Affiliation:
1. Department of Mathematics, Zhejiang University of Technology , 310014 Zhejiang , China
Abstract
Abstract
In this article, we study the following nonlinear Schrödinger system
−
Δ
u
1
+
V
1
(
x
)
u
1
=
α
u
1
u
2
+
μ
u
1
,
x
∈
R
4
,
−
Δ
u
2
+
V
2
(
x
)
u
2
=
α
2
u
1
2
+
β
u
2
2
+
μ
u
2
,
x
∈
R
4
,
\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
∫
R
4
(
u
1
2
+
u
2
2
)
d
x
=
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}}
,
V
1
(
x
)
{V}_{1}\left(x)
, and
V
2
(
x
)
{V}_{2}\left(x)
are bounded functions. Under some mild assumptions on
V
1
(
x
)
{V}_{1}\left(x)
and
V
2
(
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.