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 , andthere 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.
