❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
❯♠ Pr♦❜❧❡♠❛ ❊❧í♣t✐❝♦ ♥♦
R
N
❆ss✐♥t♦t✐❝❛♠❡♥t❡ ▲✐♥❡❛r ❡ ❆✉tô♥♦♠♦ ♥♦
■♥✜♥✐t♦
♣♦r
▼❛②r❛ ❙♦❛r❡s ❞❛ ❙✐❧✈❛ ❈♦st❛
❇r❛sí❧✐❛
Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)
CC837p
Costa, Mayra Soares da Silva
Um Problema Elíptico no R^N Assintoticamente Linear e Autônomo no Infinito / Mayra Soares da Silva Costa; orientador Ricardo Ruviaro. --Brasília, 2016.
83 p.
Dissertação (Mestrado - Mestrado em Matemática) --Universidade de Brasília, 2016.
❆❣r❛❞❡❝✐♠❡♥t♦s
❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❡✉ r❡♥❞♦ ❣r❛ç❛s ❛♦ ♠❡✉ ❙❛♥t♦ ❉❡✉s q✉❡ ❛ ❝❛❞❛ ❞✐❛ ♠❡ s✉r♣r❡❡♥❞❡ ♠❛✐s✱ s❡♠♣r❡ ♠❡ ❡♥s✐♥❛♥❞♦ q✉❡ ❡✉ ♣r❡❝✐s♦ ❝♦♥✜❛r q✉❡ t✉❞♦ ❊❧❡ ✈❛✐ ♣r♦✈❡r✱ ♠❡s♠♦ q✉❛♥❞♦ ♥ã♦ ❡st♦✉ ❝♦♠♣r❡❡♥❞❡♥❞♦ ❛s ❝✐r❝✉♥stâ♥❝✐❛s q✉❡ ♠❡ ❝❡r❝❛♠✳ ❊✉ ❥❛♠❛✐s ❝❤❡❣❛r✐❛ ❛té ❛q✉✐ s❡ ❊❧❡ ♥ã♦ ❡st✐✈❡ss❡ ❛ ♠❡ ❣✉✐❛r✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ♠✐♥❤❛ q✉❡r✐❞❛ ♠❛♠ã❡ ❘♦s❛♥❣❡❧❛ ❙♦❛r❡s✱ q✉❡ ♠❡ ❡st❡✈❡ ❛ ❛❝♦♥s❡❧❤❛r ❞✉r❛♥t❡ t♦❞♦ ❡ss❡ t❡♠♣♦ ❞❡ ❞❡❞✐❝❛çã♦ ❛♦s ❡st✉❞♦s✳ ❚❛♠❜é♠ ❛ ♠❡✉s ❛✈ós ♠❛t❡r♥♦s ▼❛r✐❛ ❞❡ ▲♦✉r❞❡s ❡ ❏♦ã♦ ❙♦❛r❡s q✉❡ ♥ã♦ tê♠ ♣♦✉♣❛❞♦ ❡s❢♦rç♦s ❛ ♠❡ ❛♣♦✐❛r ♥❡ss❡s ú❧t✐♠♦s ❞✐❛s ❞❡ ❞✐✜❝✉❧❞❛❞❡s✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❛❣r❛❞❡ç♦✱ ✐♥❝❛♥s❛✈❡❧♠❡♥t❡✱ ♣❡❧❛s ❞✐✈❡rs❛s ✈❡③❡s ♥❛s q✉❛✐s s❡ ❞✐s♣ôs ❛ ♠❡ ♦r✐❡♥t❛r ❡ ❛✉①✐❧✐❛r✱ ♥ã♦ ❛♣❡♥❛s ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦✱ ♠❛s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞❡s❞❡ ❛ ❣r❛❞✉❛çã♦✳ ❉❡✈♦ ♠✉✐t♦ ❛♦s ❡s❢♦rç♦s ❞❡❧❡✳
❊st❡♥❞♦ ♠❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛♦s ❞❡♠❛✐s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ♣❡❧♦s q✉❛✐s ♥✉tr♦ ❣r❛♥❞❡ ❛❞♠✐r❛çã♦✳ ▼✉✐t♦s ❞❡❧❡s ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✱ ❡ ♣❡❧♦ tr❛❜❛❧❤♦ ár❞✉♦ ❞❡ss❛ ❡q✉✐♣❡ tã♦ ❡✜❝✐❡♥t❡✱ ♠❡ t♦r♥❡✐ ✉♠❛ ♣r♦✜ss✐♦♥❛❧ ♠❛✐s q✉❛❧✐✜❝❛❞❛✳ ❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛♦s ❞❡♠❛✐s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ q✉❡ ♠✉✐t❛s ✈❡③❡s ❛❣✐r❛♠ ❛ ♠❡✉ ❢❛✈♦r✳ ❆✐♥❞❛ q✉❡r♦ ❣r❛❞❡❝❡r à ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ♥♦ ❞❡❝♦rr❡r ❞♦ ♠❡✉ ❝✉rs♦ ❞❡ ♠❡str❛❞♦✱ ❡ ♥ã♦ ♣♦❞❡r✐❛ ❞❡✐①❛r ❞❡ ♠❡♥❝✐♦♥❛r ♠✐♥❤❛ ❣r❛t✐❞ã♦ ❛♦ P■❈▼❊ q✉❡ ❢♦♠❡♥t♦✉ ♠✐♥❤❛ ♣❛rt✐❝✐♣❛çã♦ ♥❡ss❡ ♣r♦✲ ❣r❛♠❛ ❞❡ ❜♦❧s❛s ♣❛r❛ ❛❧✉♥♦s ♠❡❞❛❧❤✐st❛s ❞❛s ❖❧✐♠♣í❛❞❛s ❇r❛s✐❧❡✐r❛s ❞❡ ▼❛t❡♠át✐❝❛ ❞❛s ❊s❝♦❧❛s Pú❜❧✐❝❛s ✭❖❇▼❊P✮✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s q✉❡ t♦r❝❡r❛♠ ♣♦r ♠✐♠✱ ❡ ❛❝r❡❞✐t❛r❛♠ q✉❡ ❡✉ s❡r✐❛ ❝❛♣❛③ ❞❡ ❛❧❝❛♥ç❛r t❛❧ ♦❜❥❡t✐✈♦✳ ❊s♣❡❝✐❛❧♠❡♥t❡ àq✉❡❧❡s q✉❡ ♦r❛r❛♠ ♣♦r ♠✐♠✱ ❡ ❡st✐✈❡r❛♠ ♣r♦♥t♦s ❛ ♠❡ ❡s❝✉t❛r ♥♦s ❞✐❛s ❞✐❢í❝❡✐s✳
