AN OVERVIEW xv

Theorem 2. Let A0 and B0 be disjoint nonempty closed symmetric

subsets of the unit sphere S in a Banach space such that

ipA0q “ ipSzB0q ă 8,

h P CpCA0,Sq be such that hpCA0q is closed and h|A0 “ id

A0

, and let

X “ tu : u P hpCA0q, 0 ď t ď R

(

,

A “ tu : u P A0, 0 ď t ď R

(

Y u P X : }u} “ R

(

,

B “ ru : u P B0

(

with R ą r ą 0. Then A homotopically links B with respect to X.

The sign condition (10) can be removed by using a comparison of the

critical groups of Φ at zero and infinity instead of the above linking ar-

gument. First consider the nonresonant case where (9) holds with λ P

pλk , λk`1qzσp´Δpq. Then

CqpΦ,

0q «

CqpΦλ,

0q @q

by the homotopy invariance of the critical groups and hence

(13)

CkpΦ,

0q ‰ 0

by (6). On the other hand, a simple modification of an argument due to

Wang [132] shows that

CqpΦ,

8q “ 0 @q

when (11) holds (see Example 5.14). So Φ has a nontrivial critical point by

the remarks at the beginning of the chapter.

In the p-sublinear case, where Φ is bounded from below, we can use

(13) to obtain multiple nontrivial solutions of problem (7). Indeed, the

three critical points theorem (see Corollary 3.32) gives two nontrivial critical

points of Φ when k ě 2.

Note that we do not assume that there are no other eigenvalues in the

interval pλk , λk`1q, in particular, λ may be an eigenvalue. Our results hold

as long as λk ă λ ă λk`1, even if the entire interval rλk , λk`1s is contained in

the spectrum. Thus, eigenvalues that do not belong to the sequence pλk q are

not that important in this context. In fact, we will see that the cohomological

index of sublevel sets changes only when crossing an eigenvalue from this

particular sequence.

Now we consider the resonant case where (9) holds with λ P rλk , λk`1s X

σp´Δpq, and ask whether we still have (13). We will show that this is indeed

the case when a suitable sign condition holds near t “ 0. Write f as

f px, tq “ λ

|t|p´2

t ` gpx, tq,

so that

lim

tÑ0

gpx, tq

|t|p´2 t

“ 0, uniformly in x P Ω,