Universidade Federal do Espírito Santo Centro de Ciências Exatas Departamento de Matemática Dissertação de Mestrado em Matemática

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

▼✉❧t✐♣❧✐❝✐❞❛❞❡ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s

♣♦s✐t✐✈❛s ♣❛r❛ ✉♠❛ ❡q✉❛çã♦ ❡❧í♣t✐❝❛ q✉❛s✐❧✐♥❡❛r

❏♦sé ❈❛r❧♦s ❞❡ ❖❧✐✈❡✐r❛ ❏✉♥✐♦r

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛_{✳ ❉r}❛_{✳ ▼❛❣❞❛ ❙♦❛r❡s ❳❛✈✐❡r}

❆❣r❛❞❡❝✐♠❡♥t♦s

✲ ❉❡ ✉♠❛ ❢♦r♠❛ ♠✉✐t♦ ❡s♣❡❝✐❛❧✱ à ❉❡✉s✱ q✉❡ t❡♠ s✐❞♦ ✉♠ ót✐♠♦ ♣❛✐ ❡ t❡♠ ♠❡ ❝♦♥❝❡❞✐❞♦ ❣r❛♥❞❡s ♦♣♦rt✉♥✐❞❛❞❡s✳ ❙❡ ❝❤❡❣✉❡✐ ❛té ❛q✉✐✱ ✐ss♦ ❞❡✈♦ à ❊❧❡✳

✲ ❆♦ ♠❡✉ ♣❛✐ ✭♠❛✐s ❞♦ q✉❡ ❡♠ ♠❡♠ór✐❛✮✱ ❏♦sé ❈❛r❧♦s✱ ❡ à ♠✐♥❤❛ ♠ã❡✱ ▼❛r✐❛ ❞❡ ▲♦✉r❞❡s✱ ♣❡❧❛ ❡❞✉❝❛çã♦✱ ♣❡❧♦ ❢♦rt❡ ✐♥❝❡♥t✐✈♦ ❡ ♣♦r t♦❞♦s ♣r❡❝✐♦s♦s ❝♦♥s❡❧❤♦s✳ ➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣❡❧♦ ❛❝♦❧❤✐♠❡♥t♦ ❞✉r❛♥t❡ ❛ ❣r❛❞✉❛çã♦ ❡ ♦ ♠❡str❛❞♦ ❡ ♣♦r s❡ ♣r❡♦❝✉♣❛r❡♠ t❛♥t♦ ❝♦♠✐❣♦ ♥❡ss❡ ♣❡rí♦❞♦✳ ❉❡st❛❝♦ ♠✐♥❤❛ ❛✈ó✱ ◆❛s❝✐r❡♠❛✱ ❝✉❥♦ ❝❛r✐♥❤♦ ❡ ❝✉✐❞❛❞♦ ❝♦♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♠❡ ❞❡r❛♠ ❢♦rç❛s ♣r❛ ❝♦♥t✐♥✉❛r✳

✲ ➚ t♦❞♦s ♦s ❛♠✐❣♦s ❞❛ ✐❣r❡❥❛✱ ❞❡♥tr❡ ♦s q✉❛✐s ❞❡st❛❝♦ ▼❛✉rí❧✐♦ ❡ ●❛❜r✐❡❧❛✱ ♣♦r ♠❡ ♣r♦✲ ♣♦r❝✐♦♥❛r❡♠ ❞✐❛s ✐♥❞❡s❝r✐tí✈❡✐s q✉❡ s❡♠♣r❡ ❣✉❛r❞❛r❡✐ ❝♦♠✐❣♦✳ ❆♦s ❛♠✐❣♦s ❞♦ ♣♦❦❡r✱ ♣❡❧♦s ♠♦♠❡♥t♦s ú♥✐❝♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✳ ➚s ❛♠✐❣❛s ▼✐❝❤❡❧❧❡ ❡ ❏❛q✉❡❧✐♥❡✱ ♣♦r t❡r❡♠ t♦r♥❛❞♦ ❛ ❣r❛❞✉❛çã♦ ✉♠ ♣❡rí♦❞♦ ♠❛✐s ❛❣r❛❞á✈❡❧ ❡ ❞✐✈❡rt✐❞♦✳

✲ ❆♦s ♣r♦❢❡ss♦r❡s ▼❛r❝❡❧♦ ❋❡r♥❛♥❞❡s ❋✉rt❛❞♦ ❡ ❏♦ã♦ P❛❜❧♦ P✐♥❤❡✐r♦ ❞❛ ❙✐❧✈❛✱ ♣♦r t❡✲ r❡♠ ❛❝❡✐t♦ ♣❛rt✐❝✐♣❛r ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❞❡st❡ tr❛❜❛❧❤♦✳

✲ ➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ▼❛❣❞❛ ❙♦❛r❡s ❳❛✈✐❡r✱ ♣♦r s✉❛ ❡♥♦r♠❡ ♣❛❝✐ê♥❝✐❛ ❡♠ r❡s♣♦♥❞❡r ♠✐♥❤❛s ✐♥ú♠❡r❛s ♣❡r❣✉♥t❛s✱ ♣♦r s✉❛ ❞❡❞✐❝❛çã♦ ❛ ❡st❡ tr❛❜❛❧❤♦ ❡ ♣❡❧❛ ❡①♣❡r✐ê♥❝✐❛ q✉❡ ♣✉❞❡ ❛❞q✉✐r✐r ❝♦♠ s❡✉s ❡♥s✐♥❛♠❡♥t♦s✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ♣♦✲ s✐t✐✈❛s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❡♠ RN ❡♥✈♦❧✈❡♥❞♦ ♦ ♦♣❡r❛❞♦r _p✲❧❛♣❧❛❝✐❛♥♦ ❝♦♠

2 ≤ p < N✱ ✉♠❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❞♦ t✐♣♦ ♣♦tê♥❝✐❛ ❝♦♠ ❡①♣♦❡♥t❡ q s✉❜❝rít✐❝♦✱ ✉♠ ♣❛✲

râ♠❡tr♦ λ ♣♦s✐t✐✈♦ ❡ ✉♠ ♣♦t❡♥❝✐❛❧ a(x) s❛t✐s❢❛③❡♥❞♦ ❝❡rt❛s ❤✐♣ót❡s❡s✳ ❚❛❧ ♣r♦❜❧❡♠❛ ❢♦✐

✐♥✐❝✐❛❧♠❡♥t❡ ❡st✉❞❛❞♦ ♣♦r ❇❛rts❝❤ ❡ ❲❛♥❣ ❡♠ ❬✺❪ ♥♦ ❝❛s♦ ❞♦ ♦♣❡r❛❞♦r ❧❛♣❧❛❝✐❛♥♦ ✭p= 2✮✳

❆♣r❡s❡♥t❛♠♦s ❛s ✈❡rsõ❡s ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❬✺❪ ♣❛r❛ ♦ ❝❛s♦ ❞♦p✲❧❛♣❧❛❝✐❛♥♦✱ ❞❡♠♦♥str❛❞❛s

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦✱ ✇❡ st✉❞② r❡s✉❧ts ♦♥ ❡①✐st❡♥❝❡ ❛♥❞ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢♦r ❛ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ ✐♥ RN ✐♥✈♦❧✈✐♥❣ t❤❡ _p✲❧❛♣❧❛❝✐❛♥ ♦♣❡r❛t♦r ✇✐t❤ _{2 ≤ p < N}✱ ❛ s✉❜❝r✐t✐❝❛❧ ♥♦♥❧✐♥❡❛r✐t②✱ ❛ ♣♦s✐t✐✈❡ ♣❛r❛♠❡t❡r λ ❛♥❞ ❛ ♣♦t❡♥❝✐❛❧ a(x) s❛t✐s❢②✐♥❣ s♦♠❡

❤②♣♦t❤❡s❡s✳ ❙✉❝❤ ♣r♦❜❧❡♠ ✇❛s ✜rst st✉❞✐❡❞ ❜② ❇❛rts❝❤ ❛♥❞ ❲❛♥❣ ❬✺❪ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❧❛♣❧❛❝✐❛♥ ♦♣❡r❛t♦r ✭p = 2✮✳ ❲❡ ♣r❡s❡♥t ✈❡rs✐♦♥s ♦❢ t❤❡ r❡s✉❧ts ♦❢ ❬✺❪ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✺

✶✳✶ ❈♦♥❥✉♥t♦s ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✶✳✷ ❋✉♥❝✐♦♥❛❧ r❡str✐t♦ ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✶✳✸ ❆ ❝❛t❡❣♦r✐❛ ❞❡ ▲❥✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✶✳✹ ❘❡s✉❧t❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✶✳✺ ❖ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✻ ❯♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✶✳✼ ❚❡♦r❡♠❛ ❞❡ ❡①t❡♥sã♦ ❞❡ ❉✉❣✉♥❞❥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✵

✷✳✶ ❈♦♥s✐❞❡r❛çõ❡s ✐♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷✳✷ ❆ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✷✳✸ ❙♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✷✳✹ ❈♦♥❝❡♥tr❛çã♦ ❞❛s s♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✸ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✺✷

■♥tr♦❞✉çã♦

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛✱ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❡ ❝♦♥❝❡♥✲ tr❛çã♦ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❛ s❡❣✉✐♥t❡ ❝❧❛ss❡ ❞❡ ❡q✉❛çõ❡s q✉❛s✐❧✐♥❡❛r❡s ❞❡ ❙❝❤rö❞✐♥❣❡r

  

−∆pu+ (λa(x) + 1)|u|p−2_u=|u|q−2_{u, x∈R}N_, u∈W1,p(RN_),

(Sλ,q)

❡♠ q✉❡ 2 ≤ p < N, p < q < p∗_{✱ ♦♥❞❡ p}∗ _{= N p/(N −p)✱ ∆pu = ❞✐✈(|∇u|}p−2_{∇u) é}

♦ ♦♣❡r❛❞♦r p✲❧❛♣❧❛❝✐❛♥♦ ❡ λ é ✉♠ ♣❛râ♠❡tr♦ ♣♦s✐t✐✈♦✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r a ✉♠❛ ❢✉♥çã♦

s❛t✐s❢❛③❡♥❞♦

✭A1✮ a∈C(RN,R) é ♥ã♦ ♥❡❣❛t✐✈❛✱Ω = ✐♥t a−1(0) é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ ❝❧❛ss❡C2

