x AN OVERVIEW where Ω is a bounded domain in Rn, n ě 1, Δp u div ` |∇u|p´2 ∇u ˘ is the p-Laplacian of u, and p P p1, 8q. The eigenfunctions coincide with the critical points of the C1-functional Φλpuq ż Ω |∇u|p ´ λ |u|p defined on the Sobolev space W 1, p 0 pΩq with the usual norm }u} ˆż Ω |∇u|p ˙ 1 p . When λ is not an eigenvalue, zero is the only critical point of Φλ and we may take U u P W 1, p 0 pΩq : }u} ď 1 ( in the definition (1). Since Φλ is positive homogeneous, Φ0 λ X U radially contracts to the origin and Φ0 λ X U z t0u radially deformation retracts onto Φ0 λ X S Ψλ where S is the unit sphere in W 1, p 0 pΩq and Ψpuq 1 ż Ω |u|p , u P S. It follows that (2) CqpΦλ, 0q « $ &δq0 % G, Ψλ H r q´1pΨλq, Ψλ H where δ is the Kronecker delta, G is the coefficient group, and r denotes reduced cohomology. Note also that the eigenvalues coincide with the critical values of Ψ by the Lagrange multiplier rule. In the semilinear case p 2, the spectrum σp´Δq consists of isolated eigenvalues λk, repeated according to their multiplicities, satisfying 0 ă λ1 ă λ2 ď ¨ ¨ ¨ Ñ 8. If λ ă λ1 inf Ψ, then Ψλ H and hence (3) CqpΦ λ , 0q « δq0 G by (2). If λk ă λ ă λk`1, then we have the orthogonal decomposition (4) H1pΩq 0 H ´ H ` , u v ` w where H ´ is the direct sum of the eigenspaces corresponding to λ1,...,λk and H ` is its orthogonal complement, and dim H ´ k
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