G(v) ’ ’∞ as v ’ ∞, v ∈ N.

(13.32) E

To see this, let

( j)

wj = ±μk •μ ek

be a sequence in M such that ρ j = w j E ’ ∞. Take w j = w j /ρ j , and let µ > 0 be

˜

such that β+ < m + ’ µ. Then there is a constant K such that

2P(x, t, ξ )/ξ 2 < β+ + µ, |ξ | > K .

Hence,

P(x, t, w j )/ρ 2 ¤ (β+ + µ)w2 +

˜j P(x, t, w j )/ρ 2

2 j j

|w j |>K |w j |¤K

β+ + µ

¤ + C/ρ 2

j

m+ ’ γ

since w j

˜ = 1. Therefore,

E

β+ + µ ’ γ

G(w j )/ρ 2 ≥ 1 ’ ’ C/ρ 2 .

j j

m+ ’ γ

Consequently,

β+ + µ ’ γ

lim inf G(w j )/ρ 2 ≥ 1 ’ > 0.

j

m+ ’ γ

This proves (13.31). A similar argument proves (13.32). Thus, G(v + w) is convex

in w, is concave in v and satis¬es (13.31), (13.32). The theorem now follows from

Theorems 13.6, 13.7, Corollary 13.8, and Lemma 13.11.

13.8. Notes and remarks 161

Next, we prove Theorem 13.13.

Proof. First, we note that the hypotheses of Theorem 13.13 imply those of Theo-

rem 13.14. In fact, we can take β± = θ± , for we have

ξ ds

P(x, t, ξ ) = s[ p(x, t, s) ’ p(x, t, 0)] + p(x, t, 0)ξ.

s

0

Hence,

θ’ ξ 2 ¤ 2P(x, t, ξ ) ’ 2 p(x, t, 0)ξ ¤ θ+ ξ 2

by (13.26). Thus, there exists a weak solution of (13.1)“(13.3). To show that it is

unique, we note that for ¬xed v ∈ N, G(v + w) is strictly convex in w. Hence, if

there were two distinct points such that G (u 0 ) = G (u) = 0, we would have by

Lemma 13.10 both G(u) < G(u 0 ) and G(u 0 ) < G(u), an impossibility.

13.8 Notes and remarks

The problem (13.1)“(13.3) has been considered by Smiley [147] and Mawhin [90].

They assume

β0 ξ 2 ¤ ξ [ p(x, t, ξ1 ) ’ p(x, t, ξ0 )], ξ = ξ1 ’ ξ0 ∈ R,

and

| p(x, t, ξ1 ) ’ p(x, t, ξ0 )| ¤ β1 |ξ |,

where 0 < β1 and β1 < β0 . This implies 0 < β0 ¤ β1 < 1 and β1 ’ β0 <

2

2 (1 ’ β0 ). Since ρ( ) consists only of integers, our results are extensions of their

1

results. Theorems 13.13 and 13.14 were proved in [125].

Chapter 14

Type (II) Regions

14.1 Introduction

The Fuˇ´k spectrum described in Chapter 11 arises in the study of semilinear elliptic

c±

boundary-value problems of the form

Au = f (x, u),

(14.1)

where A is a selfadjoint operator having compact resolvent on L 2 ( ), ‚ Rn , and

f (x, t) is a Carath´ odory function on ¯ — R such that

e

f (x, t)/t ’ a a.e. as t ’ ’∞

(14.2)

’ b a.e. as t ’ +∞.

If |u(x)| is large, then (14.1) approximates the equation

Au = bu + ’ au ’ ,

(14.3)

where u ± = max{±u, 0}. It was ¬rst noticed by Fuˇ´k [68] that (14.3) plays an

c±

important part in the study of (14.1) when (14.2) holds. Fuˇ´k studied the problem

c±

’u = bu + ’ au ’ in (0, π), u(0) = u(π) = 0

(14.4)

and showed that there is a substantial difference in the solvability of (14.1) if (14.3) has

nontrivial solutions. We now call the set of those (a, b) ∈ R2 for which (14.3) has

nontrivial solutions the Fuˇ ´k spectrum of A. Fuˇ´k showed that for (14.4), consists

c± c±

of a sequence of decreasing curves passing through the points (k 2 , k 2 ), k = 1, 2, . . . ,

with one or two curves emanating from each of these points. Points not on these curves

are not in . He also noticed that there are two different types of regions between the

curves, namely,

(I) regions between curves passing through consecutive points

(k 2 , k 2 ), ([k + 1]2 , [k + 1]2 )

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_14,

© Birkh¤user Boston, a part of Springer Science+Business Media, LLC 2009

164 14. Type (II) Regions

and

regions between curves passing through the same point (k 2 , k 2 ).

(II)

In the type (I) regions one can solve

’u = bu + ’ au ’ + p(x), x ∈ (0, π), u(0) = u(π) = 0,

(14.5)

for arbitrary p(x) ∈ L 2 (0, π), while this is not so for regions of type (II).

No complete description of for the general case (14.1) has been found. If

0 < »0 < »1 < · · · < »k < · · ·

(14.6)

are the eigenvalues of A, it was shown in [111] and [122] that in the square

’1 , » +1 ] ,

2

[»

,1 , C ,2

there are decreasing curves C (which may coincide) passing through the point