N ) , u , v > 0 in R N ∖ { 0 } , \begin {cases}-\Delta u -\frac {\lambda _1}{|x|^2}u=u^{2^\ast -1}+ \nu \alpha u^{\alpha -1}v^\beta , \quad x\in \mathbb {R}^N,\\ -\Delta v -\frac {\lambda _2}{|x|^2}v=v^{2^\ast -1} + \nu \beta u^{\alpha }v^{\beta -1}, \quad x\in \mathbb {R}^N,\\ u,\, v\in D^{1, 2}(\mathbb {R}^N),\quad u,\, v>0\,\,\hbox {in $\mathbb {R}^N\setminus \{0\}$},\end {cases} \] where N ≥ 3 N\ge 3 , λ 1 , λ 2 ∈ ( 0 , ( N − 2 ) 2 4 ) \lambda _1, \lambda _2\in (0, \frac {(N-2)^2}{4}) , 2 ∗ = 2 N N − 2 2^\ast =\frac {2N}{N-2} and α > 1 , β > 1 \alpha >1, \beta >1 satisfying α + β = 2 ∗ \alpha +\beta =2^\ast . This problem is related to coupled nonlinear Schrödinger equations with critical exponent for Bose-Einstein condensate. For different ranges of N N , α \alpha , β \beta and ν > 0 \nu >0 , we obtain positive ground state solutions via some quite different variational methods, which are all radially symmetric. It turns out that the least energy level depends heavily on the relations among α , β \alpha , \,\beta and 2 2 . Besides, for sufficiently small ν > 0 \nu >0 , positive solutions are also obtained via a variational perturbation approach. Note that the Palais-Smale condition cannot hold for any positive energy level, which makes the study via variational methods rather complicated.