❊♥✜♠✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❡st✐✈❡r❛♠ ♣r❡s❡♥t❡s ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❞✉r❛♥t❡ ❡ss❡ ♣❡rí♦❞♦✱ ❡ q✉❡ ❞❡ ❛❧❣✉♠ ♠♦❞♦ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ q✉❡ ❡ss❡ tr❛❜❛❧❤♦ ❢♦ss❡ ❝♦♥❝r❡t✐③❛❞♦✳ ❊✉ ❧♦✉✈♦ ❛♦ ♠❡✉ ❙❡♥❤♦r ❡ ❙❛❧✈❛❞♦r ❏❡s✉s ❈r✐st♦ ♣♦r ❝❤❡❣❛r ❛té ❛q✉✐✱ ❡ ♣♦r t♦❞♦s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡ss❛ ❤✐stór✐❛✳
❉❡❞✐❝❛tór✐❛
❆ ▼❡✉ ❙❡♥❤♦r ❡ ❙❛❧✈❛❞♦r ❏❡s✉s ❈r✐st♦✱ q✉❡ ♠❡ ❡r❣✉❡✉ ❡♠ ♠♦♠❡♥t♦s ♥♦s q✉❛✐s ❡✉ ❥❛♠❛✐s ❝♦♥s❡❣✉✐r✐❛ ♠❡ ❧❡✈❛♥t❛r s♦③✐♥❤❛✳
❘❡s✉♠♦
◆❡ss❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ s♦❜r❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❛✉tô♥♦♠♦s✱ ♣❛r❛ N ≥2 :
−∆u=h(u), ❡♠ RN,
❡♠ q✉❡ hs❛t✐s❢❛③ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s ❡s♣❡❝í✜❝❛s✳ ❊♠ s❡❣✉✐❞❛✱ t❛♠❜é♠ ♣❛r❛ N ≥2✱ ❢❛③❡♠♦s ✉♠ ❡st✉❞♦
❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿
−∆u+V(x)u=f(u), ❡♠ RN,
❡♠ q✉❡ f é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r✱ ❡ s❛t✐s❢❛③✱ ❛ss✐♠ ❝♦♠♦ ♦ ♣♦t❡♥❝✐❛❧V✱ ❝❡rt❛s ❝♦♥❞✐çõ❡s ♣r❡✈✐❛♠❡♥t❡
❡st❛❜❡❧❡❝✐❞❛s✳ ◆♦ss❛ ✜♥❛❧✐❞❛❞❡ é✱ ♣♦r ♠❡✐♦ ❞❡ té❝♥✐❝❛s ✈❛r✐❛❝✐♦♥❛✐s✱ ♦❜t❡r ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❡ ✉♠❛ s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞❀ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐❀ ●❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛❀ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③❛❡✈✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦✱ ✇❡ ♣r❡s❡♥t ❛ st✉❞② ❛❜♦✉t t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ▼♦✉♥t❛✐♥ P❛ss ❧❡✈❡❧ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛ss ♦❢ ❛✉t♦♥♦♠♦✉s ♣r♦❜❧❡♠s✱ ✇❤❡♥ N≥2 :
−∆u=h(u), ❡♠ RN,
✇❤❡r❡ hs❛t✐s✜❡s s♦♠❡ s♣❡❝✐✜❝ ❤②♣♦t❤❡s✐s✳ ❆❢t❡r t❤❛t✱ ❛❧s♦ ❢♦r N≥2✱ ✇❡ st✉❞② t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿
−∆u+V(x)u=f(u), ❡♠RN,
✇❤❡r❡ f ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❧✐♥❡❛r ❛♥❞ s❛t✐s✜❡s✱ ❛s ✇❡❧❧ ❛s t❤❡ ♣♦t❡♥t✐❛❧V✱ ❝❡rt❛✐♥ ♣r❡✈✐♦✉s❧② ❡st❛❜❧✐s❤❡❞
❝♦♥❞✐t✐♦♥s✳ ❖✉r ♣✉r♣♦s❡ ✐s ✉s✐♥❣ ✈❛r✐❛t✐♦♥❛❧ t❡❝❤♥✐q✉❡s t♦ ❣❡t ❛ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥ ❛♥❞ ❛ ❧❡❛st ❡♥❡r❣② s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠✳
❑❡② ✇♦r❞s✿ ❊❦❡❧❛♥❞ ❱❛r✐❛t✐♦♥❛❧ Pr✐♥❝✐♣❧❡❀ ❈❡r❛♠✐ ❙❡q✉❡♥❝❡❀ ▼♦✉♥t❛✐♥ P❛ss ●❡♦♠❡tr②❀ P♦❤♦③❛❡✈ ■❞❡♥t✐t②✳
