Abstract
In this paper, we consider the problem
(
P
ε
)
:
Δ
2
u
=
u
n
+
4
/
n
-
4
+
ε
u
,
u
>
0
in
Ω
,
u
=
Δ
u
=
0
on
∂
Ω
, where
Ω
is a bounded and smooth domain in
R
n
,
n
>
8
and
ε
>
0
. We analyze the asymptotic behavior of solutions of
(
P
ε
)
which are minimizing for the Sobolev inequality as
ε
→
0
and we prove existence of solutions to
(
P
ε
)
which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for
ε
small,
(
P
ε
)
has at least as many solutions as the Ljusternik–Schnirelman category of
Ω
.