❡Ω = a−1_(0)❀

✭A2✮ ❡①✐st❡ M0 >0t❛❧ q✉❡

L({x∈RN _{: a(x)≤M0})<∞}✱

♦♥❞❡ L é ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡♠RN✳

❯♠ ❞♦s ♠♦t✐✈♦s ❞❡ ❡st✉❞❛r♠♦s s♦❧✉çõ❡s ❞❡(Sλ,q)é q✉❡✱ ♣❛r❛λs✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱

♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t (

−∆pu+|u|p−2_u=|u|q−2_{u ❡♠ Ω,}

u= 0 ❡♠ ∂Ω, (Dq)

s❡ t♦r♥❛ ✉♠ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛ ❧✐♠✐t❡✳

❊♠ ❬✻❪✱ ❇❡♥❝✐ ❡ ❈❡r❛♠✐ ❡st✉❞❛r❛♠ ♦ ♣r♦❜❧❡♠❛ (Dq) ♥♦ ❝❛s♦ ❞♦ ♦♣❡r❛❞♦r ❧❛♣❧❛❝✐❛♥♦✳

■♥tr♦❞✉çã♦ ✷

2∗_{✳ ❆q✉✐✱ ❝❛t(Ω) ❞❡♥♦t❛ ❛ ❝❛t❡❣♦r✐❛ ❞❡ ▲❥✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥ ❞♦ ❝♦♥❥✉♥t♦ Ω✳ P♦st❡✲}

r✐♦r♠❡♥t❡✱ ♦✉tr♦s ❛rt✐❣♦s ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s tr❛t❛♥❞♦ ❞❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ♣❛r❛

(Dq) ❝♦♠ p= 2 ❡♠ ❢✉♥çã♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞❡ Ω✱ t❛♥t♦ ♥♦ ❝❛s♦ s✉❜❝rít✐❝♦ ❬✽✱ ✶✶✱ ✼❪ q✉❛♥t♦

♥♦ ❝❛s♦ ❝rít✐❝♦ ❬✷✽✱ ✷✷❪✳ ❖ ❝❛s♦ q✉❛s✐❧✐♥❡❛r (2< p < N)❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ♣♦r ❆❧✈❡s ❡ ❉✐♥❣

❡♠ ❬✷❪ ❡ ❋✉rt❛❞♦ ❡♠ ❬✶✻❪✳

❖ ♣r♦❜❧❡♠❛ (Sλ,q) ❢♦✐ ❡st✉❞❛❞♦ ♣♦r ❇❛rts❝❤ ❡ ❲❛♥❣ ❡♠ ❬✺❪ ♥♦ ❝❛s♦ s❡♠✐❧✐♥❡❛r p= 2✳

❊❧❡s ♣r♦✈❛r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ♣❛r❛λ s✉✜❝✐❡♥t❡✲

♠❡♥t❡ ❣r❛♥❞❡✳ ▼❛✐s ❛✐♥❞❛✱ q✉❛♥❞♦λ→ ∞✱ ❡ss❛s s♦❧✉çõ❡s s❡ ❝♦♥❝❡♥tr❛♠ ❡♠ ✉♠❛ s♦❧✉çã♦

♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞♦ ♣r♦❜❧❡♠❛(Dq)✳ ❚❛♠❜é♠✱ s✉♣♦♥❞♦Ω✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦

❡ ❛♣♦✐❛❞♦s ♥♦ r❡s✉❧t❛❞♦ ❛♥t❡r✐♦r ❞❡ ❇❡♥❝✐ ❡ ❈❡r❛♠✐✱ ♦s ❛✉t♦r❡s ♣r♦✈❛r❛♠ q✉❡ (Sλ,q) ❝♦♠

p= 2 ♣♦ss✉✐ ❝❛t(Ω) s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛λ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❡ q ♣ró①✐♠♦ ❞❡2∗_✳

❊♠ ❬✶✼✱ ✶✽❪✱ ❋✉rt❛❞♦ ❡st✉❞♦✉ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿ 

     

−∆pu+ (λa(x) + 1)|u|p−2_u=|u|q−2_{u ❡♠ R}N_,

u(τ x) =−u(x) ♣❛r❛ t♦❞♦ x∈RN_,

u∈W1,p_(RN_),

❝♦♠ λ > 0✱ 2 ≤ p < N, p < q < p∗_{✱ τ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ♦rt♦❣♦♥❛❧ ❞❡ R}N _❡♠

RN s❛t✐s❢❛③❡♥❞♦ _{τ 6= Id} ❡ _τ2 _{= Id}✳ ❖ ♣♦t❡♥❝✐❛❧ _a s❛t✐s❢❛③ _{(A1), (A2)} ❡ é ✐♥✈❛r✐❛♥t❡ ♣♦r τ✱ ✐st♦ é✱ a(τ x) = a(x) ♣❛r❛ t♦❞♦ x ∈ RN✳ ❯t✐❧✐③❛♥❞♦ ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ❡❧❡ ♦❜t❡✈❡✱ ♣❛r❛ λ ❣r❛♥❞❡✱ r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s q✉❡ ♠✉❞❛♠

❞❡ s✐♥❛❧ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③✱ ❛❧é♠ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ss❛s s♦❧✉çõ❡s ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ❡q✉✐✈❛r✐❛♥t❡ ❞♦ ❝♦♥❥✉♥t♦ Ω ♦♥❞❡ ♦ ♣♦t❡♥❝✐❛❧ s❡ ❛♥✉❧❛✱ q✉❛♥❞♦ q é ♣ró①✐♠♦ ❞❡ p∗_{✳ ❆❞❛♣t❛♥❞♦ ❛s ✐❞❡✐❛s ❞❛s ❞❡♠♦♥str❛çõ❡s ❞❡ss❡s r❡s✉❧t❛❞♦s ❞❡ s♦❧✉çõ❡s ♥♦❞❛✐s ♣❛r❛ ♦}

❝❛s♦ s❡♠ ❤✐♣ót❡s❡ ❞❡ s✐♠❡tr✐❛✱ ♦ ❛✉t♦r ♣ô❞❡ ❡st❡♥❞❡r ♦s r❡s✉❧t❛❞♦s ❞❡ ❬✺❪ ♣❛r❛ ♦ ❝❛s♦ q✉❛s✐❧✐♥❡❛r✳

◆♦ss♦ ♦❜❥❡t✐✈♦ ❛q✉✐ é ❡st✉❞❛r ♦s r❡s✉❧t❛❞♦s ❞❡ ❇❛rts❝❤ ❡ ❲❛♥❣ ❬✺❪ ❡st❡♥❞✐❞♦s ♣❛r❛ ♦ ❝❛s♦ 2 ≤ p < N✱ ❞❡♠♦♥str❛❞♦s ♣♦r ❋✉rt❛❞♦ ❡♠ ❬✶✼✱ ✶✽❪✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡♠ q✉❡

tr❛❜❛❧❤❛r❡♠♦s é

E =

u∈W1,p(RN_{) :} Z

RN

a(x)|u|p _<∞

.

P❛r❛ λ≥0✱ ❞❡✜♥✐♠♦s Eλ = (E,k · kλ) ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧E ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛

kukλ =

Z

RN

(|∇u|p+ (λa(x) + 1)|u|p)

■♥tr♦❞✉çã♦ ✸

❆ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭Sλ,q✮ t❡♠♦s ♦ ❢✉♥❝✐♦♥❛❧ Iλ,qe :E →R ❞❡✜♥✐❞♦ ♣♦r

e

Iλ,q(u) = 1

p

Z

RN

(|∇u|p+ (λa(x) + 1)|u|p)− 1

q

Z

RN

|u|q.

❉✐③❡♠♦s q✉❡ u∈Eλ é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (Sλ,q) q✉❛♥❞♦

Z

RN

|∇u|p−2_{∇u· ∇φ+}Z RN

(λa(x) + 1)|u|p−2_uφ−Z RN

|u|q−2_{uφ = 0,}

♣❛r❛ t♦❞❛ ❢✉♥çã♦φ ∈Eλ✳ ❯♠❛ s♦❧✉çã♦ u❞❡ (Sλ,q) é ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ q✉❛♥❞♦

e

Iλ,q(u) = infnIλ,qe (v) :v é ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ (Sλ,q)o.

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛♠♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡ (Sλ,q) ♣❛r❛ λ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ❙ã♦ ❡❧❡s✿

❚❡♦r❡♠❛ ❆ ❙✉♣♦♥❤❛ q✉❡ (A1) ❡ (A2) s❡❥❛♠ ✈á❧✐❞❛s✳ ❊♥tã♦ ❡①✐st❡ Λ0 = Λ0(q) t❛❧ q✉❡✱

♣❛r❛ t♦❞♦ λ ≥ Λ0✱ ♦ ♣r♦❜❧❡♠❛ ✭Sλ,q✮ t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛

♠í♥✐♠❛✳

❚❡♦r❡♠❛ ❇ ❙❡❥❛♠ (λn)⊂R t❛❧ q✉❡ λn → ∞❡ (un)✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s

❞♦ ♣r♦❜❧❡♠❛ (Sλn,q) t❛❧ q✉❡ Iλen,q(un) é ❧✐♠✐t❛❞♦✳ ❊♥tã♦✱ ❛ ♠❡♥♦s ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱

un→u ❢♦rt❡ ❡♠ W1,p_(RN_{) ❝♦♠ u s❡♥❞♦ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞♦ ♣r♦❜❧❡♠❛ (Dq)✳}

❈♦r♦❧ár✐♦ ❈ ❙❡❥❛♠ (λn) ⊂ R ❝♦♠ _{λn → ∞} ❡ _(un) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞♦ ♣r♦❜❧❡♠❛ (Sλn,q)✳ ❊♥tã♦ (un) ❝♦♥✈❡r❣❡ ❡♠ W

1,p_(RN_{) ❛♦ ❧♦♥❣♦ ❞❡}

✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞❡ (Dq)✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ❡st✉❞❛♠♦s ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦Ω = ✐♥ta−1_{(0) ❡ ♦}

♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡(Sλ,q)✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡♠♦♥str❛✲

