0.7. p-LAPLACIAN 15

Proof of Theorem 0.33. Set

λk “ inf

APA

ipAqěk´1

sup

uPA

Ipuq, k ě 1.

Then pλk q 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), λk

´

ď λk ď λk

`,

in particular, λk Ñ 8.

Let λ P pλk , λk`1qzσp´Δpq. By Lemma 0.36, CkpIλ, 0q «

rk´1pIλq,

H 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

H

k´1pIλq

‰ 0 by Proposition 0.35.