inf I (v + w + y0 , a, b) > 0.

(14.29) sup

w∈M

v∈N ’1

Note that I (v + w + y0 , a, b) is strictly convex and lower semi-continuous in w ∈ M

and strictly concave and continuous in v ∈ N ’1 because (a, b) ∈ Q . Moreover,

I (w + y0 , a, b) ’ ∞ as w ’ ∞, w ∈ M.

D

To see this, let w = ρ ’ ∞. Then, for u = (w + y0 )/ρ, we have

D

I (u, a, b) ≥ 1 ’ a u ’ ’ b u+ ’ 1 ’ a w’ ’ b w+

˜ ˜ > 0,

2 2 2 2

14.2. The asymptotic equation 167

where w ¤ 1, since a, b < »

˜ +1 . Similarly,

I (v + y0 , a, b) ’ ∞ as v ’ ∞, v ∈ N.

D

Hence we may apply Theorem 13.6 to conclude that there are unique v 0 ∈ N ’1 ,

w0 ∈ M such that

I (v 0 + w0 + y0 , a, b) = inf I (v + w + y0 , a, b)

(14.30) sup

w∈M

v∈N ’1

= sup I (v + w + y0 , a, b).

inf

w∈M v∈N ’1

In particular, this shows that y0 = 0, for otherwise v 0 = 0, w0 = 0 would satisfy

(14.30) and the uniqueness would violate (14.29). We also see from (14.29) and (14.30)

that

sup I (v + w + y0 , a, b) > 0, w∈M .

(14.31)

v∈N ’1

Moreover, it follows that there is a continuous map θ from M to N such that

’1 ’1

θ (w) ¤ C w D, w∈M ’1 ,

(14.32)

θ (sw) = sθ (w), s ≥ 0,

(14.33)

and v = θ (w) is the only solution of

˜

I (w + v, a, b) ⊥ N

˜ ’1 , w∈M ’1 ,

(14.34)

and

I (w + v, a, b) = max I (w + v, a, b),

˜ w∈M ’1 .

(14.35)

v∈N ’1

Thus, (14.31) implies

I (w + sy0 + θ (w + sy0 ), a, b) ≥ 0, w ∈ M , s ≥ 0.

(14.36)

Another way of writing (14.36) is

I (u, a, b) ≥ 0, u∈S

(14.37) 2

where

S 2 , = {u ∈ D : I (u, a, b) ⊥ N ’1 , (u, y0 ) ≥ 0}.

(14.38)

Thus, we have proved

Lemma 14.4. If (a, b) ∈ Q is below the upper curve C ,2 , then there is a y0 ∈

E(» )\{0} such that (14.37) holds for all u ∈ S 2 , where S 2 is given by (14.38).

168 14. Type (II) Regions

In the same vein, if (a, b) ∈ Q is above the lower curve C 1 , then there is a

y1 ∈ N0 such that

sup I (v + w + y1 , a, b) < 0.

(14.39) inf

w∈M v∈N ’1

Using the same reasoning as before, we have y1 = 0 and (14.30) holds with y0 replaced

by y1 . This implies

inf I (v + w + y1 , a, b) < 0, v∈N ’1 .

(14.40)

w∈M

Now we use the fact that there is a continuous map „ from N to M such that

„ (v) ¤C v , v∈N,

(14.41) D

„ (sv) = s„ (v), s ≥ 0,

(14.42)

and w = „ (v) is the unique solution of

˜

I (v + w, a, b) ⊥ M ,

˜ v∈N,

(14.43)

and

I (v + w, a, b) = inf I (v + w, a, b),

˜ v∈N.

(14.44)

w∈M

Thus, (14.40) implies

I (v + sy1 + „ (v + sy1 ), a, b) ¤ 0, v∈N ’1 , s ≥ 0,

(14.45)

or

I (u, a, b) ¤ 0, u ∈ S 1,

(14.46)

where

= {u ∈ D : I (u, a, b) ⊥ M , (u, y1 ) ≥ 0}.

(14.47) S 1

We therefore have

Lemma 14.5. If (a, b) ∈ Q lies above the lower curve C ,1 , then there is a y1 ∈

E(» )\{0} such that (14.46) holds for all u ∈ S 1 , where S 1 is given by (14.47).

14.3 Local estimates

In proving Theorems 14.2 and 14.3, we shall make use of the functional

G(u) = u ’2 F(x, u) d x = I (u, a, b) ’ 2

2

(14.48) P(x, u) d x

D

for functions u ∈ D. When (14.10) holds, it is easily checked that G ∈ C 1 (D, R) and

that u ∈ D is a solution of (14.1) if, and only if, it satis¬es

G (u) = 0.

(14.49)