◆♦t❛çã♦
BR ❜♦❧❛ ❛❜❡rt❛ ❝❡♥tr❛❞❛ ❡♠ ③❡r♦ ❝♦♠ r❛✐♦R❀
BR(x) ❜♦❧❛ ❛❜❡rt❛ ❝❡♥tr❛❞❛ ❡♠x❝♦♠ r❛✐♦ R❀
BR[x] ❜♦❧❛ ❢❡❝❤❛❞❛ ❝❡♥tr❛❞❛ ❡♠x❝♦♠ r❛✐♦ R❀
un →u ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ✭❡♠ ♥♦r♠❛✮❀
un ⇀ u ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛❀
un →u,q✳t✳♣✳ ❡♠Ω ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦x❞❡Ω❀
Dαu= ∂ku
∂xa11 ...∂xaN
N
❞❡r✐✈❛❞❛ ❢r❛❝❛ ❝♦♠ ♠✉❧t✐✲í♥❞✐❝❡
α= (a1, ..., aN),❡♠ q✉❡ N
X
i=1 ai=k❀
∇u=
∂u
∂x1
, ..., ∂u ∂xN
❣r❛❞✐❡♥t❡ ❞❡u❀
∂u
∂η =η· ∇u ❞❡r✐✈❛❞❛ ♥♦r♠❛❧ ❡①t❡r✐♦r❀
∆u=
N
X
i=1 ∂2u ∂x2
i
❧❛♣❧❛❝✐❛♥♦ ❞❡ ✉❀
ω⊂⊂Ω ω¯ é ❝♦♠♣❛❝t♦ ❡ ❡stá ❝♦♥t✐❞♦ ❡♠Ω❀
|Ω| ♠❡❞✐❞❛ ❞❡Ω❀
¯
Ω ❢❡❝❤♦ ❞❡Ω❀
∂Ω ❢r♦♥t❡✐r❛ ❞❡Ω❀
❞✐❛♠(Ω) ❞✐â♠❡tr♦ ❞❡Ω❀
p′ = p
p−1 ❝♦♥❥✉❣❛❞♦ ❞♦ ❡①♣♦❡♥t❡ ❤♦❧❞❡r✐❛♥♦p❀
f =o(g)✱ q✉❛♥❞♦x→x0 lim
x→x0 |f(x)|
g(x) = 0❀
s✉♣♣f s✉♣♦rt❡ ❞❛ ❢✉♥çã♦f❀
C(X, Y) ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❞❡X ❡♠Y❀
C1(X, Y) ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♥t✐♥✉❛♠❡♥t❡
❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡X ❡♠Y❀
X′ ❡s♣❛ç♦ ❞✉❛❧ ❞❡X❀
Lp(Ω) ❡s♣❛ç♦ ❞❡ ▲❡❜❡s❣✉❡ ❞❛s ❢✉♥çõ❡sp✲✐♥t❡❣rá✈❡✐s❀
Lp❧♦❝(RN) Lp❧♦❝(Ω) ={u∈Lp(Ω′), ∀Ω′⊂⊂Ω}❀
Wk,p(Ω) Wk,p(Ω) ={u∈Lp(Ω) :Dαu∈Lp(Ω), ∀ |α| ≤k}❀
H1(Ω) ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈W1,2(Ω)❀
H−1(Ω) ❡s♣❛ç♦ ❞✉❛❧ ❞❡H1(Ω)❀
H2(Ω) ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈W2,2(Ω)❀
H2
❧♦❝(Ω) W❧♦❝2,2(Ω) ={u∈W2,2(Ω′), ∀Ω′⊂⊂Ω}❀
kukH1 =
k∇uk2 2+kuk22
1/2
♥♦r♠❛ ✉s✉❛❧ ❞❡H1(RN)❀ kuk=k∇uk2
2+kV(x)uk22 1/2
♥♦r♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛H1(RN)❀
kukp=
Z
RN|
u|pdx
1/p
, ∀p∈[1,+∞) ♥♦r♠❛ ✉s✉❛❧ ❞❡Lp(RN)❀
kukLp(Ω)=
Z
Ω| u|pdx
1/p
, ∀p∈[1,+∞) ♥♦r♠❛ ✉s✉❛❧ ❞❡Lp(Ω), p∈[1,∞)❀
kuk∞= sup
x∈RN
❡ss|u(x)| ♥♦r♠❛ ✉s✉❛❧ ❞❡L∞(RN)❀
| · | ♥♦r♠❛ ❞♦RN✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❖ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞ ✹
✶✳✶ ❙❡q✉ê♥❝✐❛s P❛❧❛✐s✲❙♠❛❧❡ ❡ ❙❡q✉ê♥❝✐❛s ❞❡ ❈❡r❛♠✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞❡ ❊♥❡r❣✐❛ ▼í♥✐♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷ ❯♠ Pr♦❜❧❡♠❛ ❆✉tô♥♦♠♦ ♥♦ RN ✶✹
✷✳✶ ❆ ●❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❆ ❊①✐stê♥❝✐❛ ❞❡ ✉♠ ❈❛♠✐♥❤♦γ∈Γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸ ❯♠❛ ❈❛r❛❝t❡r✐③❛çã♦ ❞♦ ◆í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸ ❯♠❛ ❙♦❧✉çã♦ P♦s✐t✐✈❛ ♣❛r❛ ✉♠ Pr♦❜❧❡♠❛ ❆ss✐♥t♦t✐❝❛♠❡♥t❡ ▲✐♥❡❛r ❡ ❆✉tô♥♦♠♦ ♥♦
■♥✜♥✐t♦ ✸✺
