Affiliation:
1. College of Science, China University of Petroleum Qingdao 266580 , Shandong , P. R. China
2. College of Mathematics and Statistics, Chongqing Jiaotong University , Chongqing 400074 , P.R. China
Abstract
Abstract
We are concerned with the following Kirchhoff-type equation with exponential critical nonlinearities
−
a
+
b
∫
R
2
∣
∇
u
∣
2
d
x
Δ
u
+
(
h
(
x
)
+
μ
V
(
x
)
)
u
=
K
(
x
)
f
(
u
)
in
R
2
,
-\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{2}}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+\left(h\left(x)+\mu V\left(x))u=K\left(x)f\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{2},
where
a
,
b
,
μ
>
0
a,b,\mu \gt 0
, the potential
V
V
has a bounded set of zero points and decays at infinity as
∣
x
∣
−
γ
| x{| }^{-\gamma }
with
γ
∈
(
0
,
2
)
\gamma \in \left(0,2)
, the weight
K
K
has finite singular points and may have exponential growth at infinity. By using the truncation technique and working in some weighted Sobolev space, we obtain the existence of a mountain pass solution for
μ
>
0
\mu \gt 0
large and the concentration behavior of solutions as
μ
→
+
∞
\mu \to +\infty
.
Cited by
4 articles.
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