çã♦ ❞♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✳

❚❡♦r❡♠❛ ❉ ❙✉♣♦♥❤❛ q✉❡ (A1) ❡ (A2) s❡❥❛♠ ✈á❧✐❞❛s ❡ q✉❡ Ω s❡❥❛ ❧✐♠✐t❛❞♦✳ ❊♥tã♦ ❡①✐st❡

q0 ∈ (p, p∗) ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡✱ ♣❛r❛ ❝❛❞❛ q ∈ (q0, p∗)✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ Λ(q)> 0 t❛❧

q✉❡✱ ♣❛r❛ t♦❞♦ λ≥Λ(q)✱ ♦ ♣r♦❜❧❡♠❛ (Sλ,q) t❡♠ ♣❡❧♦ ♠❡♥♦s ❝❛t(Ω) s♦❧✉çõ❡s ♣♦s✐t✐✈❛s✳

❊♠❜♦r❛ ❛s ❞❡♠♦♥str❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛ s✐❣❛♠ ❛q✉❡❧❛s ❞❡ ❇❛rts❝❤ ❡ ❲❛♥❣✱ ❡①✐st❡ ✉♠❛ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡ té❝♥✐❝❛ ❞❡✈✐❞❛ à ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❞♦ ♦♣❡r❛❞♦rp✲❧❛♣❧❛❝✐❛♥♦ ❡

♣❡❧♦ ❢❛t♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢✉♥çõ❡s ❡♠ q✉❡ s❡ ❞❡✈❡ tr❛❜❛❧❤❛r ♥ã♦ s❡r ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳ ❉❡♥tr❡ ♦s r❡s✉❧t❛❞♦s q✉❡ ❛✉①✐❧✐❛♠ ❛ tr❛♥s♣♦r ❛ ❞✐✜❝✉❧❞❛❞❡ ✐♥tr♦❞✉③✐❞❛ ♣❡❧♦ p✲❧❛♣❧❛❝✐❛♥♦✱

■♥tr♦❞✉çã♦ ✹

✉♠❛ ✈❡rsã♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡ ✭❬✸✸❪✱ ▲❡♠❛ ✶✳✹✵✮ ❞❡♠♦♥s✲ tr❛❞❛ ♣♦r ❋✉rt❛❞♦ ✭❬✶✽❪✱ ▲❡♠❛ ✷✳✻✮ ❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛s s❡q✉ê♥❝✐❛s ♠✐♥✐♠✐③❛♥t❡s ♣❛r❛S

✭❬✶✽❪✱ ▲❡♠❛ ❆✳✶✶✮✱ ♦♥❞❡Sé ❛ ♠❡❧❤♦r ❝♦♥st❛♥t❡ ❞❛ ✐♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈W₀1,p(Ω) ֒→Lp∗

❈❛♣ít✉❧♦ ✶

❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s q✉❡ ✉t✐❧✐③❛♠♦s ♥♦ ❞❡❝♦rr❡r ❞❡st❡ tr❛❜❛❧❤♦✳

✶✳✶ ❈♦♥❥✉♥t♦s ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s

❉❡✜♥✐çã♦ ✶✳✶ ❙❡❥❛♠X ❡Y ❞♦✐s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❯♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ ❛s ❛♣❧✐❝❛çõ❡s

❝♦♥tí♥✉❛s f, g : X → Y é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ H : [0,1]×X → Y t❛❧ q✉❡ ♣❛r❛ t♦❞♦ x∈X✱ t❡♠✲s❡ H(0, x) = f(x) ❡H(1, x) =g(x)✳ ❊s❝r❡✈❡✲s❡ f ≃g ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ❡①✐st❡

✉♠❛ ❤♦♠♦t♦♣✐❛ H ❡♥tr❡ f ❡ g✳

❉❡✜♥✐çã♦ ✶✳✷ ❙❡❥❛♠ X ❡ Y ❞♦✐s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❉✐③❡♠♦s q✉❡ X ❡ Y sã♦ ❤♦♠♦t♦✲

♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st❡♠ ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s f :X →Y ❡ g :Y →X t❛✐s q✉❡ f◦g ≃IdY ❡ g◦f ≃IdX✱ ♦♥❞❡ IdX é ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ X ❡ IdY ❞❡ Y✳

P❛r❛Ω⊂RN ❡_{r >0} ❞❡✜♥✐♠♦s

Ω+

r ={x∈RN : ❞✐st(x,Ω)< r} ❡ Ω−r ={x∈Ω :❞✐st(x, ∂Ω) ≥r}✳

◆❡st❛ s❡çã♦✱ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ♣r♦✈❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✱ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❞❡♠♦♥s✲ tr❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❉✳

▲❡♠❛ ✶✳✸ ❙❡❥❛ Ω ⊂ RN ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❞❡ ❝❧❛ss❡ _C2✳ ❊♥tã♦ ♣❛r❛ _{r1, r2 > 0} s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦s✱ ♦s ❝♦♥❥✉♥t♦s Ω+

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✻

❆ ❞❡♠♦♥str❛çã♦ q✉❡ ❞❛♠♦s ❛q✉✐ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❬✶✸❪ ✭Pr♦♣♦s✐çã♦ ✷✳✸✳✸✮✳ ❆♥t❡s✱ ❛♣r❡✲ s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ✉♠ r❡s✉❧t❛❞♦ s♦❜r❡ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s q✉❡ s❡rã♦ ♥❡❝❡s✲ sár✐♦s✳

❉❡✜♥✐çã♦ ✶✳✹ ❙❡❥❛ M =Mm _⊂RN _{✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C}k_{✱ k ≥1✳}

❙❡❥❛ ϕ :U0 →U ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❝♦♠ U0 ⊂Rm ✉♠ ❛❜❡rt♦✱U ⊂RN ❡ p=ϕ(x)∈M

♣❛r❛ ❛❧❣✉♠ x∈ U0✳ ❖ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛ M ♥♦ ♣♦♥t♦ p é ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦

m

TpM =ϕ′(x)·Rm_.

❉✐③❡♠♦s q✉❡ ✉♠ ✈❡t♦r u ∈ RN é ♥♦r♠❛❧ ❛ s✉♣❡r❢í❝✐❡ _M ♥♦ ♣♦♥t♦ _p q✉❛♥❞♦ _u ❢♦r ♣❡r♣❡♥❞✐❝✉❧❛r ❛ t♦❞♦s ♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ M ♥♦ ♣♦♥t♦ p✱ ✐st♦ é✱ q✉❛♥❞♦ s❡ t✐✈❡r u·v = 0 ♣❛r❛ t♦❞♦ v ∈TpM✳ ■♥❞✐❝❛♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s ♥♦r♠❛✐s ❛ M ♥♦ ♣♦♥t♦ p

♣♦r νpM✳

❉✐③❡♠♦s q✉❡ ♦ s❡❣♠❡♥t♦ [p, a] ={p+t(a−p) : 0≤t≤1} é ♥♦r♠❛❧ ❛ M ♥♦ ♣♦♥t♦ p

s❡p∈M ❡ v =a−p∈νpM✳

❉❛❞♦ε >0✱ ❛ ❜♦❧❛ ♥♦r♠❛❧B⊥_{(p;ε)é ❛ r❡✉♥✐ã♦ ❞♦s s❡❣♠❡♥t♦s ♥♦r♠❛✐s ❛M ♥♦ ♣♦♥t♦} p✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♠❡♥♦r q✉❡ ε✳ ❉✐③❡♠♦s q✉❡ ε é ✉♠ r❛✐♦ ♥♦r♠❛❧ ❛❞♠✐ssí✈❡❧ ♣❛r❛ ✉♠

s✉❜❝♦♥❥✉♥t♦ X ⊂M q✉❛♥❞♦✱ ❞❛❞♦s q✉❛✐sq✉❡r ❞♦✐s s❡❣♠❡♥t♦s[p, a] ❡[q, b]✱ ♥♦r♠❛✐s ❛ M✱

❞❡ ❝♦♠♣r✐♠❡♥t♦ ♠❡♥♦r q✉❡ε✱ ❝♦♠p6=q∈X✱ t❡♠✲s❡ [p, a]∩[q, b] =∅✳

❊♥✉♥❝✐❛♠♦s ❛❣♦r❛ ♦ t❡♦r❡♠❛ ❞❛ ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r ♣❛r❛ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✹❪✳

❚❡♦r❡♠❛ ✶✳✺ ❙❡❥❛M =Mm _⊂RN _{✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ ❞❡ ❝❧❛ss❡C}k_{✱ k ≥2✳ ❊♥tã♦✿}

✭✐✮ ❊①✐st❡ ε >0✱ r❛✐♦ ♥♦r♠❛❧ ❛❞♠✐ssí✈❡❧ ♣❛r❛ M✳

✭✐✐✮ ❆ r❡✉♥✐ã♦ Vε(M) = ∪p∈MB⊥(p;ε) ❞♦s s❡❣♠❡♥t♦s ♥♦r♠❛✐s ❛ M ❞❡ ❝♦♠♣r✐♠❡♥t♦

♠❡♥♦r q✉❡ ε é ✉♠ ❛❜❡rt♦ ❞❡ RN ❝❤❛♠❛❞♦ ❛ ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r ❞❡ _M ❞❡ r❛✐♦ _ε✳

✭✐✐✐✮ ❆ ❛♣❧✐❝❛çã♦ η : Vε(M) → M✱ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ q ∈ Vε(M) ♦ ♣é ❞♦ ú♥✐❝♦

s❡❣♠❡♥t♦ ♥♦r♠❛❧ q✉❡ ♦ ❝♦♥té♠✱ é ❞❡ ❝❧❛ss❡ Ck−1_✳

❉❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✶✳✸✿ P♦r ❤✐♣ót❡s❡✱ ∂Ω é ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ ❞❡ ❝❧❛ss❡

C2_{✳ ▲♦❣♦ ✈❛❧❡♠ (i)−(iii)❞♦ ❚❡♦r❡♠❛ ✶✳✺ ♣❛r❛ ❛❧❣✉♠ ε >0✳ ❙❡❥❛ 0< r}

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✼

g : Ω→Ω−

r1 ⊂Ω+r1 ❛ ♣r♦❥❡çã♦ ❞❛❞❛ ♣♦r

g(x) =

(

x, s❡ x∈Ω−_r1, π(x)−r1η(x), s❡ x∈Ω\Ω−r1,

♦♥❞❡η(x)é ♦ ✈❡t♦r ✉♥✐tár✐♦ ♥❛ ❞✐r❡çã♦ ❞♦ ú♥✐❝♦ s❡❣♠❡♥t♦ ♥♦r♠❛❧ q✉❡ ❝♦♥té♠x✳ ❉❡✜♥✐♠♦s

t❛♠❜é♠ f : Ω+

r1 →Ω♣♦r

f(x) =

(

g(x), s❡ x∈Ω, x−r1η(x), s❡ x∈Ω+r1 \Ω.

