12 0. MORSE THEORY AND VARIATIONAL PROBLEMS

where Ω is a bounded domain in

Rn,

n ě 1 and Δp u “ div

`

|∇u|p´2

∇u

˘

is

the p-Laplacian of u, p P p1, 8q. It is known that the first eigenvalue λ1 is

positive, simple, has an associated eigenfunction that is positive in Ω, and

is isolated in the spectrum σp´Δpq; see Anane [9] and Lindqvist [68, 69].

So the second eigenvalue λ2 “ inf σp´Δpq X pλ1, 8q is also defined; see

Anane and Tsouli [8]. In the ODE case n “ 1, where Ω is an interval,

the spectrum consists of a sequence of simple eigenvalues λk Õ 8 and the

eigenfunction associated with λk has exactly k ´1 interior zeroes; see Cuesta

[35] or Dr´ abek [46]. In the semilinear PDE case n ě 2, p “ 2 also σp´Δq

consists of a sequence of eigenvalues λk Õ 8, but in the quasilinear PDE

case n ě 2, p ‰ 2 a complete description of the spectrum is not available.

Eigenvalues of (0.2) are the critical values of the

C1-functional

Ipuq “

ż

Ω

|∇u|p,

u P S “ u P W “ W0

1, ppΩq

: }u}LppΩq “ 1

(

,

which satisfies pPSq. Denote by A the class of closed symmetric subsets of

S and by

γ`pAq

“ sup k ě 1 : D an odd continuous map

Sk´1

Ñ A

(

,

γ´pAq

“ inf k ě 1 : D an odd continuous map A Ñ

Sk´1

(

the genus and the cogenus of A P A, respectively, where

Sk´1

is the unit

sphere in

Rk.

Then

λk

˘

“ inf

APA

γ

˘

pAqěk

sup

uPA

Ipuq, k ě 1

are two increasing and unbounded sequences of eigenvalues, but, in general,

it is known whether either sequence is a complete list. The sequence

`

λk

`

˘not

was introduced by Dr´ abek and Robinson [47]; γk

´

is also called the

Krasnosel’skii genus [60].

Solutions of (0.2) are the critical points of the functional

Iλpuq “

ż

Ω

|∇u|p

´ λ

|u|p,

u P W0

1, p

pΩq.

When λ R σp´Δpq, the origin is the only critical point of Iλ and hence the

critical groups CqpIλ, 0q are defined. Again we take the coeﬃcient group to

be Z2. The following theorem is our main result on them.

Theorem 0.33 ([98, Proposition 1.1]). The spectrum of ´Δp contains a

sequence of eigenvalues λk Õ 8 such that λk

´

ď λk ď λk

`

and

λ P pλk , λk`1qzσp´Δpq ùñ CkpIλ, 0q ‰ 0.

Various applications of this sequence of eigenvalues can be found in Per-

era [99, 100], Liu and Li [75], Perera and Szulkin [105], Cingolani and

Degiovanni [30], Guo and Liu [55], Degiovanni and Lancelotti [43, 44],

Tanaka [129], Fang and Liu [50], Medeiros and Perera [82], Motreanu and

Perera [86], and Degiovanni, Lancelotti, and Perera [42].