12 0. MORSE THEORY AND VARIATIONAL PROBLEMS where Ω is a bounded domain in Rn, n ě 1 and Δp u div ` |∇u|p´2 ∇u ˘ is the p-Laplacian of u, p P p1, 8q. It is known that the first eigenvalue λ1 is positive, simple, has an associated eigenfunction that is positive in Ω, and is isolated in the spectrum σp´Δpq see Anane [9] and Lindqvist [68, 69]. So the second eigenvalue λ2 inf σp´Δpq X pλ1, 8q is also defined see Anane and Tsouli [8]. 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 associated with λk has exactly k ´1 interior zeroes see Cuesta [35] or Dr´ abek [46]. In the semilinear PDE case n ě 2, p 2 also σp´Δq consists of a sequence of eigenvalues λk Õ 8, but in the quasilinear PDE case n ě 2, p 2 a complete description of the spectrum is not available. Eigenvalues of (0.2) are the critical values of the C1-functional Ipuq ż Ω |∇u|p, u P S u P W W 1, p 0 pΩq : }u} LppΩq 1 ( , which satisfies pPSq. Denote by A the class of closed symmetric subsets of S and by γ`pAq sup k ě 1 : D an odd continuous map Sk´1 Ñ A ( , γ´pAq inf k ě 1 : D an odd continuous map A Ñ Sk´1 ( the genus and the cogenus of A P A, respectively, where Sk´1 is the unit sphere in Rk. Then λ˘ k inf APA γ ˘ pAqěk sup uPA Ipuq, k ě 1 are two increasing and unbounded sequences of eigenvalues, but, in general, it is not known whether either sequence is a complete list. The sequence λ` k ˘ was introduced by Dr´ abek and Robinson [47] γ´ k is also called the Krasnosel’skii genus [60]. Solutions of (0.2) are the critical points of the functional Iλpuq ż Ω |∇u|p ´ λ |u|p, u P W 1, p 0 pΩq. When λ R σp´Δpq, the origin is the only critical point of and hence the critical groups CqpIλ, 0q are defined. Again we take the coefficient group to be Z2. The following theorem is our main result on them. Theorem 0.33 ([98, Proposition 1.1]). The spectrum of ´Δp contains a sequence of eigenvalues λk Õ 8 such that λ´ k ď λk ď λ` k and λ P pλk, λk`1qzσp´Δpq ùñ CkpIλ, 0q 0. Various applications of this sequence of eigenvalues can be found in Per- era [99, 100], Liu and Li [75], Perera and Szulkin [105], Cingolani and Degiovanni [30], Guo and Liu [55], Degiovanni and Lancelotti [43, 44], Tanaka [129], Fang and Liu [50], Medeiros and Perera [82], Motreanu and Perera [86], and Degiovanni, Lancelotti, and Perera [42].
Previous Page Next Page