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 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].
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