An Overview Let Φ be a C1-functional defined on a real Banach space W and satisfying the pPSq condition. In Morse theory the local behavior of Φ near an isolated critical point u is described by the sequence of critical groups (1) CqpΦ,uq HqpΦc X U, Φc X U z tuuq, q ě 0 where c Φpuq is the corresponding critical value, Φc is the sublevel set tu P W : Φpuq ď cu, U is a neighborhood of u containing no other critical points, and H denotes cohomology. They are independent of U by the excision property. When the critical values are bounded from below, the global behavior of Φ can be described by the critical groups at infinity CqpΦ, 8q HqpW, Φaq, q ě 0 where a is less than all critical values. They are independent of a by the sec- ond deformation lemma and the homotopy invariance of cohomology groups. When Φ has only a finite number of critical points u1,...,uk, their critical groups are related to those at infinity by ÿk i“1 rank CqpΦ,uiq ě rank CqpΦ, 8q @q (see Proposition 3.16). Thus, if CqpΦ, 8q 0, then Φ has a critical point u with CqpΦ,uq 0. If zero is the only critical point of Φ and Φp0q 0, then taking U W in (1), and noting that Φ0 is a deformation retract of W and Φ0z t0u deformation retracts to Φa by the second deformation lemma, gives CqpΦ, 0q HqpΦ0, Φ0z t0uq « HqpW, Φ0z t0uq « HqpW, Φaq CqpΦ, 8q @q. Thus, if CqpΦ, 0q - « CqpΦ, 8q for some q, then Φ has a critical point u 0. Such ideas have been used extensively in the literature to obtain multiple nontrivial solutions of semilinear elliptic boundary value problems (see, e.g., Mawhin and Willem [81], Chang [28], Bartsch and Li [14], and their refer- ences). Now consider the eigenvalue problem $ & % ´Δp u λ |u|p´2 u in Ω u 0 on ix
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