xii AN OVERVIEW
(see Proposition 2.14) and hence
(6)
CkpΦλ,
0q 0
by (2).
The structure provided by this new sequence of eigenvalues is sufficient
to adapt many of the standard variational methods for solving semilinear
problems to the quasilinear case. In particular, we will construct new linking
sets and local linkings that are readily applicable to quasilinear problems.
Of course, such constructions cannot be based on linear subspaces since we
no longer have eigenspaces to work with. They will instead use nonlinear
splittings generated by the sub- and superlevel sets of Ψ that appear in (5),
and the indices given there will play a key role in these new topological
constructions as we will see next.
Consider the boundary value problem
(7)
$
&
%
´Δp u f px, uq in Ω
u 0 on
where the nonlinearity f is a Carath´ eodory function on Ω ˆ R satisfying the
subcritical growth condition
|f px, tq| ď C
`
|t|r´1
` 1
˘
@px, tq P Ω ˆ R
for some r P p1,
p˚q.
Here


$
&
%
np
n ´ p
, p ă n
8, p ě n
is the critical exponent for the Sobolev imbedding W0
1, ppΩq
ã Ñ
LrpΩq.
Weak solutions of this problem coincide with the critical points of the
C1-
functional
Φpuq
ż
Ω
|∇u|p
´ p F px, uq, u P W0
1, ppΩq
where
F px, tq
ż
t
0
f px, sq ds
is the primitive of f.
It is customary to roughly classify problem (7) according to the growth
of f as
piq p-sublinear if
lim
tÑ˘8
f px, tq
|t|p´2
t
0 @x P Ω,
piiq asymptotically p-linear if
0 ă lim inf
tÑ˘8
f px, tq
|t|p´2 t
ď lim sup
tÑ˘8
f px, tq
|t|p´2 t
ă 8 @x P Ω,
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