(» , » )

such that all points above or below both curves in the square are not in [and are

of type (I)] while points on the curves are in . The status of points between the

curves (when they do not coincide) is unknown in general. However, it was shown

in [72] that when » is a simple eigenvalue, points between the curves are not in .

On the other hand, C. and W. Margulies [91] have shown that there are boundary-value

emanate from a point (» , » ) when » is a

problems for which many curves in

multiple eigenvalue. (Clearly, these curves are contained in the region between the

curves C ,1 and C ,2 .)

As expected, the boundary-value problem (14.1) is more readily solved when (a, b)

is in a region of type (I) (i.e., on the same side of both curves C 1 , C 2 ). For such points,

the following was proved in [111] and [122].

Theorem 14.1. Assume that f (x, t) is of the form

f (x, t) = bt + ’ at ’ + p(x, t),

(14.7)

where p(x, t) satis¬es

| p(x, t)| ¤ V (x)1’σ |t|σ + W (x),

(14.8)

with 0 ¤ σ < 1 and V, W ∈ L 2 ( ). Then (14.1) has a solution. In particular,

Au = bu + ’ au ’ + p(x)

(14.9)

has a solution for each p ∈ L 2 ( ).

However, when it comes to points on the curves C ,1 , C ,2 , no such theorem holds.

In Chapter 11 we addressed this issue and presented suf¬cient conditions for the

existence of solutions of (14.1) when (a, b) is on either C ,1 or C ,2 . [Needless to

14.1. Introduction 165

say, much more is required of p(x, t) in Theorems 11.1 and 11.2 than in Theorem

14.1.] In the present chapter we consider the situation when (a, b) is in the region

between the curves C ,1 , C ,2 (when they do not coincide). As we saw earlier, such

points may or may not belong to . We shall be concerned with those points in this

region that are not in . Even in the case when none of these points is in (as in the

case of a simple eigenvalue), we cannot prove a theorem as comprehensive as Theorem

14.1. However, we can prove the theorems below. We assume that

| p(x, t)| ¤ C(|t| + 1), x∈ , t ∈ R,

(14.10)

and that E(» ), the subspace of eigenfunctions corresponding to » , is contained in

L ∞ ( ) (i.e., the eigenfunctions corresponding to » are bounded). We de¬ne

t

F(x, t) = f (x, s) ds

(14.11)

0

and

t

P(x, t) =

(14.12) p(x, s) ds.

0

We have

Theorem 14.2. Let (a, b) be a point in Q := (» ’1 , » +1 )

2 that lies below the upper

curve C ,2 . Assume (14.2), (14.7), and

f (x, t1 ) ’ f (x, t0 ) >» ’1 (t1 ’ t0 ), t0 < t1 , x ∈ ,

(14.13)

2P(x, t) ¤W1 (x) ∈ L 1 ( ), x∈ , t ∈ R,

(14.14)

2F(x, t) ¤» , x∈ , t ∈ R,

2

(14.15) +1 t

» t 2 ¤2F(x, t), |t| < δ for some δ > 0.

(14.16)

If (a, b) is not in , then (14.1) has a nontrivial solution.

Theorem 14.3. Let (a, b) be a point in Q \ that lies above the lower curve C ,1 .

Assume (14.2), (14.7), and

2P(x, t) ≥ ’ W1 (x) ∈ L 1 ( ), x∈ , t ∈ R,

(14.17)

f (x, t1 ) ’ f (x, t0 ) <» +1 (t1 ’ t0 ), t0 < t1 , x ∈ ,

(14.18)

» ¤2F(x, t), x∈ , t ∈ R,

2

(14.19) ’1 t

2F(x, t) ¤» t 2 , |t| < δ for some δ > 0.

(14.20)

Then (14.1) has a nontrivial solution.

It should be noted that these theorems hold even when » is a multiple eigenvalue.

In proving the theorems, we examine the functional

G(u) = u ’2 u ∈ D = D(A1/2 ),

2

(14.21) F(x, u)d x,

D

166 14. Type (II) Regions

where u D = A1/2u , and search for solutions of G (u) = 0. Even though (a, b) is

in a region of type (II), we can ¬nd a manifold S on which G is bounded from below.

Unfortunately, S has a boundary that cannot be linked with a subspace, and one must

search for another manifold that links the boundary of S and on which G is bounded

from above. Once this is achieved, we can apply the theorems of Chapter 2 to obtain

a Palais“Smale sequence. We then show that there is a convergent subsequence due to

the fact that (a, b) is not in . In Theorem 14.3 we reverse the procedure, ¬nding a

manifold on which G is bounded from above and then searching for a linking set on

which it is bounded from below.

14.2 The asymptotic equation

In this section we show how information concerning (14.3) affects the solvability of

(14.1). For each ¬xed positive integer , we let N denote the subspace of D =

D(A1/2 ) spanned by the eigenfunctions of A corresponding to the eigenvalues »0 , »1 ,

. . . , » , and we let M = N ⊥ © D. Then D = M • N . For (a, b) ∈ R2 , we de¬ne

=(Au, u),

2

(14.22) u D

’ a u’ ’ b u+ 2,

I (u, a, b) = u u ∈ D,

2 2

(14.23) D

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

(14.24)

w∈M

v∈N

w D =1

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

(14.25)

w∈M

v∈N

v D =1

ν (a) =sup{b : M (a, b) ≥ 0},

(14.26)

μ (a) = inf{b : m (a, b) ¤ 0}.

(14.27)

It follows from Lemma 11.3 that ν ’1 (a) ¤ μ (a) in Q . We let C ,1 be the (lower)

curve b = ν ’1 (a) and we let C ,2 be the (upper) curve b = μ (a). Thus, by (14.27),

if (a, b) ∈ Q is below the curve C ,2 , then

m (a, b) > 0.

(14.28)