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 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 (0.2) $ & % ´Δp u λ |u|p´2 u in Ω u 0 on
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