0.7. p-LAPLACIAN 11

The following alternative obtained in Perera [95] gives a nontrivial crit-

ical point with a nontrivial critical group produced by a local linking.

Theorem 0.31. If Φ has a local linking near the origin with dim W1 “ k,

HkpΦb, Φaq

“ 0 where ´8 ă a ă 0 ă b ď `8 are regular values, and Φ

has only a finite number of critical points in Φa

b

and satisfies pCqc for all

c P ra, bs X R, then Φ has a critical point u1 ‰ 0 with either

a ă Φpu1q ă 0, Ck´1pΦ,u1q ‰ 0

or

0 ă Φpu1q ă b, Ck`1pΦ,u1q ‰ 0.

When Φ is bounded from below, taking a ă inf ΦpW q and b “ `8

gives the following three critical points theorem; see also Krasnosel’skii [60],

Chang [27], Li and Liu [72], and Liu [71].

Corollary 0.32. If Φ has a local linking near the origin with dim W1 “ k ě

2, is bounded from below, has only a finite number of critical points, and

satisfies pCq, then Φ has a global minimizer u0 ‰ 0 with

Φpu0q ă 0, CqpΦ,u0q “ δq0 G

and a critical point u1 ‰ 0,u0 with either

Φpu1q ă 0, Ck´1pΦ,u1q ‰ 0

or

Φpu1q ą 0, Ck`1pΦ,u1q ‰ 0.

Theorems 0.30 and 0.31 and Corollary 0.32 also hold under the more

general notion of homological local linking introduced in Perera [96].

0.7. p-Laplacian

The p-Laplacian operator

Δp u “ div

`

|∇u|p´2

∇u

˘

, p P p1, 8q

arises in non-Newtonian fluid flows, turbulent filtration in porous media,

plasticity theory, rheology, glacelogy, and in many other application areas;

see, e.g., Esteban and V´ azquez [48] and Padial, Tak´c, aˇ and Tello [90]. Prob-

lems involving the p-Laplacian have been studied extensively in the literature

during the last fifty years. In this section we present a result on nontrivial

critical groups associated with the p-Laplacian obtained in Perera [98]; see

also Dancer and Perera [40].

Consider the nonlinear eigenvalue problem

(0.2)

$

&

%

´Δp u “ λ

|u|p´2

u in Ω

u “ 0 on BΩ