0.7. p-LAPLACIAN 11
The following alternative obtained in Perera  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,
“ 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
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
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 ,
Chang , Li and Liu , and Liu .
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
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 .
The p-Laplacian operator
Δp u “ div
, 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  and Padial, Tak´c, aˇ and Tello . 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 ; see
also Dancer and Perera .
Consider the nonlinear eigenvalue problem
´Δp u “ λ
u in Ω
u “ 0 on BΩ