Kenneth Kuttler

24.1. MOUNTAIN PASS THEOREM IN HILBERT SPACE 817

Corollary 24.1.11 Suppose f : X → X is continuous and f is locally Lipschitz on U, anopen subset of X, a Banach space. Suppose also that f (x) = 0 for all x /∈ U and that∥ f (x)∥< M for all x ∈ X . Then there exists a solution to the IVP

y′ = f (y) , y(0) = y0

for t ∈ [0,T ] for any T > 0.

Proof: Let T be given. If y0 /∈ U, there is nothing to show. The solution is y(t) ≡y0. Suppose then that y0 ∈ U . Then by Theorem 24.1.8, there exists a unique solutionto the initial value problem on an interval [0, T̂ ) of maximal length. If T̂ = T, then astn→ T,{y(tn)} must converge. This is because for tm < tn,

∥y(tn)− y(tm)∥ ≤M |tn− tm|

showing that this is a Cauchy sequence. Since all such sequences lead to a Cauchy se-quence, it must be the case that limt→T y(t) exists. Thus it equals

y0 +∫ T

0f (y(t))dt

We let y(T ) equal the above and it follows from Gronwall’s inequality that there is a uniquesolution to the IVP on [0,T ] so the claim is true in this case.

Otherwise, if T̂ < T, then one can define

y(T̂)≡ y0 +

∫ T̂

0f (y(s))ds

If y(T̂)∈U, then by the assumption that f is bounded, one could consider a new initial

condition and extend the solution further violating the maximality of the length of [0, T̂ ).Therefore, it must be the case that y

(T̂)∈UC. Then the solution is

ŷ(t) ={

y(t) , t < T̂y(T̂), t > T̂

because f(y(T̂))

= 0 by assumption.One could also change the above argument for Corollary 24.1.11 to include the case

that f has linear growth.

24.1.2 Mountain Pass Theorem In Banach SpaceIn this section, is a more general version of the mountain pass theorem. It is generalized intwo ways. First, the space is not a Hilbert space and second, the derivative of the functionalis not assumed to be Lipschitz. Instead of using I′ one uses the pseudogradient in anappropriate differential equation. This is a significant generalization because there is noconvenient projection map from X ′ to X like there is in Hilbert space. This is why the useof the psedogradient is so interesting. For many more considerations of this sort of thing,see [55]. First is a deformation theorem. Here I will be defined on a Banach space X andI′ (x) ∈ X ′. First recall the Palais Smale conditions.

24.1. MOUNTAIN PASS THEOREM IN HILBERT SPACE 817Corollary 24.1.11 Suppose f : X — X is continuous and f is locally Lipschitz on U, anopen subset of X, a Banach space. Suppose also that f (x) = 0 for all x € U and that|| f (x) || < M for all x € X. Then there exists a solution to the IVPy=fl(y), y)=yofort € [0,7] for any T > 0.Proof: Let T be given. If yo ¢ U, there is nothing to show. The solution is y(t) =yo. Suppose then that yp € U. Then by Theorem 24.1.8, there exists a unique solutionto the initial value problem on an interval [0,7) of maximal length. If 7 = T, then ast, + T,{y (tn) } must converge. This is because for ty < ty,lly (tn) —y(tn)|| < M |ty —tn|showing that this is a Cauchy sequence. Since all such sequences lead to a Cauchy se-quence, it must be the case that lim,_,7 y(t) exists. Thus it equalsw+ [ rowatWe let y (T) equal the above and it follows from Gronwall’s inequality that there is a uniquesolution to the IVP on [0,7] so the claim is true in this case.Otherwise, if 7 < T, then one can definey(f)=v0+ | FoUs)asIf y (T) € U, then by the assumption that f is bounded, one could consider a new initialcondition and extend the solution further violating the maximality of the length of [0,7).Therefore, it must be the case that y (T) € UC. Then the solution isaay (t),t<Ts)=f y(f).t >tbecause f (y(7)) =0 by assumption. fOne could also change the above argument for Corollary 24.1.11 to include the casethat f has linear growth.24.1.2 Mountain Pass Theorem In Banach SpaceIn this section, is a more general version of the mountain pass theorem. It is generalized intwo ways. First, the space is not a Hilbert space and second, the derivative of the functionalis not assumed to be Lipschitz. Instead of using J’ one uses the pseudogradient in anappropriate differential equation. This is a significant generalization because there is noconvenient projection map from X’ to X like there is in Hilbert space. This is why the useof the psedogradient is so interesting. For many more considerations of this sort of thing,see [55]. First is a deformation theorem. Here J will be defined on a Banach space X andI' (x) € X". First recall the Palais Smale conditions.