Abstract
In this paper, we study the following Schrödinger–Poisson equations:
{
-
ε
2
Δ
u
+
V
(
x
)
u
+
K
(
x
)
ϕ
u
=
|
u
|
p
-
2
u
,
x
∈
ℝ
3
,
-
ε
2
Δ
ϕ
=
K
(
x
)
u
2
,
x
∈
ℝ
3
,
\left\{\begin{aligned} &\displaystyle{-}\varepsilon^{2}\Delta u+V(x)u+K(x)\phi
u%
=\lvert u\rvert^{p-2}u,&\hskip 10.0ptx&\displaystyle\in\mathbb{R}^{3},\\
&\displaystyle{-}\varepsilon^{2}\Delta\phi=K(x)u^{2},&\hskip 10.0ptx&%
\displaystyle\in\mathbb{R}^{3},\end{aligned}\right.
where
p
∈
(
4
,
6
)
{p\in(4,6)}
,
ε
>
0
{\varepsilon>0}
is a parameter, and V and K are nonnegative potential functions which satisfy the critical frequency conditions in the sense that
inf
ℝ
3
V
=
inf
ℝ
3
K
=
0
{\inf_{\mathbb{R}^{3}}V=\inf_{\mathbb{R}^{3}}K=0}
. By using a penalization method, we show the existence of multi-bump solutions for the above problem, with several local maximum points whose corresponding values are of different scales with respect to
ε
→
0
{\varepsilon\rightarrow 0}
. Moreover, under suitable local assumptions on V and K, we prove the uniqueness of multi-bump solutions concentrating around zero points of V and K via the local Pohozaev identity.
Investigation of a Control Function Determining the Location of the Zeros as a Possible Proof of the Riemann Hypothesis.