Abstract
In this paper, we consider the following Schrödinger–Poisson system with perturbation:
{
-
Δ
u
+
u
+
λ
ϕ
(
x
)
u
=
|
u
|
p
-
2
u
+
g
(
x
)
,
x
∈
ℝ
3
,
-
Δ
ϕ
=
u
2
,
x
∈
ℝ
3
,
\left\{\begin{aligned} \displaystyle-\Delta u+u+\lambda\phi(x)u&\displaystyle=%
|u|^{p-2}u+g(x),&&\displaystyle x\in\mathbb{R}^{3},\\
\displaystyle-\Delta\phi&\displaystyle=u^{2},&&\displaystyle x\in\mathbb{R}^{3%
},\end{aligned}\right.
where
λ
>
0
{\lambda>0}
,
p
∈
(
3
,
6
)
{p\in(3,6)}
and the radial general perturbation term
g
(
x
)
∈
L
p
p
-
1
(
ℝ
3
)
{g(x)\in L^{\frac{p}{p-1}}(\mathbb{R}^{3})}
. By establishing a new abstract perturbation theorem based on the Bolle’s method, we prove the existence of infinitely many radial solutions of the above system. Moreover, we give the asymptotic behaviors of these solutions as
λ
→
0
{\lambda\to 0}
. Our results partially solve the open problem addressed in [Y. Jiang, Z. Wang and H.-S. Zhou,
Multiple solutions for a nonhomogeneous Schrödinger–Maxwell system in
ℝ
3
\mathbb{R}^{3}
,
Nonlinear Anal. 83 2013, 50–57] on the existence of infinitely many solutions of the Schrödinger–Poisson system for
p
∈
(
2
,
4
]
{p\in(2,4]}
and a general perturbation term g.
Homogenization for stochastic reaction-diffusion equations with singular perturbation term