If N is unbounded, assume that there is a w0 ∈ M such that

G(v, w0 ) ’ ’∞ as v ’ ∞, v ∈ N.

(13.10)

[If M is bounded, then (13.9) is automatically satis¬ed; the same is true for (13.10)

when N is bounded.] We have

Theorem 13.6. Under the above hypotheses, G has at least one saddle point.

Proof. Assume ¬rst that M, N are bounded. Then, for each v ∈ N, there is at least

one point w where G(v, w) achieves its minimum (Lemma 13.1). Let

J (v) = min G(v, w).

w∈M

Since J (v) is the minimum of a family of functionals that are concave and u.s.c., it is

also concave and u.s.c. In fact, if

v t = (1 ’ t)v 0 + tv 1 , t ∈ [0, 1],

then

G(v t , w) ≥ (1 ’ t) min G(v 0 , w) + t min G(v 1 , w),

ˆ ˆ w ∈ M.

w∈M

ˆ w∈M

ˆ

13.3. Existence of saddle points 153

Since this is true for each w ∈ M, we see that

J (v t ) ≥ (1 ’ t)J (v 0 ) + t J (v 1 ).

(13.11)

Similarly, if v k ’ v ∈ N, then we have

J (v k ) ¤ G(v k , w), w ∈ M.

Thus,

lim sup J (v k ) ¤ lim sup G(v k , w) ¤ G(v, w), w ∈ M.

Since this is true for each w ∈ M, we have

lim sup J (v k ) ¤ inf G(v, w) = J (v).

(13.12)

w∈M

Therefore, J (v) is concave and u.s.c. Consequently, J (v) has a maximum point v

¯

satisfying

J (v) ¤ J (v), v ∈ N

¯

(Lemma 13.1). In particular, we have

J (v) = min G(v, w) ¤ G(v, w),

¯ ¯ˆ ¯ w ∈ M.

(13.13)

w∈M

ˆ

Let v be an arbitrary point in N, and let

v θ = (1 ’ θ )v + θ v,

¯ 0 ¤ θ ¤ 1.

Since G is concave in v, we have

G(v θ , w) ≥ (1 ’ θ )G(v, w) + θ G(v, w).

¯

Consequently,

J (v) ≥ J (v θ )

¯

= G(v θ , wθ )

≥ (1 ’ θ )G(v, wθ ) + θ G(v, wθ )

¯

≥ (1 ’ θ )J (v) + θ G(v, wθ ),

¯

where wθ is any point in M such that

G(v θ , wθ ) = min G(v θ , w).

w∈M

This gives

J (v) ≥ G(v, wθ ),

¯ v ∈ N, 0 < θ ¤ 1.

(13.14)

Let {θk } be a sequence converging to 0, and let v k = v θk , wk = wθk . Then v k ’ v.

¯

Since M is bounded, there is a renamed subsequence such that wk w. Since

¯

(1 ’ θ )G(v, wθ ) + θ G(v, wθ ) ¤ G(v θ , wθ ) ¤ G(v θ , w),

¯ w ∈ M,

154 13. Semilinear Wave Equations

we have

(1 ’ θk )G(v, wk ) + θk J (v) ¤ G(v k , w),

¯ w ∈ M.

In the limit this gives

G(v, w) ¤ G(v, w),

¯¯ ¯ w∈M

(cf. Lemma 13.3). Since

J (v) ≥ G(v, wk ),

¯

we have

G(v, w) ¤ J (v) ¤ G(v, w),

¯ ¯ ¯ v ∈ N, w ∈ M,

in view of (13.13) and (13.14). Take v = v and w = w. Then

¯ ¯

G(v, w) ¤ J (v) ¤ G(v, w),

¯¯ ¯ ¯¯

showing that

G(v, w) = J (v)

¯¯ ¯

and

G(v, w) ¤ G(v, w) ¤ G(v, w),

¯ ¯¯ ¯ v ∈ N, w ∈ M.

Thus, (v, w) is a saddle point.

¯¯

Now, we remove the restriction that M, N are bounded. Let R be so large that

v 0 < R, w0 < R. The sets

M R = {w ∈ M : w ¤ R}, N R = {v ∈ N : v ¤ R}

are closed, convex, and bounded. By what we have already proved, there is a saddle

point (v R , w R ) such that

¯¯

G(v, w R ) ¤ G(v R , w R ) ¤ G v R , w),

¯ ¯¯ ¯ v ∈ NR , w ∈ MR .

(13.15)

In particular, we have

G(v 0 , w R ) ¤ G(v R , w R ) ¤ G(v R , w0 ).

¯ ¯¯ ¯

Since G(v 0 , w) is convex, is l.s.c., and satis¬es (13.9), it is bounded from below on M

(Lemma 13.5). Thus,

G(v 0 , w R ) ≥ A > ’∞.

¯

Similarly, G(v, w0 ) is bounded from above. Hence,