✸✳✶ ❆ ●❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷ ❆ ▲✐♠✐t❛çã♦ ❞❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✸ ❯♠ P♦♥t♦ ❈rít✐❝♦ ◆ã♦✲❚r✐✈✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✹ ❯♠❛ ❙♦❧✉çã♦ ❞❡ ❊♥❡r❣✐❛ ▼í♥✐♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
❆ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞♦s ❋✉♥❝✐♦♥❛✐s ❏ ❡ ■ ✻✹
❇ ❘❡s✉❧t❛❞♦s ■♠♣♦rt❛♥t❡s ✼✺
❇✳✶ ■♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ❇✳✷ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③❛❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ❇✳✸ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ❇✳✹ ❚❡♦r❡♠❛ ❞❡ ❚♦♥❡❧❧✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ❇✳✺ ❋✉♥çõ❡s ❘❡❣✉❧❛r✐③❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ❇✳✻ ❈❛r❛❝t❡r✐③❛çã♦ ❊s♣❡❝tr❛❧ ❞❡ ✉♠ ❖♣❡r❛❞♦r ❆✉t♦❛❞❥✉♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽
❇✳✼ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ❇✳✽ ▲❡♠❛ ❞❡ ▲✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ❇✳✾ ❚❡♦r❡♠❛ ❞❡ ❱❛✐♥❜❡r❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ❇✳✶✵ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ❋♦rt❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ❇✳✶✶ ▲❡♠❛ ❞❡ ❋❛t♦✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶
■♥tr♦❞✉çã♦
❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ♦ r❡s✉❧t❛❞♦ ❞❡✈✐❞♦ ❛ ❏❡❛♥❥❡❛♥ ❡ ❚❛♥❛❦❛ ❬✶✻❪ q✉❡✱ s♦❜ ❝❡rt❛s ❝♦♥❞✐çõ❡s✱ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛u∈H1(RN)♣❛r❛ ♦ ♣r♦❜❧❡♠❛
−∆u+V(x)u=f(u), ❡♠RN, ✭✶✮
❝✉❥❛s ♣r✐♥❝✐♣❛✐s ✈❛♥t❛❣❡♥s ❡stã♦ ❡♠ ❛ss✉♠✐r ❤✐♣ót❡s❡s q✉❡ t♦r♥❛♠ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r ❡ ♦ ♣r♦❜❧❡♠❛ ❛ss♦❝✐❛❞♦ ❛♦ ✏✐♥✜♥✐t♦✑ ❛✉tô♥♦♠♦✳ ❚r❛❜❛❧❤❛♠♦s ♣♦r ♠❡✐♦ ❞❡ ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s✱ ✐st♦ é✱ ❛ss♦❝✐❛♥❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭✶✮ ♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ♥❛t✉r❛❧I:H1(RN)→R❞❡✜♥✐❞♦ ♣♦r
I(u) =1 2
Z
RN
|∇u|2+V(x)u2dx−
Z
RN
F(u)dx,
❡♠ q✉❡ F(u) =
Z u
0
f(s)ds✱ ❡ ❝♦♠ ✐ss♦✱ ❛ ✜♠ ❞❡ ♦❜t❡r s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ✭✶✮✱ ♦ ♦❜❥❡t✐✈♦ é ❡♥❝♦♥tr❛r ✉♠
♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ I✳ P❛r❛ t❛♥t♦✱ s♦❜ ❛s ❤✐♣ót❡s❡s ❞♦ ♣r♦❜❧❡♠❛✱ ♠♦str❛♠♦s q✉❡I ♣♦ss✉✐ ❛
❣❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛❀ q✉❡ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ I✱ q✉❡ é ❧✐♠✐t❛❞❛ ❡♠ H1(RN)❀ ❡ q✉❡ t❛❧ s❡q✉ê♥❝✐❛ ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ✉♠
♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ I✳ ❉❡ss❡ ♠♦❞♦ ♥♦ss♦s ♠❛✐♦r❡s ❞❡s❛✜♦s sã♦ ♠♦str❛r ❛ ❧✐♠✐t❛çã♦ ❞❛ s❡q✉ê♥❝✐❛
❞❡ ❈❡r❛♠✐ ❡ ❣❛r❛♥t✐r ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ♣❛r❛ ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧✳
❆ ♥♦ss❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ♣r♦✈❛r ❛ ❧✐♠✐t❛çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ♦ ❢❛t♦ ❞❡ ❝♦♥s✐❞❡r❛r♠♦s ✉♠ ♣r♦❜❧❡♠❛ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r✳ ●❡r❛❧♠❡♥t❡✱ ♣❛r❛ ❣❛r❛♥t✐r ❛ ❧✐♠✐t❛çã♦ ❞❛ s❡q✉ê♥✲ ❝✐❛ ❞❡ ❈❡r❛♠✐✱ ❛ ♠❛✐♦r✐❛ ❞♦s ❛✉t♦r❡s ❛ss✉♠❡ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ ❞❡ s✉♣❡r❧✐♥❡❛r✐❞❛❞❡ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❆♠❜r♦s❡tt✐ ❡ ❘❛❜✐♥♦✇✐t③ ❬✸❪✿
∃µ >2 : 0< µF(s)≤f(s)s, ♣❛r❛ t♦❞♦ s >0. ✭✷✮
❖❜s❡r✈❡ q✉❡ ❛ ❝♦♥❞✐çã♦ ❞❛❞❛ ❡♠ ✭✷✮ ✐♠♣❧✐❝❛ ❡♠
lim inf
s→+∞
f(s)
sµ−1 >0,
❝♦♥t✉❞♦✱ t❛❧ ❤✐♣ót❡s❡ ❞❡ ❝r❡s❝✐♠❡♥t♦ é ❝♦♥trár✐❛ àq✉❡❧❛s ❝♦♠ ❛s q✉❛✐s tr❛❜❛❧❤❛♠♦s✳
■♥tr♦❞✉çã♦ ✷
❡ss❛ ❝♦♥❞✐çã♦✱ ❞❡ ❝❡rt♦ ♠♦❞♦✱ ♦ ♣r♦❜❧❡♠❛ ✜❝❛ ❞❡✜♥✐❞♦ ❡♠ R✱ ♦ q✉❡ ❣❛r❛♥t❡ ❛ ❝♦♠♣❛❝✐❞❛❞❡✳ ❖ ❡st✉❞♦
♠❛✐s ♣ró①✐♠♦ ❛♦ q✉❡ s❡rá ❞❡s❡♥✈♦❧✈✐❞♦ ♥❡ss❡ tr❛❜❛❧❤♦✱ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ♣♦r ❏❡❛♥❥❡❛♥ ❬✶✺❪ ❡ s❡ tr❛t❛ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❛ ❢♦r♠❛
∆u+Ku=f(x, u), ❡♠ RN,
❡♠ q✉❡ K >0 é ✉♠❛ ❝♦♥st❛♥t❡ ❡f(x, s)é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r ❡♠s❡ ♣❡r✐ó❞✐❝❛ ❡♠x∈RN✳ ❉❡♣♦✐s
❞✐ss♦✱ ✈❛❧❡ ♠❡♥❝✐♦♥❛r q✉❡ ❢❛③❡♥❞♦ ✉s♦ ❞❡ ❛❧❣✉♠❛s té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ❡♠ ❬✶✺❪✱ ❙t✉❛rt ❡ ❩❤♦✉ ❛♣r❡s❡♥t❛r❛♠ ✉♠ ❡st✉❞♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ s♦❜r❡ ♣r♦❜❧❡♠❛s r❛❞✐❛❧♠❡♥t❡ s✐♠étr✐❝♦s ♥♦ RN ✭❝❢✳ ❬✷✹❪✮✳ ❊ t❛♠❜é♠ ♥ã♦
♣♦❞❡♠♦s ❞❡✐①❛r ❞❡ ❝✐t❛r ❝♦♠♦ ✐♥s♣✐r❛çã♦ ♦ tr❛❜❛❧❤♦ ❡♠ ❬✷✺❪✱ ❞❡✈✐❞♦ ❛ ❙③✉❧❦✐♥ ❡ ❩♦✉✱ q✉❡ tr❛t❛ s✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♠ ✉♠❛ ♣❛rt❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r✳
❙❡❣✉✐♥❞♦ ❛ ❧✐♥❤❛ ❞❡ r❛❝✐♦❝í♥✐♦ ❞❡s❡♥✈♦❧✈✐❞❛ ❡♠ ❬✶✺❪✱ ❛ ❧✐♠✐t❛çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ❞❡♠♦♥s✲ tr❛❞❛ ♥❡ss❡ tr❛❜❛❧❤♦ é ❜❛s❡❛❞❛ ❡♠ ❬✶✽❪✳ ❈♦♥t✉❞♦✱ ❞❡ss❛ ✈❡③ ❞❡✈✐❞♦ à ❡str✉t✉r❛ ❡s♣❡❝tr❛❧ ❡ ❛ ❢❛❧t❛ ❞❡ ✐♥✈❛r✐â♥❝✐❛ ♣♦r tr❛♥s❧❛çõ❡s✱ ♦ ❛r❣✉♠❡♥t♦ é ✉♠ ♣♦✉❝♦ ♠❛✐s s♦✜st✐❝❛❞♦✳
