AN OVERVIEW xiii
piiiq p-superlinear if
lim
tÑ˘8
f px, tq
|t|p´2 t
8 @x P Ω.
Consider the asymptotically p-linear case where
lim
tÑ˘8
f px, tq
|t|p´2 t
λ, uniformly in x P Ω
with λk ă λ ă λk`1, and assume λ R σp´Δpq to ensure that Φ satisfies the
pPSq condition.
In the semilinear case p 2, let
A v P H
´
: }v} R
(
, B H
`
with H
˘
as in (4) and R ą 0. Then
(8) max ΦpAq ă inf ΦpBq
if R is sufficiently large, and A cohomologically links B in dimension k ´ 1
in the sense that the homomorphism
r
H
k´1pH0 1pΩqzBq
Ñ
r
H
k´1pAq
induced by the inclusion is nontrivial. So it follows that problem (7) has a
solution u with
CkpΦ,uq
0 (see Proposition 3.25).
We may ask whether this well-known argument can be modified to obtain
the same result in the quasilinear case p 2 where we no longer have the
splitting given in (4). We will give an affirmative answer as follows. Let
A Ru : u P
Ψλk
(
, B tu : u P Ψλk`1 , t ě 0
(
with R ą 0. Then (8) still holds if R is sufficiently large, and A cohomolog-
ically links B in dimension k ´ 1 by (5) and the following theorem proved
in Section 3.7, so problem (7) again has a solution u with
CkpΦ,uq
0.
Theorem 1. Let A0 and B0 be disjoint nonempty closed symmetric
subsets of the unit sphere S in a Banach space such that
ipA0q ipSzB0q k
where i denotes the cohomological index, and let
A Ru : u P A0
(
, B tu : u P B0, t ě 0
(
with R ą 0. Then A cohomologically links B in dimension k ´ 1.
Now suppose f px, 0q 0, so that problem (7) has the trivial solution
upxq 0. Assume that
(9) lim
tÑ0
f px, tq
|t|p´2 t
λ, uniformly in x P Ω,
λk ă λ ă λk`1, and the sign condition
(10) p F px, tq ě λk`1
|t|p
@px, tq P Ω ˆ R
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