Affiliation:
1. School of Mathematics and Statistics & Chongqing Key Laboratory of Economic and Social Application Statistics , Chongqing Technology and Business University , Chongqing 400067 , P. R. China
2. School of Mathematical and Statistical Sciences , National University of Ireland , Galway , Ireland
Abstract
Abstract
In this paper, we study the existence and multiplicity of solutions for the Schrödinger–Bopp–Podolsky system
{
-
Δ
u
+
V
(
x
)
u
+
ϕ
u
=
f
(
u
)
+
λ
|
u
|
4
u
in
ℝ
3
,
-
Δ
ϕ
+
a
2
Δ
2
ϕ
=
4
π
u
2
in
ℝ
3
,
\left\{\begin{aligned} \displaystyle{-}\Delta u+V(x)u+\phi u&\displaystyle=f(u%
)+\lambda|u|^{4}u&&\displaystyle\phantom{}\text{in }\mathbb{R}^{3},\\
\displaystyle{-}\Delta\phi+a^{2}\Delta^{2}\phi&\displaystyle=4\pi u^{2}&&%
\displaystyle\phantom{}\text{in }\mathbb{R}^{3},\end{aligned}\right.
where
x
∈
ℝ
3
{x\in\mathbb{R}^{3}}
,
a
>
0
{a>0}
,
V
(
x
)
∈
𝒞
(
ℝ
3
,
ℝ
)
{V(x)\in\mathcal{C}(\mathbb{R}^{3},\mathbb{R})}
.
Using variational methods and the symmetric mountain pass theorem, we establish the existence of multiple solutions for this system.