0.7. p-LAPLACIAN 15 Proof of Theorem 0.33. Set λk “ inf APA ipAqěk´1 sup uPA Ipuq, k ě 1. Then pλkq is an increasing sequence of critical points of I, and hence eigen- values of ´Δp, by a standard deformation argument (see [98, Proposition 3.1]). By (0.3), λ´ k ď λk ď λ`, k in particular, λk Ñ 8. Let λ P pλk, λk`1qzσp´Δpq. By Lemma 0.36, CkpIλ, 0q « r k´1 pIλq, and Iλ P A since I is even. Since λ ą λk, there is an A P A with ipAq ě k ´ 1 such that I ď λ on A. Then A Ă Iλ and hence ipIλq ě ipAq ě k ´ 1 by Proposition 0.34. On the other hand, ipIλq ď k ´ 1 since I ď λ ă λk`1 on Iλ. So ipIλq “ k ´ 1 and hence r k´1 pIλq ‰ 0 by Proposition 0.35.
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