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
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