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
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 [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
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
, 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 azquez [48] and Padial, Tak´c, 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
´Δp u λ
u in Ω
u 0 on
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