xvi AN OVERVIEW
set
Gpx, tq
ż
t
0
gpx, sq ds,
and assume that either
λ λk, Gpx, tq ě 0 @x P Ω, |t| small,
or
λ λk`1, Gpx, tq ď 0 @x P Ω, |t| small.
In the semilinear case p 2, let
A v P H
´
: }v} ď r
(
, B w P H
`
: }w} ď r
(
with H
˘
as in (4) and r ą 0. Then
(14) Φ|A ď 0 ă Φ|Bzt0u
if r is sufficiently small (see Li and Willem [67]), so Φ has a local linking
near zero in dimension k and hence (13) holds (see Liu [71]).
So we may ask whether the notion of a local linking can be generalized to
apply in the quasilinear case p 2 as well. We will again give an affirmative
answer. Let
A tu : u P
Ψλk
, 0 ď t ď r
(
, B tu : u P Ψλk`1 , 0 ď t ď r
(
with r ą 0. Then (14) still holds if r is sufficiently small (see Degiovanni,
Lancelotti, and Perera [42]), so Φ has a cohomological local splitting near
zero in dimension k in the sense of the following definition given in Section
3.11. Hence (13) holds again (see Proposition 3.34).
Definition 3. We say that a
C1-functional
Φ defined on a Banach space
W has a cohomological local splitting near zero in dimension k if there is an
r ą 0 such that zero is the only critical point of Φ in
U u P W : }u} ď r
(
and there are disjoint nonempty closed symmetric subsets A0 and B0 of BU
such that
ipA0q ipSzB0q k
and
Φ|A ď 0 ă Φ|Bzt0u
where
A tu : u P A0, 0 ď t ď 1
(
, B tu : u P B0, 0 ď t ď 1
(
.
These constructions, which were based on the existence of a sequence of
eigenvalues satisfying (5), can be extended to situations involving indefinite
eigenvalue problems such as
$
&
%
´Δp u λ V pxq
|u|p´2
u in Ω
u 0 on
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