❖❜s❡r✈❛♠♦s q✉❡f ❡g❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s ❡ sã♦ ❝♦♥tí♥✉❛s✳ ❱❛♠♦s ♠♦str❛r q✉❡f◦g ≃IdΩ

❡ g◦f ≃Id_Ω+

r₁✳ P❛r❛ ✐ss♦✱ t♦♠❛♠♦s H1 : [0,1]×Ω→Ω ❞❛❞❛ ♣♦r

H1(t, x) = (1−t)f◦g(x) +tx= (1−t)g(x) +tx.

❙❡ x ∈ Ω−

r1✱ H1(t, x) =x ∈ Ω−r1 ⊂ Ω✳ ❙❡ x ∈ Ω\Ω−r1✱ H1(t, x) ❡stá ❝♦♥t✐❞♦ ♥♦ s❡❣♠❡♥t♦ [x, g(x)] ⊂ [π(x), g(x)] ❞❡ Vε(∂Ω)✳ ❉❡s❞❡ q✉❡ ❡ss❡ s❡❣♠❡♥t♦ ♥ã♦ ❝♦♥té♠ ♦✉tr♦ ♣♦♥t♦ ❞❡

❢r♦♥t❡✐r❛ ❛❧é♠ ❞❡ π(x)✱ s❡❣✉❡ q✉❡ H1(t, x) ⊂ Ω✳ ▲♦❣♦ H1 ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ é ✉♠❛

❤♦♠♦t♦♣✐❛ ❡♥tr❡ f◦g ❡ IdΩ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡✜♥✐♠♦s H2 : [0,1]×Ω+r1 →Ω+r1 ♣♦r

H2(t, x) = (1−t)g◦f(x) +tx.

❖❜s❡r✈❛♠♦s q✉❡

g◦f(x) =

(

g(x), s❡ x∈Ω, π(x)−r1η(x), s❡ x∈Ω+r1 \Ω.

▲♦❣♦ H2(t, x) ❡stá ❝♦♥t✐❞♦ ♥♦ s❡❣♠❡♥t♦ ♥♦r♠❛❧ ❞❡ Vε(∂Ω) q✉❡ ❝♦♥té♠ x✱ s❡❣♠❡♥t♦ ❡ss❡

❝♦♥t✐❞♦ ❡♠Ω+

r1✳ ❆ss✐♠H2é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡g◦f ❡IdΩ+r₁✳ ❉❛íΩ

+

r1 é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡

❡q✉✐✈❛❧❡♥t❡ ❛ Ω✳ ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ♠♦str❛✲s❡ q✉❡ Ω−

r2 t❛♠❜é♠ ♦ é✳

✶✳✷ ❋✉♥❝✐♦♥❛❧ r❡str✐t♦ ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡

❈♦♥s✐❞❡r❡ (X,k · k) ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ψ ∈C1_(X,R)✱

V ={v ∈X :ψ(v) = 0} ❡ ψ′_{(v)6= 0✱ ♣❛r❛ t♦❞♦ v ∈V✳}

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✽

TzV ={y ∈X: hψ′_{(z), yi= 0}}

♦✉ s❡❥❛✱ TzV é ♦ ♥ú❝❧❡♦ ❞♦ ❢✉♥❝✐♦♥❛❧ ψ′(z)✳

❉❡✜♥✐çã♦ ✶✳✼ ❙❡❥❛ ϕ ∈ C1_{(X,R)✳ ❆ ♥♦r♠❛ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ϕ|}

V := ϕV ❡♠ v ∈ V é

❞❡✜♥✐❞❛ ♣♦r

kϕ′_V(v)k_∗ = sup{hϕ′(v), yi:y ∈TvV e kyk= 1}.

▲❡♠❛ ✶✳✽ ❙❡❥❛♠ f ❡ g ❢✉♥❝✐♦♥❛✐s ❧✐♥❡❛r❡s ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❋✳ ❙❡ N(f) ⊂ N(g)

❡♥tã♦ g ≡ kf ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ k ∈ R✱ ♦♥❞❡ _N(f) ❡ _N(g) ❞❡♥♦t❛♠ ♦ ♥ú❝❧❡♦ ❞❛s ❢✉♥çõ❡s f ❡ g✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❉❡♠♦♥str❛çã♦✿ ❙❡ f ≡ 0 ❡♥tã♦ g ≡ 0 ❡ ♦ ❧❡♠❛ ❡stá ♣r♦✈❛❞♦ ♣❛r❛ q✉❛❧q✉❡r ❡s❝♦❧❤❛ ❞❡

k ∈ R✳ ❈❛s♦ _{f 6= 0}✱ ❡①✐st❡ _{v ∈ F} t❛❧ q✉❡ _{f(v) 6= 0}✳ ❈♦♥s✐❞❡r❡ _{k =} g(v)

f(v) ❡ ♦ ❢✉♥❝✐♦♥❛❧

❧✐♥❡❛r h(x) = g(x)−kf(x) ♣❛r❛ x∈F✳ ❱❛♠♦s ♠♦str❛r q✉❡h≡0❡✱ ❝♦♠ ✐ss♦✱ ❝♦♥❝❧✉✐r ❛

❞❡♠♦s♥tr❛çã♦ ❞♦ ❧❡♠❛✳ P❛r❛ t❛♥t♦✱ s❡❥❛ w∈F✳ ❊s❝♦❧❤❡♥❞♦ s=s(w) = f(w)

f(v) ✈❡♠♦s q✉❡

w =sv+u ♦♥❞❡ u= w−sv ∈N(f)✳ ❈♦♠♦ N(f) ⊂ N(g) t❡♠♦s h(u) = 0 ❡✱ ♣♦rt❛♥t♦✱

h(w) =sh(v) +h(u) = 0✳ ▲♦❣♦✱ h≡0✳

❖ ❧❡♠❛ ❛ s❡❣✉✐r ❝♦rr❡s♣♦♥❞❡ à Pr♦♣♦s✐çã♦ ✺✳✶✷ ❞❡ ❬✸✸❪✱ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ t❛♠✲ ❜é♠ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✸❪✳

▲❡♠❛ ✶✳✾ ❙❡❥❛ u∈V✳ ❊♥tã♦

kϕ′

V(u)k∗ = min t∈R kϕ

′_(u)−tψ′_(u)k X′✳

❉❡♠♦♥str❛çã♦✿ P❛r❛t ∈R✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❉❡✜♥✐çõ❡s ✶✳✼ ❡ ✶✳✻✱

kϕ′_V(u)k∗ = sup{hϕ′(u), yi:y ∈TuV ❡ kyk= 1}

= sup{hϕ′(u)−tψ′(u), yi:y∈TuV ❡ kyk= 1}

≤ sup{hϕ′_(u)−tψ′_{(u), yi:y∈X ❡kyk= 1}}

= kϕ′(u)−tψ′(u)kX′.

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✾

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❍❛♥❤✲❇❛♥❛❝❤✱ ❡①✐st❡ Φ :X →R ❧✐♥❡❛r ❝♦♥tí♥✉❛ t❛❧ q✉❡

Φ|TuV =ϕ

′

V(u) ❡ kΦkX′ =kϕ′

V(u)k∗

❈♦♠♦ N(ψ′_{(u)) = TuV ⊂N(ϕ}′_{(u)−Φ)✱ ♣❡❧♦ ▲❡♠❛ ✶✳✽✱ ❡①✐st❡t}

0 ∈R t❛❧ q✉❡

ϕ′(u)−Φ = t0ψ′(u).

❆ss✐♠✱

kϕ′(u)−t0ψ′(u)kX′ =kΦk_X′ =kϕ′

V(u)k∗.

❈♦♠❜✐♥❛♥❞♦ ❝♦♠ ✭✶✳✶✮✱ ❝♦♥❝❧✉í♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✳

✶✳✸ ❆ ❝❛t❡❣♦r✐❛ ❞❡ ▲❥✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❛ ❝❛t❡❣♦r✐❛ ❞❡ ▲✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥ ❜❡♠ ❝♦♠♦ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ✉s❛❞❛s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❉✳ P❛r❛ ✉♠ ♠❡❧❤♦r ❡st✉❞♦ s♦❜r❡ ♦ ❛ss✉♥t♦✱ ✈❡❥❛ ❬✷✾✱ ✸✸✱ ✶✷✱ ✸❪ ❡ t❛♠❜é♠ ❬✶✸❪✳

❉❡✜♥✐çã♦ ✶✳✶✵ ❉✐③❡♠♦s q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❝♦♥trát✐❧

❡♠ X q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ h: [0; 1]×A→X t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ x∈A✱ h(0, x) = x ❡ h(1, x) = x0 ♣❛r❛ ❛❧❣✉♠ x0 ∈ X✳ ❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❡①✐st❡ ✉♠❛ ❤♦♠♦t♦✲

♣✐❛ ❡♥tr❡ ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ A ❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡✳ ❚❛❧ h é ❝❤❛♠❛❞❛ ❞❡

❞❡❢♦r♠❛çã♦ ❞❡ A ❡♠ X✳

❉❡✜♥✐çã♦ ✶✳✶✶ ❙❡❥❛ A ⊂ X✱ ♦♥❞❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❆ ❝❛t❡❣♦r✐❛ ❞❡ A ❡♠ X✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r ❝❛tX(A)✱ é ♦ ♠❡♥♦r ✐♥t❡✐r♦ k t❛❧ q✉❡ A ♣♦❞❡ s❡r ❝♦❜❡rt♦ ♣♦r k

s✉❜❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s ❡ ❝♦♥trát❡✐s ❡♠X✳ ❙❡ ♥ã♦ ❡①✐st✐r t❛❧ ✐♥t❡✐r♦✱ ❞✐③❡♠♦s q✉❡ ❝❛tX(A) =

