Abstract
We are concerned with the following Schrödinger system with coupled quadratic nonlinearity
−
ε
2
Δ
v
+
P
(
x
)
v
=
μ
v
w
,
x
∈
R
N
,
−
ε
2
Δ
w
+
Q
(
x
)
w
=
μ
2
v
2
+
γ
w
2
,
x
∈
R
N
,
v
>
0
,
w
>
0
,
v
,
w
∈
H
1
R
N
,
$$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right),\end{array}\right. \end{equation}$$
which arises from second-harmonic generation in quadratic media. Here ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive function potentials. By applying reduction method, we prove that if x
0 is a non-degenerate critical point of Δ(P + Q) on some closed N − 1 dimensional hypersurface, then the system above has a single peak solution (vε
, wε
) concentrating at x
0 for ε small enough.
Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearity
