xiv AN OVERVIEW

holds. In the p-superlinear case it is customary to also assume the following

Ambrosetti-Rabinowitz type condition to ensure that Φ satisfies the pPSq

condition:

(11) 0 ă μ F px, tq ď tf px, tq @x P Ω, |t| large

for some μ ą p.

In the semilinear case p “ 2, we can then obtain a nontrivial solution

of problem (7) using the well-known saddle point theorem of Rabinowitz as

follows. Fix a w0 P H `z t0u and let

X “ u “ v ` s w0 : v P H

´,

s ě 0, }u} ď R

(

,

A “ v P H

´

: }v} ď R

(

Y u P X : }u} “ R

(

,

B “ w P H

`

: }w} “ r

(

with H

˘

as in (4) and R ą r ą 0. Then

(12) max ΦpAq ď 0 ă inf ΦpBq

if R is suﬃciently large and r is suﬃciently small, and A homotopically links

B with respect to X in the sense that

γpXq X B ‰ H @γ P Γ

where

Γ “ γ P CpX, H0

1pΩqq

: γ|A “ id

A

(

.

So it follows that

c :“ inf

γPΓ

sup

uPγpX q

Φpuq

is a positive critical level of Φ (see Proposition 3.21).

Again we may ask whether linking sets that would enable us to use this

argument in the quasilinear case p ‰ 2 can be constructed. In Perera and

Szulkin [105] the following such construction based on the piercing property

of the index (see Proposition 2.12) was given. Recall that the cone CA0

on a topological space A0 is the quotient space of A0 ˆ r0, 1s obtained by

collapsing A0 ˆ t1u to a point. We identify A0 ˆ t0u with A0 itself. Fix an

h P

CpCΨλk

, Sq such that

hpCΨλk

q is closed and h|Ψλk “ id

Ψλk

, and let

X “ tu : u P

hpCΨλk

q, 0 ď t ď R

(

,

A “ tu : u P

Ψλk

, 0 ď t ď R

(

Y u P X : }u} “ R

(

,

B “ ru : u P Ψλk`1

(

with R ą r ą 0. Then (12) still holds if R is suﬃciently large and r is

suﬃciently small, and A homotopically links B with respect to X by (5)

and the following theorem proved in Section 3.6, so Φ again has a positive

critical level.