Solutions to the coupled Schrödinger systems with steep potential well and critical exponent

Abstract

In the present paper, we consider the coupled Schrödinger systems with critical exponent: − Δ u i + λ V i ( x ) + a i u i = ∑ j = 1 d β i j u j 3 u i u i in R 3 , u i ∈ H 1 ( R N ) , i = 1,2 , … , d , $$\begin{cases}-{\Delta}{u}_{i}+\left(\lambda {V}_{i}\left(x\right)+{a}_{i}\right){u}_{i}=\sum _{j=1}^{d}{\beta }_{ij}{\left\vert {u}_{j}\right\vert }^{3}\left\vert {u}_{i}\right\vert {u}_{i}\quad \,\text{in}\,{\mathbb{R}}^{3},\quad \hfill \\ {u}_{i}\in {H}^{1}\left({\mathbb{R}}^{N}\right),\quad i=1,2,\dots ,d,\quad \hfill \end{cases}$$ where d ≥ 2, β ii > 0 for every i, β ij = β ji when i ≠ j, λ > 0 is a parameter and 0 ≤ V i ∈ L loc ∞ R N $0\le {V}_{i}\in {L}_{\text{loc\,}}^{\infty }\left({\mathbb{R}}^{N}\right)$ have a common bottom int V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ composed of ℓ 0 ℓ 0 ≥ 1 ${\ell }_{0}\left({\ell }_{0}\ge 1\right)$ connected components Ω k k = 1 ℓ 0 ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{{\ell }_{0}}$ , where int V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ is the interior of the zero set V i − 1 ( 0 ) = x ∈ R N ∣ V i ( x ) = 0 ${V}_{i}^{-1}\left(0\right)=\left\{x\in {\mathbb{R}}^{N}\mid {V}_{i}\left(x\right)=0\right\}$ of V i . We study the existence of least energy positive solutions to this system which are trapped near the zero sets int V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ for λ > 0 large for weakly cooperative case β i j > 0 s m a l l $\left({\beta }_{ij}{ >}0 \mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\right)$ and for purely competitive case β i j ≤ 0 $\left({\beta }_{ij}\le 0\right)$ . Besides, when d = 2, we construct a one-bump fully nontrivial solution which is localised at one prescribed components Ω k k = 1 ℓ ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{\ell }$ for large λ.