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.
