DOI: 10.1515/dema-2025-0272 ISSN: 2391-4661
On multi-bump solutions for Choquard equation involving the biharmonic operator in
R
N
${\mathbb{R}}^{N}$
Jiaying Ma, Yueqiang Song Abstract
This article studies the following Choquard equation involving the biharmonic operator in
R
N
${\mathbb{R}}^{N}$
:
Δ
2
u
+
(
λ
A
(
x
)
+
1
)
u
=
(
I
μ
∗
G
(
u
)
)
g
(
u
)
in
R
N
,
u
∈
H
2
(
R
N
)
,
$$\begin{cases}{{\Delta}}^{2}u+\left(\lambda \mathcal{A}\left(x\right)+1\right)u=\left({I}_{\mu }\ast G\left(u\right)\right)g\left(u\right)\hfill & \text{in} {\mathbb{R}}^{N},\hfill \\ u\in {H}^{2}\left({\mathbb{R}}^{N}\right),\hfill \end{cases}$$
where
N
≥ 1, 0 <
μ
<
N
, Δ
2
is the biharmonic operator,
g
is a continuous function with subcritical growth,
A
:
R
N
→
R
$\mathcal{A} : {\mathbb{R}}^{N}\to \mathbb{R}$
is a continuous function verifying some hypotheses. Assuming that the nonnegative function
A
$\mathcal{A}$
has a potential well int
(
A
−
1
(
{
0
}
)
)
$\left({\mathcal{A}}^{-1}\left(\left\{0\right\}\right)\right)$
composed of
k
disjoint components Ω
1
, Ω
2
, ⋯, Ω
k
. By using the variational methods and Morse iteration technique, the existence and multiplicity of positive multi-bump solutions are obtained if the parameter
λ
> 0 is large enough.