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
, 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
0q «
0q @q
by the homotopy invariance of the critical groups and hence
0q 0
by (6). On the other hand, a simple modification of an argument due to
Wang [132] shows that
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 ` gpx, tq,
so that
gpx, tq
|t|p´2 t
0, uniformly in x P Ω,
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