The mountain pass theorem is an existence theorem from the calculus of variations. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points.
The assumptions of the theorem are:
I
is a functional from a Hilbert space H to the reals,
I
∈
C
1
(
H
,
R
)
and
I
′
is Lipschitz continuous on bounded subsets of H,
I
satisfies the Palais-Smale compactness condition,
I
[
0
]
=
0
,
there exist positive constants r and a such that
I
[
u
]
≥
a
if
∥
u
∥
=
r
, and
there exists
v
∈
H
with
∥
v
∥
>
r
such that
I
[
v
]
≤
0
.
If we define:
Γ
=
{
g
∈
C
(
[
0
,
1
]
;
H
)
|
g
(
0
)
=
0
,
g
(
1
)
=
v
}
and:
c
=
inf
g
∈
Γ
max
0
≤
t
≤
1
I
[
g
(
t
)
]
,
then the conclusion of the theorem is that c is a critical value of I.
The intuition behind the theorem is in the name "mountain pass." Consider I as describing elevation. Then we know two low spots in the landscape: the origin because
I
[
0
]
=
0
, and a far-off spot v where
I
[
v
]
≤
0
. In between the two lies a range of mountains (at
∥
u
∥
=
r
) where the elevation is high (higher than a>0). In order to travel along a path g from the origin to v, we must pass over the mountains — that is, we must go up and then down. Since I is somewhat smooth, there must be a critical point somewhere in between. (Think along the lines of the mean-value theorem.) The mountain pass lies along the path that passes at the lowest elevation through the mountains. Note that this mountain pass is almost always a saddle point.
For a proof, see section 8.5 of Evans.
Let
X
be Banach space. The assumptions of the theorem are:
Φ
∈
C
(
X
,
R
)
and have a Gâteaux derivative
Φ
′
:
X
→
X
∗
which is continuous when
X
and
X
∗
are endowed with strong topology and weak* topology respectively.
There exists
r
>
0
such that one can find certain
∥
x
′
∥
>
r
with
max
(
Φ
(
0
)
,
Φ
(
x
′
)
)
<
inf
∥
x
∥
=
r
Φ
(
x
)
=:
m
(
r
)
.
Φ
satisfies weak Palais-Smale condition on
{
x
∈
X
∣
m
(
r
)
≤
Φ
(
x
)
}
.
In this case there is a critical point
x
¯
∈
X
of
Φ
satisfying
m
(
r
)
≤
Φ
(
x
¯
)
. Moreover, if we define
Γ
=
{
c
∈
C
(
[
0
,
1
]
,
X
)
∣
c
(
0
)
=
0
,
c
(
1
)
=
x
′
}
then
Φ
(
x
¯
)
=
inf
c
∈
Γ
max
0
≤
t
≤
1
Φ
(
c
(
t
)
)
.
For a proof, see section 5.5 of Aubin and Ekeland.