+∞✳ ❆❧é♠ ❞✐ss♦✱ catX(∅) = 0 ❡ r❡♣r❡s❡♥t❛♠♦s ❝❛tX(X) ♣♦r ❝❛t(X)✳

❈♦♠♦ ❡①❡♠♣❧♦s✱ t❡♠♦s q✉❡catRN(SN−1) =catRN(B1(0)) = 1✳ ❊①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s

q✉❡ t❡♠ ❝❛t❡❣♦r✐❛ ♠❛✐♦r q✉❡ ✶ é ❛ ❡s❢❡r❛ N✲❞✐♠❡♥s✐♦♥❛❧SN _⊂RN+1 _{❝✉❥❛ cat}

SN(SN) = 2

❡ ♦ t♦r♦N✲❞✐♠❡♥s✐♥♦♥❛❧TN _{❝✉❥❛cat}

TN(TN) =N+ 1✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡s ❢❛t♦s ♣♦❞❡

s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✾❪✳

Pr♦♣♦s✐çã♦ ✶✳✶✷ ❙❡❥❛♠ A ❡ B s✉❜❝♦♥❥✉♥t♦s ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X✳ ❆ ❝❛t❡❣♦r✐❛

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✵

✭✐✮ ❙❡ A⊂B✱ ❡♥tã♦ catX(A)≤catX(B);

✭✐✐✮ catX(A∪B)≤catX(A) +catX(B)❀

✭✐✐✐✮ ❙❡ B é ❢❡❝❤❛❞♦ ❡♠ X✱ ❡♥tã♦ catX(B)≤catB(B)❀

✭✐✈✮ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ♠♦❞❡❧❛❞❛ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡ K ✉♠ s✉❜❝♦♥❥✉♥t♦

❝♦♠♣❛❝t♦ ❞❡ M✳ ❊♥tã♦✱ catM(K) <+∞ ❡ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ K t❛❧ q✉❡ catM(U) =catM(K)❀

✭✈✮ ❙❡ X é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ Y✱ ❡♥tã♦ catX(X) = catY(Y)✳

❚❡♦r❡♠❛ ✶✳✶✸ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ M ⊂ X ✉♠❛ C1_{✲✈❛r✐❡❞❛❞❡ ❡ I ∈}

C1_{(X,R) ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ M✳ ❙✉♣♦♥❤❛ q✉❡ I s❛t✐s❢❛③ (P S)}

c

♣❛r❛ t♦❞♦ c≤d ❡ ❝♦♥s✐❞❡r❡ Id _{={u∈M :I(u)≤d}✳ ❊♥tã♦ ♦ ❢✉♥❝✐♦♥❛❧ I r❡str✐t♦ à M}

t❡♠ ♣❡❧♦ ♠❡♥♦s cat(Id_{) ♣♦♥t♦s ❝rít✐❝♦s u t❛✐s q✉❡ I(u)≤d✳}

✶✳✹ ❘❡s✉❧t❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡

❊♥✉♥❝✐❛♠♦s ❛q✉✐ r❡s✉❧t❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡ q✉❡ s❡rã♦ út❡✐s ♥♦ ❈❛♣í✲ t✉❧♦ ✸✳ ❆♥t❡s✱ ❞❛♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ✉♠ r❡s✉❧t❛❞♦ ❞❛ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ q✉❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✶✺❪✳

❉❡✜♥✐çã♦ ✶✳✶✹ ❙❡❥❛ µ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❇♦r❡❧ ❡♠ RN ❡ _{B ⊂} RN ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ❇♦r❡❧✳ µ é r❡❣✉❧❛r ❡①t❡r✐♦r ❡♠B s❡ µ(B) = inf{µ(U) :U ⊃B, U é ❛❜❡rt♦}❡ µé r❡❣✉❧❛r

✐♥t❡r✐♦r ❡♠ B s❡ µ(B) = sup{µ(K) :K ⊂B, K é ❝♦♠♣❛❝t♦}✳ ❉✐③❡♠♦s q✉❡ µ é r❡❣✉❧❛r

s❡ µ é r❡❣✉❧❛r ❡①t❡r✐♦r ❡ r❡❣✉❧❛r ✐♥t❡r✐♦r ❡♠ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❞❡ ❇♦r❡❧✳

❉❡✜♥✐çã♦ ✶✳✶✺ ❯♠❛ ♠❡❞✐❞❛ ❞❡ ❘❛❞♦♥ ❡♠ RN é ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❇♦r❡❧ q✉❡ é ✜♥✐t❛ ❡♠ ❝♦♥❥✉♥t♦s ❝♦♠♣❛❝t♦s✱ r❡❣✉❧❛r ❡①t❡r✐♦r ❡♠ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❞❡ ❇♦r❡❧ ❡ r❡❣✉❧❛r ✐♥t❡r✐♦r ❡♠ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✳

❉❡♥♦t❛♠♦s ♣♦r M(RN₎ ♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❞❛s ♠❡❞✐❞❛s ❞❡ ❘❛❞♦♥ s♦❜r❡ RN ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛

|ω|= sup

Z

RN

φ(x)dω

:φ∈C0(RN), |φ|∞≤1

.

❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ (ωn) ⊂ M(RN₎ ❝♦♥✈❡r❣❡ ❢r❛❝❛♠❡♥t❡ ♣❛r❛ _ω ❡♠ _M(RN₎ s❡ Z

RN

φ dωn→

Z

RN

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✶

Pr♦♣♦s✐çã♦ ✶✳✶✻ ❙❡❥❛ µ✉♠❛ ♠❡❞✐❞❛ ♣♦s✐t✐✈❛ ❞❡ ❘❛❞♦♥ ❡♠ RN ❡ _(fn)⊂L1₍RN_{, µ)} ❝♦♠ Z

RN

|fn|dµ ≤ C ♣❛r❛ t♦❞♦ n ∈ N ❡ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ C✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ♠❡❞✐❞❛

♣♦s✐t✐✈❛ ❞❡ ❘❛❞♦♥ µ0 t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ µn =|fn|dµ ⇀ µ0 ❢r❛❝❛♠❡♥t❡ ♥♦

s❡♥t✐❞♦ ❞❛s ♠❡❞✐❞❛s✳

❉❡s❞❡ ♦s tr❛❜❛❧❤♦s ❞❡ P✳▲✳ ▲✐♦♥s ❡♠ ❬✷✻✱ ✷✼❪✱ ♦ ♠ét♦❞♦ ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ❝♦♠♣❛❝✐✲ ❞❛❞❡ t❡♠ s✐❞♦ ❧❛r❣❛♠❡♥t❡ ✉t✐❧✐③❛❞♦ ♣♦r ✈ár✐♦s ❛✉t♦r❡s ♣❛r❛ ❝♦♠♣❡♥s❛r ♣r♦❜❧❡♠❛s ❞❡ ❢❛❧t❛ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡✳ ❖ ❧❡♠❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ❝♦rr❡s♣♦♥❞❡ ❛♦ ▲❡♠❛ ✷✳✻ ❡♠ ❬✶✽❪✳ ❊❧❡ ❢♦✐ ♣r♦✈❛❞♦ ♥♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r p = 2✱ qn ≡ 2∗ ♣♦r ❲✐❧❧❡♠ ❡♠ ❬✸✸❪ ♥♦ ▲❡♠❛ ✶✳✹✵✳ ❆

✈❡rsã♦ ❞❡♠♦♥str❛❞❛ ♣♦r ❋✉rt❛❞♦✱ ❜❡♠ ❝♦♠♦ s✉❛ ❞❡♠♦♥str❛çã♦✱ ❢♦✐ ✐♥s♣✐r❛❞❛ ♥❡ss❡ ú❧t✐♠♦ tr❛❜❛❧❤♦ ❡ t❛♠❜é♠ ♣❡❧♦ ❞❡ ❙♠❡ts ❡♠ ❬✸✶❪ ✭✈❡❥❛ ▲❡♠❛ ✷✳✶ ❡ ❖❜s❡r✈❛çã♦ ✷✳✷✮✱ ♦♥❞❡ ♦ ❛✉t♦r ❝♦♥s✐❞❡r❛ ♦ ❝❛s♦ 1 < p < N✱ qn ≡ p∗ _{❡ ♣❡r♠✐t❡ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ✉♠ ♣♦♥t❡♥❝✐❛❧ V q✉❡}

♣♦❞❡ s❡r s✐♥❣✉❧❛r✳

▲❡♠❛ ✶✳✶✼ ❙❡❥❛ (qn) ⊂ [p, p∗] ✉♠❛ s❡q✉ê♥❝✐❛ ♥ã♦✲❞❡❝r❡s❝❡♥t❡ t❛❧ q✉❡ qn → p∗✳ ❙❡❥❛

(un)⊂W1,p_(RN_{) s❛t✐s❢❛③❡♥❞♦}

un ⇀ u f racamente em D1,p_(RN_),

|∇(un−u)|p _{⇀ ω f racamente em M(R}N_),

|un−u|qn ⇀ ν f racamente em M(RN), un(x)→u(x) q.t.p. x∈RN_,

∇un(x)→ ∇u(x) q.t.p. x∈RN_, ❡ ❞❡✜♥❛

ω∞= lim

R→∞lim sup_n→∞

Z

|x|>R

|∇un|p, ν∞= lim

R→∞lim sup_n→∞

Z

|x|>R

|un|qn_.

