x AN OVERVIEW
where Ω is a bounded domain in
Rn,
n ě 1,
Δp u div
`
|∇u|p´2
∇u
˘
is the p-Laplacian of u, and p P p1, 8q. The eigenfunctions coincide with the
critical points of the
C1-functional
Φλpuq
ż
Ω
|∇u|p
´ λ
|u|p
defined on the Sobolev space W0
1, ppΩq
with the usual norm
}u}
ˆż
Ω
|∇u|p
˙
1
p
.
When λ is not an eigenvalue, zero is the only critical point of Φλ and we
may take
U u P W0
1, p
pΩq : }u} ď 1
(
in the definition (1). Since Φλ is positive homogeneous, Φλ
0
X U radially
contracts to the origin and Φλ
0
X U z t0u radially deformation retracts onto
Φλ
0
X S
Ψλ
where S is the unit sphere in W0
1, ppΩq
and
Ψpuq
1
ż
Ω
|u|p
, u P S.
It follows that
(2)
CqpΦλ,
0q «
$
&δq0
%
G,
Ψλ
H
r
H q´1pΨλq, Ψλ H
where δ is the Kronecker delta, G is the coefficient group, and
r
H denotes
reduced cohomology. Note also that the eigenvalues coincide with the critical
values of Ψ by the Lagrange multiplier rule.
In the semilinear case p 2, the spectrum σp´Δq consists of isolated
eigenvalues λk, repeated according to their multiplicities, satisfying
0 ă λ1 ă λ2 ď ¨ ¨ ¨ Ñ 8.
If λ ă λ1 inf Ψ, then
Ψλ
H and hence
(3)
CqpΦλ,
0q « δq0 G
by (2). If λk ă λ ă λk`1, then we have the orthogonal decomposition
(4) H0
1pΩq
H
´
H
`,
u v ` w
where H
´
is the direct sum of the eigenspaces corresponding to λ1,...,λk
and H
`
is its orthogonal complement, and
dim H
´
k
Previous Page Next Page