Affiliation:
1. School of Mathematics and Statistics, and Jiangxi Provincial Center for Applied Mathematics, Jiangxi Normal University , Nanchang , Jiangxi 330022 , People’s Republic of China
2. School of Mathematics and Statistics, Central China Normal University , Wuhan , Hubei 430079 , People’s Republic of China
Abstract
Abstract
In this article, we consider the following Schrödinger-Poisson problem:
−
ε
2
Δ
u
+
V
(
y
)
u
+
Φ
(
y
)
u
=
∣
u
∣
p
−
1
u
,
y
∈
R
3
,
−
Δ
Φ
(
y
)
=
u
2
,
y
∈
R
3
,
\left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V(y)u+\Phi (y)u={| u| }^{p-1}u,& y\in {{\mathbb{R}}}^{3},\\ -\Delta \Phi (y)={u}^{2},& y\in {{\mathbb{R}}}^{3},\end{array}\right.
where
ε
>
0
\varepsilon \gt 0
is a small parameter,
1
<
p
<
5
1\lt p\lt 5
, and
V
(
y
)
V(y)
is a potential function. We construct multi-peak solution concentrating at the critical points of
V
(
y
)
V(y)
through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schödinger-Poisson system.
Subject
General Mathematics,Statistical and Nonlinear Physics