❊♥tã♦

|ν|p/p∗ ≤S−1|ω|,

lim sup

n→∞

|un|p_p,RN =|∇u|

p

p,RN +|ω|+ω∞,

❡

lim sup

n→∞

|un|q_qn_n,RN =|∇u|

p∗

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✷

❆❧é♠ ❞♦ ♠❛✐s✱ s❡u= 0❡|ν|p/p∗

=S−1_{|ω|✱ ❡♥tã♦ ❝❛❞❛ ✉♠❛ ❞❛s ♠❡❞✐❞❛sω❡ν s❡ ❝♦♥❝❡♥tr❛}

❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦✳

❆❜❛✐①♦ ❡♥✉♥❝✐❛♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈♦♥❝❡♥tr❛çã♦✲❈♦♠♣❛❝✐❞❛❞❡ ❞❡✈✐❞♦ ❛ ▲✐♦♥s ✭❬✷✼❪✱ ▲❡♠❛ ■✳✶✮ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ t❛♠❜é♠ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✸✷❪ ✭▲❡♠❛ ✹✳✽✮✳

▲❡♠❛ ✶✳✶✽ ❙❡❥❛ (un) ⊂ D1,p_(RN_{) ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣✐♥❞♦ ❢r❛❝❛♠❡♥t❡ ♣❛r❛ u ❡♠} D1,p_(RN_{)✳ ❊♥tã♦ ❡①✐st❡♠ ❞✉❛s ♠❡❞✐❞❛s ✜♥✐t❛s ♥ã♦✲♥❡❣❛t✐✈❛s µ, ν ∈ M(R}N_{)✱ ✉♠ ❝♦♥✲}

❥✉♥t♦ ♥♦ ♠á①✐♠♦ ❡♥✉♠❡rá✈❡❧ ❏✱ ✉♠❛ ❢❛♠✐❧í❛ (xj)j∈J ❞❡ ♣♦♥t♦s ❞✐st✐♥t♦s ❡♠ RN ❡ ❞✉❛s

s❡q✉ê♥❝✐❛s (µj)j∈J,(νj)j∈J ❝♦♥t✐❞❛s ❡♠ (0,∞) t❛✐s q✉❡

ν = |u|p∗

+X

j∈J νjδxj,

µ ≥ |∇u|p₊X j∈J

µjδxj,

µj ≥ Sνp/p

∗

j ,

♣❛r❛ t♦❞♦ j ∈ J✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ P_j∈Jν_jp/p∗ < ∞✳ ❆❧é♠ ❞♦ ♠❛✐s✱ s❡ ν(RN₎1/p∗ _≥

Sµ(RN₎1/p ❡♥tã♦ _{ν =γδx0 =γ}−p/q_Cp

0µ♣❛r❛ ❛❧❣✉♠ x0 ∈RN ❡ γ ≥0✳

❯t✐❧✐③❛♥❞♦ ♦ ▲❡♠❛ ✶✳✶✽ ❡ ✐❞❡✐❛s ❞❡ ❬✸✵❪✱ ❋✉rt❛❞♦ ❬✶✽❪ ❞❡♠♦♥str♦✉ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡✲ ❞❛❞❡ ❞❛s s❡q✉ê♥❝✐❛s ♠✐♥✐♠✐③❛♥t❡s ♣❛r❛ S✳

▲❡♠❛ ✶✳✶✾ ❙❡❥❛ (vn)⊂W01,p(Ω) t❛❧ q✉❡

R

Ω|vn|

p∗

dx = 1 ❡ kvnkp →S✳ ❊♥tã♦ ❡①✐st❡ v ∈ W01,p(Ω) t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ vn ⇀ v ❢r❛❝❛♠❡♥t❡ ❡♠ W01,p(Ω) ❡ ∇vn(x)→

∇v(x) q✳t✳♣✳ ❡♠ Ω✳

✶✳✺ ❖ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜

◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛♠♦s ♦ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ ❬✶✵❪ ❡ t❛♠❜é♠ ✉♠❛ ✈❡rsã♦ ✈❡t♦r✐❛❧ ❞♦ ♠❡s♠♦✱ ❞❡✈✐❞♦ ❛ ❬✶❪✳ ❆ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ q✉❡ ❞❛♠♦s ❛q✉✐ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✶❪ ✭▲❡♠❛ ✹✳✻✮✳

▲❡♠❛ ✶✳✷✵ ✭❇ré③✐s✲▲✐❡❜✮ ❙❡❥❛♠ 1≤p <∞✱ D ⊂ RN ❛❜❡rt♦ ❡ _(fn)n∈N ✉♠❛ s❡q✉ê♥❝✐❛

❧✐♠✐t❛❞❛ ❞❡ ❢✉♥çõ❡s ❡♠ Lp_{(D) q✉❡ ❝♦♥✈❡r❣❡ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ D ♣❛r❛ f✳ ❊♥tã♦} f ∈Lp_{(D) ❡}

|f|p_p,D = lim

n→∞(|fn| p

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✸

♦♥❞❡ | · |p,D ❞❡♥♦t❛ ❛ ♥♦r♠❛ ✉s✉❛❧ ❡♠ Lp(D)✳

❉❡♠♦♥str❛çã♦✿ P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ C > 0 t❛❧ q✉❡ |fn|p_p,D ≤ C✱ ♣❛r❛ t♦❞♦ n ∈ N✳ P❡❧♦ ▲❡♠❛ ❞❡ ❋❛t♦✉✱ _Z

D

|f|p ₌Z D

lim inf

n→∞ |fn|

p _{≤lim inf} n→∞

Z

D

|fn|p _≤C.

■ss♦ ♠♦str❛ q✉❡ f ∈Lp_(D)✳

❙❡❥❛ε >0✳ ❆✜r♠❛♠♦s q✉❡ ❡①✐st❡ Cε >0q✉❡ ❞❡♣❡♥❞❡ ❞❡ ε ❡ ❞❡ pt❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ s∈R✱ s❡ t❡♠

|s+ 1|p _{− |s|}p₋₁

≤ε|s|p_{+Cε. ✭✶✳✷✮}

❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ lim

|s|→∞

|s+ 1|p_{− |s|}p₋₁

|s|p = 0✱ ❡①✐st❡ Aε >0 t❛❧ q✉❡

|s+ 1|p _{− |s|}p ₋₁

≤ε|s|p_{, s❡♠♣r❡ q✉❡ |s|> A}

ε. ✭✶✳✸✮

P❛r❛ |s| ≤Aε t❡♠♦s

|s+ 1|p − |s|p−1≤2p|s|p+ 2p+|s|p+ 1 ≤2pAp_ε+ 2p +Ap_ε+ 1.

❉❡✜♥✐♥❞♦Cε = 2p_Ap

ε+ 2p+Apε+ 1>0✱ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✸✮ ♠♦str❛♠

♥♦ss❛ ❛✜r♠❛çã♦✳ ❉❛❞♦b ∈R✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✮ ♣♦r _|b|p ♦❜t❡♠♦s

|sb+b|p _{− |sb|}p_{− |b|}p

≤ε|sb|p_+Cε|b|p_{, ♣❛r❛ t♦❞♦s∈R,}

❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r a, b∈R t❡♠✲s❡

|a+b|p _{− |a|}p _{− |b|}p

≤ε|a|p_+Cε|b|p_{. ✭✶✳✹✮}

P❛r❛ ❝❛❞❛ n∈N✱ ❝♦♥s✐❞❡r❡

un =

|fn|p − |fn−f|p− |f|p

❡ Zn = (un−ε|fn−f|p)+,

♦♥❞❡ u+ _{= max{u,0} ♣❛r❛ q✉❛❧q✉❡r ❢✉♥çã♦ r❡❛❧ u✳ ❱❛♠♦s ♠♦str❛r q✉❡}

Z

D

un →0✳ P❛r❛

✐ss♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✹✮ ❝♦♠a=fn−f ❡ b =f ♦❜t❡♠♦s

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✹

❡✱ ♣♦rt❛♥t♦✱

0≤Zn= max{(un−ε|fn−f|p),0} ≤Cε|f|p.

❈♦♠♦ fn(x)→f(x) q✳t✳♣✳ ❡♠ D✱ t❡♠♦s q✉❡ Zn→ 0q✳t✳♣✳ ❡♠ D✳ ❙❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ❞❛

❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡ ✭❚✳❈✳❉✳▲✳✮ q✉❡ Z

D

Zn→0 q✉❛♥❞♦ n→ ∞. ✭✶✳✺✮

❚❡♠♦s t❛♠❜é♠ q✉❡

un=u+

n = (ε|fn−f|p+un−ε|fn−f|p)+ ≤ ε|fn−f|p+ (un−ε|fn−f|p)+

= ε|fn−f|p_+Zn.

P♦r ❤✐♣ót❡s❡✱(fn) é ❧✐♠✐t❛❞❛ ❡♠Lp_{(D)❡✱ ♣♦rt❛♥t♦✱ ❡①✐st❡ M >0 t❛❧ q✉❡|fn−f|}p p ≤M✳

❉❛í ❡ ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ s❡❣✉❡ q✉❡ Z

D

un=ε|fn−f|p p +

Z

D

Zn ≤εM +

Z

D Zn.

❋❛③❡♥❞♦n → ∞ ❡ε→0✱ ♣♦r ✭✶✳✺✮ ♦❜t❡♠♦s

Z

D

un→0✳ ❈♦♠♦

|fn|p_p,D− |fn−f|p_p,D− |f|p_p,D≤

Z

D un,

♦ ❧❡♠❛ ❡stá ♣r♦✈❛❞♦✳

❖❜s❡r✈❛çã♦ ✶✳✷✶ ❆❞❛♣t❛♥❞♦ ❧✐❣❡✐r❛♠❡♥t❡ ❛ ❞❡♠♦♥str❛çã♦ ❛❝✐♠❛✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡✱ ♣❛r❛ 1< p <∞✱

Z

D

|∇(un−u)|p =

Z

D

|∇un|p−

Z

D

|∇u|p+o(1) ✭✶✳✻✮

q✉❛♥❞♦ n → ∞✱ s❡♠♣r❡ q✉❡ (un) é ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡ (|∇un|) é ❧✐♠✐t❛❞❛ ❡♠ Lp_{(D) ❡}

∇un(x) ❝♦♥✈❡r❣❡ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ D ♣❛r❛ ∇u(x)✳

❉❡ ❢❛t♦✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ❋❛t♦✉✱ |∇u|p _{∈ L}1_(RN_{)✳ ❚❛♠❜é♠ ♣♦❞❡♠♦s ♠♦str❛r q✉❡✱ ♣❛r❛} ε >0❞❛❞♦✱ ❡①✐st❡ Cε >0 q✉❡ ❞❡♣❡♥❞❡ ❞❡ ε ❡ ❞❡ p t❛❧ q✉❡

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✺

❈♦♠ ❡❢❡✐t♦✱ s❡❥❛♠ A, B ∈ RN ❡ ❝♦♥s✐❞❡r❡ _{F(x) = |x|}p_{, x ∈} RN✳ P❡❧♦ ❚❡♦r❡♠❛ ❋✉♥❞❛✲ ♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦✱

|F(A+B)−F(A)| =

Z 1 0 d

dtF(A+tB)dt

= Z 1 0

∇F(A+tB)·B dt

✭✶✳✽✮ ≤ N X i=1 Z 1 0

_∂x∂F_i(A+tB)

|Bi|dt ≤

N

X

i=1

Z 1

0

p|A+tB|p−1|Bi|dt.

