Affiliation:
1. School of Mathematical Sciences , Beijing Normal University , Beijing 100875 , P. R. China
2. Department of Mathematics , University of Connecticut , Storrs , CT 06269 , USA
Abstract
Abstract
In this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in
ℝ
4
{\mathbb{R}^{4}}
. We also give a new Sobolev compact embedding which states
W
2
,
2
(
ℝ
4
)
{W^{2,2}(\mathbb{R}^{4})}
is compactly embedded into
L
p
(
ℝ
4
,
|
x
|
-
β
d
x
)
{L^{p}(\mathbb{R}^{4},|x|^{-\beta}\,dx)}
for
p
≥
2
{p\geq 2}
and
0
<
β
<
4
{0<\beta<4}
. As applications, we establish the existence
of ground state solutions to the following bi-Laplacian equation with critical nonlinearity:
Δ
2
u
+
V
(
x
)
u
=
f
(
x
,
u
)
|
x
|
β
in
ℝ
4
,
\displaystyle\Delta^{2}u+V(x)u=\frac{f(x,u)}{|x|^{\beta}}\quad\mbox{in }%
\mathbb{R}^{4},
where
V
(
x
)
{V(x)}
has a positive lower bound and
f
(
x
,
t
)
{f(x,t)}
behaves like
exp
(
α
|
t
|
2
)
{\exp(\alpha|t|^{2})}
as
t
→
+
∞
{t\to+\infty}
. In the case
β
=
0
{\beta=0}
, because of the loss of Sobolev compact embedding, we use the principle
of symmetric criticality to obtain the existence of ground state solutions by assuming
f
(
x
,
t
)
{f(x,t)}
and
V
(
x
)
{V(x)}
are radial with respect to x and
f
(
x
,
t
)
=
o
(
t
)
{f(x,t)=o(t)}
as
t
→
0
{t\rightarrow 0}
.
Funder
National Natural Science Foundation of China
Subject
General Mathematics,Statistical and Nonlinear Physics
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