(10.130)

= min{ , νρ 2 }. By (10.93), there is an R > ρ such that

and 1

sup G ¤ 0,

(10.131)

A

where A = N © ‚ B R . We can now apply Theorem 10.2 to conclude that there is

a sequence in D satisfying (10.3). As in the proof of Theorem 10.7, this leads to a

solution of (10.86) satisfying G(u) = c ≥ 1 . Hence, u = 0, and we have a nontrivial

solution of (10.76), (10.77). It therefore remains only to prove (10.128). Clearly, ν ≥ 0.

If ν = 0, then there is a sequence {wk } ‚ M such that

G(0, wk ) ’ 0, b(wk ) = 1.

(10.132)

Thus, there is a renamed subsequence such that wk ’ w weakly in M, strongly in N,

and a.e. in . Consequently,

G(0, wk ) ≥ 1 ’ μ(x)wk d x ≥ [μ0 ’ μ(x)]wk d x ’ 0

2 2

(10.133)

and

μ(x)w2 d x ¤ μ0 w 2

1= ¤ b(x) ¤ 1,

(10.134)

which means that we have equality throughout. It follows that we must have w ∈

E(μ0 ), the eigenspace of μ0 . Since w ≡ 0, we have w = 0 a.e. But

[μ0 ’ μ(x)]w2 d x = 0

(10.135)

implies that the integrand vanishes identically on , and consequently μ(x) ≡ μ0 ,

violating (10.127). This establishes (10.128) and completes the proof of the theorem.

10.5 Notes and remarks

The concept of weak linking was begun in [19] and [78]. Further work in this direction

was carried out in [106]. The main thrust was to consider the case when G can be

written in the form

1

G(u) = (Lu, u) + b(u)

(10.136)

2

10.5. Notes and remarks 127

where L is a self-adjoint operator and b (u) is weakly continuous and E is a Hilbert

space. With this method, very few sets have been found to link. The results of [19],

[78], and [122] do not require E to be separable. However, they require G to be of

the form (10.136) with b (u) compact. In [78] the compactness of b (u) is replaced

by the assumption that it be weak-to-weak continuous, and b(u) itself is required to

be bounded from below and be weakly lower semicontinuous. A theorem is given

in [78] that requires only (10.5) but also requires G to be „ -upper semi-continuous

(the „ -topology is specially constructed to accommodate the splitting of E into sub-

spaces). In all of the results mentioned only two examples of linking sets A, B are

given. The presentation given here is from [117], [118], [140]. It has several advan-

tages. It does not require G to be of the form (10.136) [which indeed satis¬es (10.5)].

It does not require hypotheses on each of an exhausting sequence of ¬nite-dimensional

subspaces. Moreover, all sets A, B known to link when one of the subspaces M, N is

¬nite-dimensional will link now as well.

Chapter 11

Fuˇ ´k Spectrum: Resonance

c±

11.1 Introduction

Let be a bounded domain in Rn and let A ≥ »0 > 0 be a self-adjoint operator on

L 2 ( ) with compact resolvent and eigenvalues

0 < »0 < »1 < · · · < »k < · · · .

(11.1)

If f (x, t) is a Carath´ odory function on — R, then the semilinear problem

e

Au = f (x, u), u ∈ D(A)

(11.2)

is said to have asymptotic resonance at in¬nity if

f (x, t)/t ’ »k as |t| ’ ∞,

(11.3)

where »k is one of the eigenvalues of A. Interest in resonant problems began with

the pioneering work of Landesman and Lazer [79] and has continued until the present

because such problems are more dif¬cult to solve than nonresonant problems. The

reason is that for |u(x)| large, (11.2) approximates the eigenvalue problem

Au = »k u

(11.4)

with its inherent instabilities. Even if (11.3) does not occur, but

f (x, t)/t ’ a a.e. as t ’ ’∞

(11.5)

’ b a.e. as t ’ +∞,

one encounters the same dif¬culties if

Au = bu + ’ au ’ , u ± = max{±u, 0}

(11.6)

has a nontrivial solution. In fact, the dif¬culties are compounded because there is no

eigenspace with which to work in this case. We call the set of those (a, b) ∈ R2 for

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_11,

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

130 11. Fuˇ ´k Spectrum: Resonance

c±

which (11.6) has nontrivial solutions the Fuˇ´k spectrum of A. For each , it was shown

c±

in [111, 122] that in the square [» ’1 , » +1 ]2 there are decreasing curves C ,1 , C ,2

(which may coincide) passing through the point (» , » ) such that all points above or

below both curves in the square are not in . We denote the lower curve by C ,1 and the

upper curve by C ,2 . Further discussions concerning the Fuˇ´k spectrum can be found

c±

in Chapter 14.

In the present chapter we shall study resonance problems with respect to the Fuˇ´k

c±

spectrum. We shall allow (a, b) to be on either of the curves C , j in Q =

(» ’1 , » +1 )2 . We do not presently consider the case when (a, b) is a point between

the curves (when there are points in in that region). Such points will be discussed in

Chapter 14. We have

Theorem 11.1. Let f (x, t) be a Carath´ odory function such that

e

| f (x, t)| ¤ C(|t| + 1), x∈ , t ∈ R,

(11.7)

and (11.6) has a nontrivial solution with (a, b) ∈ C ,1 . Assume that

2

» ¤ 2F(x, t) + W1 (x), x∈ , t ∈ R,

(11.8) ’1 t

and

H (x, t) ¤ W0 (x), x∈ , t ∈ R,

(11.9)

where Wi (x) ∈ L 1 ( ),

t

f (x, s)ds,

(11.10) F(x, t) :=

0

and

H (x, t) := 2F(x, t) ’ t f (x, t).