An Overview
Let Φ be a
C1-functional
defined on a real Banach space W and satisfying
the pPSq condition. In Morse theory the local behavior of Φ near an isolated
critical point u is described by the sequence of critical groups
(1)
CqpΦ,uq
“
HqpΦc
X U,
Φc
X U z tuuq, q ě 0
where c “ Φpuq is the corresponding critical value,
Φc
is the sublevel set
tu P W : Φpuq ď cu, U is a neighborhood of u containing no other critical
points, and H denotes cohomology. They are independent of U by the
excision property. When the critical values are bounded from below, the
global behavior of Φ can be described by the critical groups at infinity
CqpΦ,
8q “
HqpW, Φaq,
q ě 0
where a is less than all critical values. They are independent of a by the sec-
ond deformation lemma and the homotopy invariance of cohomology groups.
When Φ has only a finite number of critical points u1,...,uk, their
critical groups are related to those at infinity by
ÿk
i“1
rank
CqpΦ,uiq
ě rank
CqpΦ,
8q @q
(see Proposition 3.16). Thus, if
CqpΦ,
8q ‰ 0, then Φ has a critical point u
with
CqpΦ,uq
‰ 0. If zero is the only critical point of Φ and Φp0q “ 0, then
taking U “ W in (1), and noting that
Φ0
is a deformation retract of W and
Φ0z
t0u deformation retracts to
Φa
by the second deformation lemma, gives
CqpΦ,
0q “
HqpΦ0, Φ0z
t0uq «
HqpW, Φ0z
t0uq
«
HqpW, Φaq
“
CqpΦ,
8q @q.
Thus, if
CqpΦ,
0q - «
CqpΦ,
8q for some q, then Φ has a critical point u ‰ 0.
Such ideas have been used extensively in the literature to obtain multiple
nontrivial solutions of semilinear elliptic boundary value problems (see, e.g.,
Mawhin and Willem [81], Chang [28], Bartsch and Li [14], and their refer-
ences).
Now consider the eigenvalue problem
$
&
%
´Δp u “ λ
|u|p´2
u in Ω
u “ 0 on BΩ
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