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