AN OVERVIEW xi

is called the Morse index of zero. It is easy to check that

ηpu, tq “

v ` p1 ´ tq w

}v ` p1 ´ tq w}

, pu, tq P

Ψλ

ˆ r0, 1s

is a deformation retraction of

Ψλ

onto H

´

X S, so

CqpΦλ,

0q «

r

H

q´1pH ´

X Sq « δqk G

by (2).

The quasilinear case p ‰ 2 is far more complicated. Very little is known

about the spectrum σp´Δpq itself. The first eigenvalue λ1 is positive, simple,

and has an associated eigenfunction ϕ1 that is positive in Ω (see Anane [9]

and Lindqvist [68, 69]). Moreover, λ1 is isolated in the spectrum, so the

second eigenvalue λ2 “ inf σp´ΔpqXpλ1, 8q is also well-defined. In the ODE

case n “ 1, where Ω is an interval, the spectrum consists of a sequence of

simple eigenvalues λk Õ 8, and the eigenfunction ϕk associated with λk has

exactly k ´ 1 interior zeroes (see, e.g., Dr´ abek [46]). In the PDE case n ě 2,

an increasing and unbounded sequence of eigenvalues can be constructed

using a standard minimax scheme involving the Krasnoselskii’s genus, but

it is not known whether this gives a complete list of the eigenvalues.

If λ ă λ1, then (3) holds as before. It was shown in Dancer and Perera

[40] that

CqpΦλ,

0q « δq1 G

if λ1 ă λ ă λ2 and that

CqpΦλ,

0q “ 0, q “ 0, 1

if λ ą λ2. Thus, the question arises as to whether there is a nontrivial

critical group when λ ą λ2. An aﬃrmative answer was given in Perera

[98] where a new sequence of eigenvalues was constructed using a minimax

scheme involving the Z2-cohomological index of Fadell and Rabinowitz [49]

as follows.

Let F denote the class of symmetric subsets of S, let ipM q denote the

cohomological index of M P F, and set

λk :“ inf

M PF

ipM qěk

sup

uPM

Ψpuq.

Then λk Õ 8 is a sequence of eigenvalues, and if λk ă λk`1, then

(5)

ipΨλk

q “ ipSzΨλk`1 q “ k

where

Ψλk

“ u P S : Ψpuq ď λk

(

, Ψλk`1 “ u P S : Ψpuq ě λk`1

(

(see Theorem 4.6). Thus, if λk ă λ ă λk`1, then

ipΨλq

“ k

by the monotonicity of the index, which implies that

r

H

k´1pΨλq

‰ 0