◆♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦ s❡❣✉♥❞♦ ❞❡s❛✜♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ ✐st♦ é✱ ♠♦str❛r q✉❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧✱ ❛ s✐t✉❛çã♦ é ✉♠ ♣♦✉❝♦ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛✳ ❆ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s ❛✉t♦r❡s tr❛❜❛❧❤❛ s♦❜ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ ❛ ❢✉♥çã♦s7→f(s)s−1é ♥ã♦ ❞❡❝r❡s❝❡♥t❡✱ ❡
❛ss✐♠✱ s♦❜ t❛❧ ❝♦♥❞✐çã♦✱ ❢❛③❡♠ ✉s♦ ❞❡ ✉♠❛ r❡str✐çã♦ ♥❛t✉r❛❧ ❞♦ ❡s♣❛ç♦ ❛♠❜✐❡♥t❡✳ ❈♦♠♦ ❡①❡♠♣❧♦ ♣♦❞❡♠♦s ❝✐t❛r ❬✷✵❪ ❡ ❬✷✷❪✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ s❡❣✉✐♠♦s ❡ss❛ ❧✐♥❤❛✱ ❡♠ ✈❡③ ❞✐ss♦✱ t✐r❛♠♦s ✈❛♥t❛❣❡♠ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❞✐❧❛t❛çã♦ ❞❛ ❢✉♥çã♦ t7→u(x/t)✳
❆ ✐❞❡✐❛ é ❡①♣❧♦r❛r ♦ ♣r♦❜❧❡♠❛ ♥♦ ✏✐♥✜♥✐t♦✑ q✉❡ é ❛✉tô♥♦♠♦✳ P❛r❛ ✐ss♦✱ ❛♥❛❧✐s❛♠♦s ♣r♦❜❧❡♠❛s ❛✉tô♥♦♠♦s ❞❛ ❢♦r♠❛
−∆u=h(u), ❡♠ RN. ✭✸✮
❆ss✐♠✱ ❛ ❝❤❛✈❡ ♣❛r❛ ❛✈❛♥ç❛r ❡stá ♥♦s r❡s✉❧t❛❞♦s s♦❜r❡ ♣r♦❜❧❡♠❛s ❛✉tô♥♦♠♦s ❡st❛❜❡❧❡❝✐❞♦s ♣♦r ✲❡st②❝❦✐ ❡ ▲✐♦♥s ❬✻❪ q✉❛♥❞♦ N ≥ 3✱ ❡ ♣♦r ❇❡r❡st②❝❦✐✱ ●❛❧❧♦✉ët ❡ ❑❛✈✐❛♥ ❬✺❪✱ ♣❛r❛ N = 2✳ P♦r ♠❡✐♦ ❞❡ss❡s
r❡s✉❧t❛❞♦s ♦❜t❡♠♦s ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ ✏♥♦ ✐♥✜♥✐t♦✑ ♣♦ss✉❛ s♦❧✉çã♦✳ ❊♥tã♦ r❡❧❛❝✐♦♥❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❛❞♦ ❡♠ ✭✶✮ ❝♦♠ ❛q✉❡❧❡ ❛ss♦❝✐❛❞♦ ✏♥♦ ✐♥✜♥✐t♦✑✱ ❞❡s❡♥✈♦❧✈❡♠♦s ✉♠ ❛r❣✉♠❡♥t♦ q✉❡✱ ♣♦r ❝♦♥tr❛❞✐çã♦✱ ♣r♦✈❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ I✳
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡stá ❡str✉t✉r❛❞♦ ❝♦♠♦ s❡❣✉❡✳ ◆♦ ❈❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❜r❡✈❡ ❡st✉❞♦ s♦❜r❡ ♦ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✱ t❛❧ ❢❡rr❛♠❡♥t❛ é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ♦❜t❡♥çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ I✱ ✉t✐❧✐③❛❞❛ ♥❛ ❛r❣✉♠❡♥t❛çã♦ ❞♦ ❈❛♣ít✉❧♦ ✸✳ ❊①✐❜✐♠♦s
✉♠❛ ♣r♦✈❛ ♣❛r❛ ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❜❛s❡❛❞❛ ❡♠ ❬✶✵❪✱ ❬✾❪ ❡ ❬✶✷❪✳ ❊♠ s❡❣✉✐❞❛✱ ❛✐♥❞❛ ❝♦♠ ❜❛s❡ ♥❛s r❡❢❡rê♥❝✐❛s ❛♥t❡r✐♦r❡s✱ ❛ss✐♠ ❝♦♠♦ ❡♠ ❬✶✹❪✱ ♠♦str❛♠♦s ❝♦♠♦ ❛♣❧✐❝❛r t❛❧ r❡s✉❧t❛❞♦ ♣❛r❛ ♦❜t❡r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♣❛r❛ ❞❡t❡r♠✐♥❛❞♦ ❢✉♥❝✐♦♥❛❧✱ t❛♥t♦ ♥♦ ♥í✈❡❧ ❞❡ ♠í♥✐♠♦✱ q✉❛♥t♦ ♥♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳
◆♦ ❈❛♣ít✉❧♦ ✷✱ ❡st✉❞❛♠♦s ♣r♦❜❧❡♠❛s ❛✉tô♥♦♠♦s ❝♦♠♦ ❞❛❞♦ ❡♠ ✭✸✮✳ P❛r❛ ✐ss♦ t♦♠❛♠♦s ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❬✻❪ ❡ ❬✺❪ ♣❛r❛✱ s♦❜ ❝❡rt❛s ❤✐♣ót❡s❡s✱ ❣❛r❛♥t✐r ✉♠❛ s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ s❛t✐s❢❛③❡♥❞♦ ❛ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③❛❡✈ ✭❝❢✳ ❆♣ê♥❞✐❝❡ ❇✳✷✮✳ ❈♦♠ ✐ss♦✱ ❡①♣❧❛♥❛♠♦s ♦ r❡s✉❧t❛❞♦ ❛♣r❡s❡♥t❛❞♦ ♣♦r ❏❡❛♥❥❡❛♥ ❡ ❚❛♥❛❦❛ ❡♠ ❬✶✼❪✳ ◆❡st❡ tr❛❜❛❧❤♦ ♦s ❛✉t♦r❡s ❝❛r❛❝t❡r✐③❛♠ ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ ♥❛t✉r❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❛❞♦ ❡♠ ✭✸✮✱ ✐st♦ é✱J :H1(RN)→R❞❛❞♦ ♣♦r
J(u) = 1 2
Z
RN|∇
u|2dx−
Z
RN
H(u)dx,
❡♠ q✉❡ H(u) =
Z u
0
h(s)ds✱ ♠♦str❛♥❞♦ q✉❡ t❛❧ ♥í✈❡❧ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ♥í✈❡❧ ❞❡ ♠í♥✐♠♦ ❞♦ ♠❡s♠♦
■♥tr♦❞✉çã♦ ✸
q✉❡ ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ J ❡st❡❥❛ ❜❡♠ ❞❡✜♥✐❞♦✱ ❡ss❡ ❝❛♠✐♥❤♦ é t❛♠❜é♠ ❛ ❢❡rr❛♠❡♥t❛
♣r✐♥❝✐♣❛❧ ♣❛r❛ ♣r♦✈❛r ❛ ✐❣✉❛❧❞❛❞❡ ❞♦s ♥í✈❡✐s s✉♣r❛❝✐t❛❞♦s✳ ❆❧é♠ ❞✐ss♦✱ ❛❧❣♦ q✉❡ ✈❛❧❡ ❛ ♣❡♥❛ r❡ss❛❧t❛r é q✉❡ ♣❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ss❡ ❝❛♠✐♥❤♦ ❡s♣❡❝í✜❝♦✱ r❡❧❛❝✐♦♥❛♠♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ❡♠ ❬✶✻❪ ❡ ❬✶✼❪✳ ◆❛ ✈❡r❞❛❞❡✱ ♦ q✉❡ ❢❛③❡♠♦s é ♣r♦✈❛r ✉♠ r❡s✉❧t❛❞♦ q✉❡ é ♠❛✐s ❣❡r❛❧ q✉❡ ♦ ❡①✐❣✐❞♦ ❡♠ ❬✶✼❪✱ ♠❛s q✉❡ é ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✶✻❪ ❡ ♣♦rt❛♥t♦ ✉t✐❧✐③❛❞♦ ♥❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❛❞♦ ❡♠ ✭✶✮✳
P♦r ✜♠✱ ♥♦ ❈❛♣ít✉❧♦ ✸✱ ❝♦♠❡ç❛♠♦s ♣♦r ❣❛r❛♥t✐r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ I ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭✶✮
♣♦ss✉✐ ❛ ●❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❡ ♣❛r❛ t❛♥t♦ ✉t✐❧✐③❛♠♦s ❛s ❤✐♣ót❡s❡s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❛ss✉♠✐❞❛s s♦❜r❡ ❛ ❢✉♥çã♦ f ❡ ♦ ♣♦t❡♥❝✐❛❧ V✳ ❉❡♣♦✐s ❞❡ ❝❡rt✐✜❝❛r q✉❡ I t❡♠ ❛ ❣❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱
❢❛③❡♠♦s ✉s♦ ❞♦s r❡s✉❧t❛❞♦s ❞♦ ❈❛♣ít✉❧♦ ✶ ❡ ❣❛r❛♥t✐♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ (un)✱ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐
♣❛r❛I♥♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳ ▲♦❣♦ ❛♣ós✱ ❜❛s❡❛❞♦s ♥♦ Pr✐♥❝í♣✐♦ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡ ❈♦♠♣❛❝✐❞❛❞❡
❞❡s❡♥✈♦❧✈✐❞♦ ❡♠ ❬✶✽❪✱ s✉♣♦♠♦s ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡ (un)é ✐❧✐♠✐t❛❞❛✱ ❛ss✐♠ ✉t✐❧✐③❛♠♦s ✉♠ ❛r❣✉♠❡♥t♦ ❞❡
❛♥✉❧❛♠❡♥t♦ ♦✉ ♥ã♦ ❛♥✉❧❛♠❡♥t♦ ❞❛ s❡q✉ê♥❝✐❛ unkunk−1❡ ♣♦r ♠❡✐♦ ❞❛ t❡♦r✐❛ ❡s♣❡❝tr❛❧ ♦❜t❡♠♦s ✉♠❛
❝♦♥tr❛❞✐çã♦✱ ♦ q✉❡ ♣r♦✈❛ ❛ ❧✐♠✐t❛çã♦ ❞❡ (un)✳
❊♠ s❡❣✉✐❞❛✱ ❛ ✜♠ ❞❡ ♠♦str❛r q✉❡ ♦ ❧✐♠✐t❡ ❢r❛❝♦ ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ (un) é ✉♠ ♣♦♥t♦
❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧ I✱ ❧❛♥ç❛♠♦s ♠ã♦ ❞♦s r❡s✉❧t❛❞♦s ❞♦ ❈❛♣ít✉❧♦ ✷ ♣❛r❛ ❡st✉❞❛r ♦ ♣r♦❜❧❡♠❛ ❛ss♦❝✐❛❞♦ ❛♦
