xvi AN OVERVIEW set Gpx, tq ż t 0 gpx, sq ds, and assume that either λ λk, Gpx, tq ě 0 @x P Ω, |t| small, or λ λk`1, Gpx, tq ď 0 @x P Ω, |t| small. In the semilinear case p 2, let A v P H ´ : }v} ď r ( , B w P H ` : }w} ď r ( with H ˘ as in (4) and r ą 0. Then (14) Φ| A ď 0 ă Φ| Bzt0u if r is sufficiently small (see Li and Willem [67]), so Φ has a local linking near zero in dimension k and hence (13) holds (see Liu [71]). So we may ask whether the notion of a local linking can be generalized to apply in the quasilinear case p 2 as well. We will again give an affirmative answer. Let A tu : u P Ψλk, 0 ď t ď r ( , B tu : u P Ψλ k`1 , 0 ď t ď r ( with r ą 0. Then (14) still holds if r is sufficiently small (see Degiovanni, Lancelotti, and Perera [42]), so Φ has a cohomological local splitting near zero in dimension k in the sense of the following definition given in Section 3.11. Hence (13) holds again (see Proposition 3.34). Definition 3. We say that a C1-functional Φ defined on a Banach space W has a cohomological local splitting near zero in dimension k if there is an r ą 0 such that zero is the only critical point of Φ in U u P W : }u} ď r ( and there are disjoint nonempty closed symmetric subsets A0 and B0 of BU such that ipA0q ipSzB0q k and Φ| A ď 0 ă Φ| Bzt0u where A tu : u P A0, 0 ď t ď 1 ( , B tu : u P B0, 0 ď t ď 1 ( . These constructions, which were based on the existence of a sequence of eigenvalues satisfying (5), can be extended to situations involving indefinite eigenvalue problems such as $ & % ´Δp u λ V pxq |u|p´2 u in Ω u 0 on
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