Existence of normalized peak solutions for a coupled nonlinear Schrödinger system

In this article, we study the following nonlinear Schrödinger system −Δu1+V1(x)u1=αu1u2+μu1,x∈R4,−Δu2+V2(x)u2=α2u12+βu22+μu2,x∈R4,\left\{\begin{array}{ll}-\Delta {u}_{1}+{V}_{1}\left(x){u}_{1}=\alpha {u}_{1}{u}_{2}+\mu {u}_{1},& x\in {{\mathbb{R}}}^{4},\\ -\Delta {u}_{2}+{V}_{2}\left(x){u}_{2}=\frac{\alpha }{2}{u}_{1}^{2}+\beta {u}_{2}^{2}+\mu {u}_{2},& x\in {{\mathbb{R}}}^{4},\end{array}\right. with the constraint ∫R4(u12+u22)dx=1{\int }_{{{\mathbb{R}}}^{4}}\left({u}_{1}^{2}+{u}_{2}^{2}){\rm{d}}x=1, where α>0\alpha \gt 0 and α>β\alpha \gt \beta , μ∈R\mu \in {\mathbb{R}}, V1(x){V}_{1}\left(x), and V2(x){V}_{2}\left(x) are bounded functions. Under some mild assumptions on V1(x){V}_{1}\left(x) and V2(x){V}_{2}\left(x), we prove the existence of normalized peak solutions by using the finite dimensional reduction method, combined with the local Pohozaev identities. Because of the interspecies interaction between the components, we aim to obtain some new technical estimates.