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.
