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 BΩ

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

p˚

“

$

&

%

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,