Abstract
This paper deals with the following logarithmic Schrödinger–Bopp–Podolsky system
{
−
Δ
u
+
λ
V
(
x
)
u
+
q
(
x
)
ϕ
u
=
u
log
u
2
,
in
ℝ
3
,
−
Δ
ϕ
+
Δ
2
ϕ
=
4
π
q
(
x
)
u
2
,
in
ℝ
3
,
where
λ
>
0
,
V
(
x
)
∈
C
(
ℝ
3
,
ℝ
)
and
q
(
x
)
≥
0
. Under suitable conditions on potentials
V
(
x
)
and
q
(
x
)
, we demonstrate the existence of a positive solution
u
λ
∈
H
1
(
ℝ
3
)
of the above problem for
λ
>
0
large enough via the variational method developed by Szulkin for the functional that is the sum of a smooth and a convex lower semicontinuous term, and we also study the concentration behavior of positive solutions
{
u
λ
}
as
λ
→
+
∞
.