Schechter Minimax Systems and Critical Point Theory (Springer, 2009)

3.5 Examples of linking sets

We now discuss various subsets of a Banach space E with respect to linking. First we have

Proposition 3.9. [122] Let A, B be two closed, bounded subsets of E such that E\ A is path connected. If A links B [hm], then B links A [hm].

The next proposition gives a very useful method of checking the linking of two sets.

Proposition 3.10. [122] Let F be a continuous map from E to Rn, and let Q E be such that F0 = F|Q is a homeomorphism of Q onto the closure of a bounded open subset of Rn. If p , then F0−1(∂ ) links F−1( p) [hm].

Proposition 3.11. [122] If H is a homeomorphism of E onto itself and A links B [hm], then H A links H B [hm].

The following examples were given in [122].

Example 1. Let B be an open set in E, and let A consist of two points e1, e2 with

Example 2. Let M, N be closed subspaces such that dim N < ∞ and E = M N. Let

Example 3. We take M, N as in Example 2. Let w0 = 0 be an element of M, and take

A= {v N : v ≤ R} {sw0 + v : v N, s ≥ 0, sw0 + v = R},

B= ∂ Bδ ∩ M, 0 < δ < R.

Then A and B link each other [hm].

Example 4. Take M, N as before and let v0 = 0 be an element of N. We write N = {v0} N . We take

A= {v N : v ≤ R} {sv0 + v : v N , s ≥ 0, sv0 + v = R},

B= {w M : w ≥ δ} {sv0 + w : w M, s ≥ 0, sv0 + w = δ},

where 0 < δ < R. Then A links B [hm].

Example 5. This is the same as Example 4 with A replaced by A = ∂ BR ∩ N.

Example 6. Let M, N be as in Example 2. Take A = ∂ Bδ ∩ N, and let v0 be any element in ∂ B1 ∩ N. Take B to be the set of all u of the form

u = w + sv0, w M,

3.6 Various geometries

We now apply the theorems of the preceding sections to various geometries in Banach space. As before, we assume that G C1(E, R) and that ψ satisfies the hypotheses of Theorem 2.4.

Theorem 3.13. Assume that there is a δ > 0 such that

Then, for each function ψ(t) satisfying the hypotheses of Theorem 2.4, there is a

Then, for each function ψ(t) satisfying the hypotheses of Theorem 2.4, there is a sequence {uk } E such that

(3.21) G(uk ) → c, α ≤ c ≤ γ , G (uk )/ψ(d(uk , M)) → 0.

Proof. This time we take A and B as in Example 2 above. Thus, A links B [hm]. Again, aR given by (2.6) is finite for each R since

Again we see that we can apply Theorem 2.24 to conclude that the desired sequence exists.

Theorem 3.15. Let M, N be as in Theorem 3.14, and let G C1(E, R) satisfy

for some w0 ∂ B1 ∩ M, where 0 < δ < R0. Then, for each function ψ(t) satisfying the hypotheses of Theorem 2.4, there is a sequence {uk } E such that (3.12) holds.

Proof. Here we take A, B as in Example 3 above. Thus, A and B link each other [hm].

Here

A = {sw0 + v : s ≥ 0, v N, sw0 + v = R}.

Again, for each R, the quantity a given by (2.6) is finite since

aR ≤ max G,

Q

where

Q = {sw0 + v : s ≥ 0, v N, sw0 + v ≤ R}.

We now apply Theorem 2.24 to conclude that the desired sequence exists.

Theorem 3.16. Let M, N be as in Theorem 3.14, and let v0 ∂ B1 ∩ N. Take N = {v0} N . Let G C1(E, R) be such that

where 0 < δ < R0. Then, for each function ψ(t) satisfying the hypotheses of Theorem 2.4, there is a sequence {uk } E such that (3.12) holds.

26 3. Examples of Minimax Systems

Proof. We take A, B as in Example 5 above. Thus, A links B [hm]. As before, we note that aR < ∞ for each R. Hence, (3.12) holds by Theorem 2.24.

3.7 A sandwich theorem

We now discuss a very useful theorem that allows one to consider functionals that are bounded from below on one subspace and bounded from above on another with no correlation between the bounds. This provides such functionals with the same advantages as those that are semibounded. One drawback is the requirement that one of the subspaces be finite-dimensional. This condition will be removed in Chapter 15 if we assume that the functional satisfies more than the mere continuity of its derivative.

Theorem 3.17. Let N be a closed subspace of a Hilbert space E and let M = N . Assume that at least one of the subspaces M, N is finite-dimensional. Let G be a C1-functional on E such that

Let ψ(t) be a positive, nonincreasing, locally Lipschitz continuous function on [0, ∞) such that (2.8) holds. Then there are a constant c R and a sequence {uk } E such that

where P is the (orthogonal) projection onto M.

Proof. We may assume dim N < ∞; otherwise, we can consider −G in place of G. Let A be the set ∂ BR ∩ N, and take B = M, where R > 0 is arbitrary. Then A links B [hm] by Example 2 above. We now apply Theorem 2.21. Note that a0 ≤ m1, m0 = b0, and (2.31) holds for R sufficiently large. We also note that

aR ≤ sup G ≤ m1

BR ∩N

by taking (s)u = (1 − s)u, u E. Hence, by Theorem 2.21, for each δ > 0, there is a u E such that

m0 − δ ≤ G(u) ≤ m1 + δ, G (u) < ψ(d(u, B )).

Since this is true for each δ > 0, we obtain the desired conclusion.

Since ε was arbitrary, we see that (3.35) holds.

Here are some consequences.

Theorem 3.21. Let G be a C1-functional on E such that

If ψ satisfies the hypotheses of Theorem 2.4, then there is a sequence {uk } E such that

Proof. We refer to Theorem 2.24. We take a sequence of points such that G(vk ) >

where βk = vk . We then take

ψk (t) = ck ψ(t + βk )

in Theorem 2.24. Then (2.53) holds. We used the fact that ∂ BRk +βk links {vk } for each k (Theorem 3.12). The conclusion follows since

u ≤ d(u, vk ) + βk .

In the second case, we replace G with −G.

Corollary 3.22. If (3.36) holds, then there is a sequence satisfying

3.8 Notes and remarks

The results of Section 3.2 are from [136], [114], and [120] (cf. also [122]). Sections 3.5 and 3.6 are from [122]. The results of Section 3.7 come from [143], [109], [108], [129], and [132].