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

ix