This paper deals with embedding theorems involving the Sobolev spaces
H
m,p
(Ω) and
H
0
m
,
p
(
Ω
)
on an unbounded domain
Ω
in
R
n
(
n
>
1
)
. It is shown, for example, that the Sobolev space
H
0
1
,
n
(
Ω
)
is continuously embedded in the Orlicz space L
Φ*
(Ω), where
Φ
(
t
)
=
|
t
|
n
exp
(
|
t
|
n
/
(
n
−
1
)
)
; and that multiplication by suitable functions acts as a compact map of
H
0
1
,
n
(
Ω
)
to
L
Ψ
∗
(
Ω
)
for any Orlicz function
Ψ
subordinate to
Φ
in a certain sense. These results extend the earlier work of Trudinger, who dealt with the case in which
Ω
is bounded. Examples are given of unbounded domains
Ω
for which the natural embedding of in
H
1
,
p
(
Ω
)
in
L
p
(
Ω
)
(
1
<
p
<
∞
)
is a
k
-set contraction for some
k
< 1: the case
k
= 0 corresponds to a compact embedding. Applications are made to the Dirichlet problem in an unbounded domain for elliptic equations with violent non-linearities.
Compact embedding theorems and a Lions' type Lemma for fractional Orlicz–Sobolev spaces
