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:−Δui+λVi(x)+aiui=∑j=1dβijuj3uiui in R3,ui∈H1(RN),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≤Vi∈Lloc ∞RN $0\le {V}_{i}\in {L}_{\text{loc\,}}^{\infty }\left({\mathbb{R}}^{N}\right)$ have a common bottom int Vi−1(0) ${V}_{i}^{-1}\left(0\right)$ composed of ℓ0ℓ0≥1 ${\ell }_{0}\left({\ell }_{0}\ge 1\right)$ connected components Ωkk=1ℓ0 ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{{\ell }_{0}}$ , where int Vi−1(0) ${V}_{i}^{-1}\left(0\right)$ is the interior of the zero set Vi−1(0)=x∈RN∣Vi(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 Vi−1(0) ${V}_{i}^{-1}\left(0\right)$ for λ > 0 large for weakly cooperative case βij>0small $\left({\beta }_{ij}{ >}0 \mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\right)$ and for purely competitive case βij≤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 Ωkk=1ℓ ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{\ell }$ for large λ.

Keywords

• schrödinger system
• critical exponent
• least energy positive solutions
• asymptotic behavior
• one-bump solution
• 35j20
• 35j60
• 35b09