Abstract
In this paper, we investigate the logarithmic Kirchhoff-type equation
-
(
a
+
b
∫
R
3
|
∇
u
|
2
d
x
)
Δ
u
+
V
(
x
)
u
=
|
u
|
p
-
2
u
log
|
u
|
,
x
∈
R
3
where
a
,
b
>
0
are constants, and
V
:
R
3
→
R
is continuous and may change sign. Using Nehari manifold method and the concentration-compactness principle, we prove the existence of nontrivial and nonnegative (weak) solutions under some assumptions on the potential function
V
without radial symmetry or compactness hypotheses.