Affiliation:
1. East China Jiaotong University
2. Shandong University of Science and Technology
Abstract
In this article, we consider the concentration of positive solutions for the following equation with Trudinger–Moser nonlinearity:
{
−
Δ
N
u
−
Δ
q
u
+
V
(
ε
x
)
(
|
u
|
N
−
2
u
+
|
u
|
q
−
2
u
)
=
f
(
u
)
,
x
∈
R
N
,
u
∈
W
1
,
N
(
R
N
)
∩
W
1
,
q
(
R
N
)
,
x
∈
R
N
,
where
V
is a positive continuous function and has a local minimum,
ε
>
0
is a small parameter,
2
≤
N
<
q
<
+
∞
,
f
is
C
1
with subcritical growth. When
V
and
f
satisfy some appropriate assumptions, we construct the solution
u
ε
that concentrates around any given isolated local minimum of
V
by applying the penalization method for the above equation.
Funder
National Natural Science Foundation of China