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ d(ε)>0 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ 0≤t ≤1✱

|A+tB|p−1_|B

i| ≤ ε|A+tB|p+d(ε)|Bi|p ≤ ε2p|A|p+ (2pε+d(ε))|B|p.

❙✉❜st✐t✉✐♥❞♦ ❡♠ ✭✶✳✽✮ ♦❜t❡♠♦s ❝♦♥st❛♥t❡sC, k(ε)>0 t❛✐s q✉❡

|A+B|p _{− |A|}p

≤εC|A|p_+k(ε)|B|p_.

❉❛í s❡❣✉❡ ✭✶✳✼✮✳ ❆❣♦r❛ ❡s❝♦❧❤❛A=∇un− ∇u ❡ B =∇u✳ ❊♥tã♦

|∇un|p − |∇un− ∇u|p− |∇u|p

≤ε|∇un− ∇u|p +Cε|∇u|p

♣❛r❛ t♦❞♦ x ∈ RN ❡ _{n ∈} N✳ ❈♦♥s✐❞❡r❛♥❞♦ _{eun = |∇un|}p _{− |∇un− ∇u|}p_{− |∇u|}p✱ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ ✭✶✳✻✮ s❡❣✉❡ ♦s ♠❡s♠♦s ♣❛ss♦s ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r✳

❆♣r❡s❡♥t❛♠♦s ❛❜❛✐①♦ ✉♠❛ ✈❡rsã♦ ✈❡t♦r✐❛❧ ❞♦ ❧❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ q✉❡ ❢♦✐ ♣r♦✈❛❞❛ ♣♦r ❆❧✈❡s ❡♠ ❬✶❪✳

▲❡♠❛ ✶✳✷✷ ❙❡❥❛♠ K ≥ 1✱ s ≥ 2 ❡ A(y) = |y|s−2_{y✱ ♣❛r❛ y ∈ R}K_{✳ ❈♦♥s✐❞❡r❡ ✉♠❛}

s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ✈❡t♦r✐❛✐s ηn : RN →RK t❛❧ q✉❡ (ηn) ⊂ (Ls(RN))K ❡ ηn → 0 q✳t✳♣✳

❡♠ RN✳ ❊♥tã♦✱ s❡ _|ηn|(Ls₍RN₎₎K é ❧✐♠✐t❛❞♦✱ t❡♠♦s

lim

n→∞

Z

RN

|A(ηn) +A(ω)−A(ηn+ω)|s/(s−1) = 0✳

♣❛r❛ ❝❛❞❛ ω ∈(Ls_(RN₎₎K _{✜①❛❞♦✳}

❉❡♠♦♥str❛çã♦✿ P❛r❛ y ∈ RK ❡ _{x ∈} RN ❡ _{1 ≤ i ≤ K}✱ s❡❥❛♠ _Ai(y) ❡ _ωi(x) ❛ _i✲és✐♠❛ ❝♦♠♣♦♥❡♥t❡ ❞♦s ✈❡t♦r❡sA(y) ❡ ω(x)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦

Ai(ηn+ω)−Ai(ηn) =

Z 1

0

d

dtAi(ηn+tω)

dt =

Z 1

0

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✻

❖❜s❡r✈❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ y∈RK ❝♦♠ _{y6= 0} ❡_{1≤j ≤K}✱

∂Ai(y)

∂yj =

(

(s−2)yiyj|y|s−4_{, s❡ j 6=i,}

|y|s−2_{+ (s−2)y}2

i|y|s−4, s❡ j =i.

❈♦♠♦ |yj| ≤ |y|✱ s❡❣✉❡ q✉❡

∂Ai(y)

∂yj

≤(s−1)|y|s−2. ▲♦❣♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C1 >0 t❛❧ q✉❡

|Ai(ηn+ω)−Ai(ηn)| ≤ |ω|

Z 1

0

|∇Ai(ηn+tω)|dt ≤ C1|ω|

Z 1

0

|ηn+tw|s−2_dt

≤ C1|ω|(|ηn|+|ω|)s−2.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡ C2, C3 >0 t❛✐s q✉❡

|Ai(ηn+ω)−Ai(ηn)| ≤C2|ω|s−1+C3|ω||ηn|s−2.

P❛r❛ s = 2✱ ♦ ❧❡♠❛ é ✈❡r❞❛❞❡✐r♦✳ P❛r❛ s > 2✱ 0 < ε < 1 ✜①❛❞♦ ❡ θ = s−2✱ ✉s❛♥❞♦ ❛

❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ ❝♦♠ ❡①♣♦❡♥t❡s s−1

s−2 >1❡ s−1 ♥❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s

|Ai(ηn+ω)−Ai(ηn)| ≤ C2|ω|s−1+

C3

εθ|ω|ε θ_|η

n|s−2

≤ C2|ω|s−1+

C3

εθ

|ω|s−1+εθss−−12|ηn|s−1

▲♦❣♦

|Ai(ηn+ω)−Ai(ηn)| ≤Cεb |ω|s−1_+ε|ηn|s−1_,

♦♥❞❡ Cεb =C2+

C3

εs−2 >0✳ P♦rt❛♥t♦ ❡①✐st❡Cε >0 t❛❧ q✉❡

|A(ηn+ω)−A(ηn)| ≤Cε|ω|s−1+ε|ηn|s−1. ✭✶✳✾✮

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❛ ❢✉♥çã♦Gε,n :RN _→R ❞❡✜♥✐❞❛ ♣♦r

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✼

P♦r ❤✐♣ót❡s❡✱ ηn(x) →0 q✳t✳♣✳ ❡♠ RN✳ ❉❛í✱ ❝♦♠♦ _A é ❝♦♥tí♥✉❛✱ s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦_Gε,n q✉❡

lim

n→∞Gε,n(x) = 0, q✳t✳♣✳ ❡♠ R N_.

❉❡ ✭✶✳✾✮ t❡♠♦s

0≤Gε,n(x)≤(Cε+ 1)|ω|s−1 ∈Ls/(s−1)(RN).

▲♦❣♦ ♣♦❞❡♠♦s ✉s❛r ♦ ❚✳❈✳❉✳▲✳ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡

lim

n→∞

Z

RN

|Gε,n(x)|s/(s−1)_{dx= 0. ✭✶✳✶✵✮}

❚❛♠❜é♠ ❞❛ ❞❡✜♥✐çã♦ ❞❡Gε,n✱ t❡♠♦s

|A(ηn+ω)−A(ηn)−A(ω)| ≤ε|ηn|s−1 +Gε,n.

❈♦♠♦0< ε <1✱ t❡♠♦sεs/(s−1) _{< ε❡ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ✐♠♣❧✐❝❛ q✉❡ ❡①✐st❡C}

5 >0t❛❧ q✉❡

|A(ηn+ω)−A(ηn)−A(ω)|s/(s−1) ≤C5ε|ηn|s+C5|Gε,n|s/(s−1).

P♦r ✭✶✳✶✵✮ ❡ ♣❡❧♦ ❢❛t♦ ❞❡ (ηn)s❡r ❧✐♠✐t❛❞❛ ❡♠ (Ls_(RN₎₎K_{✱ ♦❜t❡♠♦s C}

6 >0t❛❧ q✉❡

lim sup

n→∞

Z

RN

|A(ηn+ω)−A(ηn)−A(ω)|s/(s−1)_{dx ≤C}

5εlim sup

n→∞

Z

RN

|ηn|s_dx≤C

6ε.

❋❛③❡♥❞♦ε →0✱ s❡❣✉❡ q✉❡

0 ≤ lim inf

n→∞

Z

RN

|A(ηn+ω)−A(ηn)−A(ω)|s/(s−1)_dx

≤ lim sup

n→∞

Z

RN

|A(ηn+ω)−A(ηn)−A(ω)|s/(s−1)dx≤0,

♦ q✉❡ ♠♦str❛ ♦ ❧❡♠❛✳

✶✳✻ ❯♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛

❆q✉✐ ❛♣r❡s❡♥t❛♠♦s ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ ❡s♣❛ç♦s Lp _{❝✉❥❛ ❞❡♠♦♥str❛çã♦}

❡♥❝♦♥tr❛✲s❡ ❡♠ ❬✷✶❪ ✭✈❡❥❛ ▲❡♠❛ ✹✳✽✮✳

▲❡♠❛ ✶✳✷✸ ❙❡❥❛♠ Ω ⊂ RN ✉♠ ❛❜❡rt♦ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧✐♠✐t❛❞♦✱ _{1 < p < ∞} ❡

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✽

f✳ ❊♥tã♦ fn ⇀ f ❢r❛❝❛♠❡♥t❡ ❡♠ Lp_(Ω)✳

❉❡♠♦♥str❛çã♦✿ P❡❧♦ ▲❡♠❛ ❞❡ ❋❛t♦✉✱ f ∈ Lp_{(Ω)✳ ❈♦♠♦ (fn) é ❧✐♠✐t❛❞❛ ❡♠ L}p_{(Ω) ❡} Lp_{(Ω) é r❡✢❡①✐✈♦✱ ❡①✐st❡ g ∈L}p_{(Ω) t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ fn ⇀ g ❢r❛❝❛♠❡♥t❡}

❡♠ Lp_{(Ω)✳ ❱❛♠♦s ♠♦str❛r q✉❡ g(x) = f(x) q✳t✳♣✳ ❡♠ Ω✳ ❈♦♥s✐❞❡r❛♠♦s ♣❛r❛ j ∈ N ♦}

❝♦♥❥✉♥t♦

Ωj :={x∈Ω :|fn(x)−f(x)| ≤1, ♣❛r❛ t♦❞♦n ≥j}.

