AN OVERVIEW xi is called the Morse index of zero. It is easy to check that ηpu, tq v ` p1 ´ tq w }v ` p1 ´ tq w} , pu, tq P Ψλ ˆ r0, 1s is a deformation retraction of Ψλ onto H ´ X S, so CqpΦ λ , 0q « r q´1 pH ´ X Sq « δqk G by (2). The quasilinear case p 2 is far more complicated. Very little is known about the spectrum σp´Δpq itself. The first eigenvalue λ1 is positive, simple, and has an associated eigenfunction ϕ1 that is positive in Ω (see Anane [9] and Lindqvist [68, 69]). Moreover, λ1 is isolated in the spectrum, so the second eigenvalue λ2 inf σp´ΔpqXpλ1, 8q is also well-defined. In the ODE case n 1, where Ω is an interval, the spectrum consists of a sequence of simple eigenvalues λk Õ 8, and the eigenfunction ϕk associated with λk has exactly k ´ 1 interior zeroes (see, e.g., Dr´ abek [46]). In the PDE case n ě 2, an increasing and unbounded sequence of eigenvalues can be constructed using a standard minimax scheme involving the Krasnoselskii’s genus, but it is not known whether this gives a complete list of the eigenvalues. If λ ă λ1, then (3) holds as before. It was shown in Dancer and Perera [40] that CqpΦλ, 0q « δq1 G if λ1 ă λ ă λ2 and that CqpΦλ, 0q 0, q 0, 1 if λ ą λ2. Thus, the question arises as to whether there is a nontrivial critical group when λ ą λ2. An affirmative answer was given in Perera [98] where a new sequence of eigenvalues was constructed using a minimax scheme involving the Z2-cohomological index of Fadell and Rabinowitz [49] as follows. Let F denote the class of symmetric subsets of S, let ipM q denote the cohomological index of M P F, and set λk :“ inf M PF ipM qěk sup uPM Ψpuq. Then λk Õ 8 is a sequence of eigenvalues, and if λk ă λk`1, then (5) ipΨλk q ipSzΨλ k`1 q k where Ψλk u P S : Ψpuq ď λk ( , Ψλ k`1 u P S : Ψpuq ě λk`1 ( (see Theorem 4.6). Thus, if λk ă λ ă λk`1, then ipΨλq k by the monotonicity of the index, which implies that r k´1 pΨλq 0
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