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 BΩ