❋✐①❛❞♦ j ≥ 1✱ s❡❥❛ ϕj ∈ C∞

0 (Ωj)✳ P♦r ❤✐♣ót❡s❡✱ fn(x) → f(x) q✳t✳♣✳ ❡♠ Ω✳ P❛r❛ n ≥ j✱

t❡♠✲s❡

|fn(x)ϕj(x)| ≤ C|fn(x)| ≤ C(|fn(x)−f(x)|+|f(x)|)

≤ C+C|f(x)| := h(x)

♣❛r❛ t♦❞♦x∈Kj :=s✉♣♣ϕj ⊂Ωj ❡ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡C >0✳ ❈♦♠♦ Kj é ❝♦♠♣❛❝t♦

❡ f ∈ Lp_{(Ω)✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ h ∈ L}1_{(Kj)✳ P❡❧♦}

❚✳❈✳❉✳▲✳✱

lim

n→∞

Z

Ω

fn(x)ϕj(x) = lim n→∞

Z

Kj

fn(x)ϕj(x) =

Z

Kj

f(x)ϕj(x) =

Z

Ω

f(x)ϕj(x). ✭✶✳✶✶✮

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ϕj ∈ Lp′

(Ω)✱ T : Lp_{(Ω) → R ❞❡✜♥✐❞❛ ♣♦r T(f) =} R

Ωf ϕj é ❧✐♥❡❛r

❝♦♥tí♥✉❛✳ P❡❧❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛ fn⇀ g ❡♠ Lp_{(Ω) s❡❣✉❡ q✉❡ T(fn)→T(g)✱ ✐st♦ é✱}

lim

n→∞

Z

Ω

fn(x)ϕj(x) =

Z

Ω

g(x)ϕj(x). ✭✶✳✶✷✮

❈♦♠❜✐♥❛♥❞♦ ✭✶✳✶✶✮ ❡ ✭✶✳✶✷✮ ♦❜t❡♠♦s Z

Ωj

(g(x)−f(x))ϕj(x) = 0, ♣❛r❛ t♦❞❛ ϕj ∈C₀∞(Ωj).

P❡❧♦ ▲❡♠❛ ✹✳✷✹ ❡♠ ❬✾❪✱

g(x) =f(x) q✳t✳♣✳ ❡♠ Ωj ♣❛r❛ t♦❞♦j ∈N.

❉❛í✱ ♣❛r❛ ❝❛❞❛ j ∈ N✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ _{Zj ⊂ Ωj} ❞❡ ♠❡❞✐❞❛ ♥✉❧❛ t❛❧ q✉❡ _{g(x) = f(x)} ♣❛r❛ t♦❞♦ x ∈ Ωj \Zj✳ ❈♦♥s✐❞❡r❡ A = ∪∞j=1Ωj ❡ N = ∪∞j=1Zj✳ ❉❛❞♦ x ∈ A\N t❡♠✲s❡

x∈Ωj \Zj ♣❛r❛ ❛❧❣✉♠ j ∈N ❡✱ ♣♦rt❛♥t♦✱ g(x) = f(x)✳ ▼♦str❛♠♦s ❝♦♠ ✐ss♦ q✉❡

❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✾

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ fn(x) → f(x) q✳t✳♣✳ ❡♠ Ω✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ Z ⊂ Ω ❞❡ ♠❡❞✐❞❛

♥✉❧❛ t❛❧ q✉❡ fn(x) → f(x) s❡❥❛ q✉❛❧ ❢♦r x ∈ Ω\Z✳ ■ss♦ ❞✐③ q✉❡ ♣❛r❛ ❝❛❞❛ x ∈ Ω\Z✱

❡①✐st❡ j0 ∈Nt❛❧ q✉❡

|fn(x)−f(x)| ≤1 s❡♠♣r❡ q✉❡ n≥j0.

❆ss✐♠ x∈Ωj0 ⊂A✳ ❈♦♠ ✐ss♦✱ Ω\Z ⊂ A ❡✱ ❞❛í✱ Ω\(Z ∪N)⊂ A\N✳ ❙❡❣✉❡ ❞❡ ✭✶✳✶✸✮

q✉❡

g(x) =f(x) ♣❛r❛ t♦❞♦ x∈Ω\(Z∪N).

❈♦♠♦ Z∪N t❡♠ ♠❡❞✐❞❛ ♥✉❧❛✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

✶✳✼ ❚❡♦r❡♠❛ ❞❡ ❡①t❡♥sã♦ ❞❡ ❉✉❣✉♥❞❥✐

❆ s❡❣✉✐r ❡♥✉♥❝✐❛♠♦s ✉♠ t❡♦r❡♠❛ q✉❡ tr❛t❛ ❞❛ ❡①t❡♥sã♦ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✱ ❞❡✈✐❞♦ ❛ ❉✉❣✉♥❞❥✐✳ ❙✉❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✹❪ ✭❚❡♦r❡♠❛ ✻✳✶✮✳

❉❡✜♥✐çã♦ ✶✳✷✹ ❙❡❥❛ L ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❊♥tã♦ L é ✉♠ ❡s♣❛ç♦ ❛✜♠ ❞♦ t✐♣♦ ♠ s❡✱

♣❛r❛ ❝❛❞❛ ❡s♣❛ç♦ ♠étr✐❝♦ X ❡ ♣❛r❛ t♦❞❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ f : X → L✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡

♣r♦♣r✐❡❞❛❞❡✿ ♣❛r❛ t♦❞♦ x ∈ X ❡ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ W ⊃ f(x)✱ ❡①✐st❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛

U ⊃x ❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ C ⊂L t❛❧ q✉❡ f(U)⊂C⊂W✳

❚❡♦r❡♠❛ ✶✳✷✺ ✭❉✉❣✉♥❞❥✐✮ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ A ⊂ X ✉♠ s✉❜❝♦♥❥✉♥t♦ ❢❡✲

❝❤❛❞♦ ❡ L✉♠ ❡s♣❛ç♦ ❛✜♠ ❞♦ t✐♣♦ m✳ ❊♥tã♦✱ ❝❛❞❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛f :A→L ♣♦ss✉✐ ✉♠❛

❈❛♣ít✉❧♦ ✷

❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇

✷✳✶ ❈♦♥s✐❞❡r❛çõ❡s ✐♥✐❝✐❛✐s

❊st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❡st✉❞❛r ❛ ❡①✐stê♥❝✐❛ ❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s s♦❧✉çõ❡s ❞❛

❡q✉❛çã♦ _

 

−∆pu+ (λa(x) + 1)|u|p−2_u=|u|q−2_{u, x∈R}N_,

u∈W1,p_(RN_), (Sλ,q)

❡♠ q✉❡ 2≤p < N, p < q < p∗_✱∆

pu= ❞✐✈(|∇p−2u|∇u) é ♦ ♦♣❡r❛❞♦r p✲▲❛♣❧❛❝✐❛♥♦ ❡λ é

✉♠ ♣❛râ♠❡tr♦ ♣♦s✐t✐✈♦✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛ra ✉♠❛ ❢✉♥çã♦ s❛t✐s❢❛③❡♥❞♦

✭A1✮ a∈C(RN,R)é ♥ã♦ ♥❡❣❛t✐✈❛✱ Ω =✐♥ta−1(0) é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❝♦♠ ❢r♦♥t❡✐r❛

s✉❛✈❡ ❡ Ω =a−1_(0)❀

✭A2✮ ❡①✐st❡ M0 >0t❛❧ q✉❡

L({x∈RN _{: a(x)≤M0})<∞}✱ ♦♥❞❡ L é ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡♠RN✳

❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥❛t✉r❛❧ ❡♠ q✉❡ tr❛❜❛❧❤❛♠♦s é

E =

u∈W1,p(RN_{) :} Z

RN

a(x)|u|p _<∞

.

P❛r❛ λ≥0✱ ❞❡✜♥✐♠♦s Eλ = (E,k · kλ) ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧E ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛

kukλ =

Z

RN

(|∇u|p_{+ (λa(x) + 1)|u|}p_)dx

❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✶

❉✉r❛♥t❡ t♦❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❞❡♥♦t❛♠♦s ♣♦rk·k❛ ♥♦r♠❛ ❡♠W1,p_(RN₎

❞❛❞❛ ♣♦r

kuk=

Z

RN

(|∇u|p+|u|p)

1/p .

❉❛❞♦s 1 ≤ s < ∞ ❡ D ⊂ RN✱ ✐♥❞✐❝❛♠♦s ♣♦r _{| · |s,D} ❛ ✉s✉❛❧ ❞❡_Ls_(D)✳ P❛r❛ s✐♠♣❧✐✜❝❛r✱ q✉❛♥❞♦ D=RN✱ ❡s❝r❡✈❡♠♦s_|u|s ❛♦ ✐♥✈és ❞❡ _|u|s,RN✳

❖ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛(Sλ,q) é Iλ,qe :Eλ →R ❞❡✜♥✐❞♦ ♣♦r

e

Iλ,q(u) = 1

p

Z

RN

(|∇u|p _{+ (λa(x) + 1)|u|}p₎₋ 1 q

Z

RN

|u|q ₌ 1 pkuk

p λ−

1

q|u| q q.

➱ ♣♦ssí✈❡❧ ♠♦str❛r ✭✈❡❥❛ Pr♦♣♦s✐çã♦ ✶✳✶✷ ❡♠ ❬✸✸❪✮ q✉❡Iλ,qe ∈C1_{(Eλ,R) ❝♦♠}

hIeλ,q′ (u), vi=

Z

RN

|∇u|p−2_{∇u· ∇v+ (λa(x) + 1)|u|}p−2_uv−

Z

RN

|u|q−2_uv,

♣❛r❛ t♦❞♦u, v ∈Eλ✳ ❉✐③❡♠♦s q✉❡ u∈Eλ é s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (Sλ,q) q✉❛♥❞♦

hIe_λ,q′ (u), vi= 0, ♣❛r❛ t♦❞♦ v ∈Eλ.

❯♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (Sλ,q) é ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ q✉❛♥❞♦

e

Iλ,q(u) = inf

n e

Iλ,q(v) :v é ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ (Sλ,q)

o

.

❉❡✜♥✐♠♦s ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧Ieλ,q ♣♦r

e

Nλ,q =

n

u∈Eλ \ {0}:hIe′

λ,q(u), ui= 0

o

=u∈Eλ\ {0}:kukp_λ =|u|q q .

❖❜s❡r✈❛♠♦s q✉❡Neλ,q ❝♦♥té♠ t♦❞❛s ❛s s♦❧✉çõ❡s ❢r❛❝❛s ❞❡(Sλ,q)✳ ◗✉❛♥❞♦ u∈Neλ,q t❡♠✲s❡

e

Iλ,q(u) =

1

p−

1

q

kukp_λ,

❡✱ ♣♦rt❛♥t♦✱ ♦ ❢✉♥❝✐♦♥❛❧ Iλ,qe é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ Neλ,q✳ ❆ss✐♠✱ é ✜♥✐t♦ ♦ ♥ú♠❡r♦

e

cλ,q = inf u∈Neλ,q

e

Iλ,q(u).