✏✐♥✜♥✐t♦✑✳ ❈♦♠♦H1(RN)é r❡✢❡①✐✈♦✱ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛ ❞❡(u
n)✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ é ❣❛r❛♥t✐❞❛✱
❡ ❛✐♥❞❛ s♦❜ ❛s ❤✐♣ót❡s❡s ❞♦ ♣r♦❜❧❡♠❛ ✭✶✮ ❝♦♥s❡❣✉✐♠♦s ♣r♦✈❛r ❛ ♥ã♦ ♥❡❣❛t✐✈✐❞❛❞❡ ❞♦ ❧✐♠✐t❡ ❢r❛❝♦ u✳
P♦rt❛♥t♦ r❡st❛ ♠♦str❛r q✉❡ ué ♥ã♦ ♥✉❧♦✳ ❖✉tr❛ ✈❡③ ❛r❣✉♠❡♥t❛♥❞♦ ♣♦r ❝♦♥tr❛❞✐çã♦✱ ❛ ✐❞❡✐❛ é ♠♦str❛r
✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡str✐t❛ ❡♥tr❡ ❛ ❡♥❡r❣✐❛ ❞♦ ❢✉♥❝✐♦♥❛❧ I ❡ ❛ ❡♥❡r❣✐❛ ❞♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛
✏♥♦ ✐♥✜♥✐t♦✑✱ q✉❛♥❞♦ ❡st❡s sã♦ ❛♣❧✐❝❛❞♦s ❛♦ ❝❛♠✐♥❤♦ ❝♦♥str✉í❞♦ ♥♦ ❈❛♣ít✉❧♦ ✷✳ ❉❡ ❢❛t♦✱ ❞❡s❡♥✈♦❧✈❡♥❞♦ ❡ss❡ r❛❝✐♦❝í♥✐♦ ❝❤❡❣❛♠♦s ❛ ✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ t✐♣♦ c < c✱ ❡♠ q✉❡c é ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛
♣❛r❛ I✳ ❉❡ss❡ ♠♦❞♦✱ ❝♦♥❝❧✉í♠♦s ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡u 6= 0 é ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳ ❋✐♥❛❧✐③❛♥❞♦ ❡ss❡ tr❛❜❛❧❤♦✱ ❛✐♥❞❛ ♥♦ ❈❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ r❡s✉❧t❛❞♦ ♣❛rt✐❝✉❧❛r q✉❡ ❣❛r❛♥t❡ ✉♠❛ s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛❞♦ ❡♠ ✭✶✮✱ ♣❛r❛ ✐ss♦ ❛r❣✉♠❡♥t❛♠♦s ❛ss✉♠✐♥❞♦ ♦ r❡s✉❧t❛❞♦ ♣r♦✈❛❞♦ ❛♦ ❧♦♥❣♦ ❞♦ ❝❛♣ít✉❧♦✱ ✐st♦ é✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳
❆❧é♠ ❞✐ss♦✱ ❛ ✜♠ ❞❡ ❢❛❝✐❧✐t❛r ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ tr❛❜❛❧❤♦✱ ❛❝r❡s❝❡♥t❛♠♦s ♦s ❆♣ê♥❞✐❝❡s ❆ ❡ ❇✳ ◆♦ ♣r✐♠❡✐r♦✱ ❞❡♠♦♥str❛♠♦s ❛ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞♦s ❢✉♥❝✐♦♥❛✐s I ❡ J ❞❡✜♥✐❞♦s ❛❝✐♠❛✳ ❊ ♥♦ s❡❣✉♥❞♦
❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s q✉❡ sã♦ ✉s❛❞♦s ❢♦rt❡♠❡♥t❡ ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳ P❛r❛ ✉♠ ❡st✉❞♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦ ❞❡ t❛✐s r❡s✉❧t❛❞♦s✱ ❞❡✐①❛♠♦s s✉❛s r❡s♣❡❝t✐✈❛s r❡❢❡rê♥❝✐❛s✳
❋✐♥❛❧♠❡♥t❡ r❡ss❛❧t❛♠♦s q✉❡ ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦ ❛ ❧❡tr❛ C✱ ❜❡♠ ❝♦♠♦ ❛ ❧❡tr❛ M ❡ ❛❧❣✉♠❛s
❈❛♣ít✉❧♦
1
❖ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞
◆❡ss❡ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ✈❛♠♦s ❛❜♦r❞❛r ♦ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✱ q✉❡ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ✈❛r✐❛❝✐♦♥❛❧ ❜❛st❛♥t❡ ✉s❛❞❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s✱ ❛ ✜♠ ❞❡ ♦❜t❡r ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ ❞❡t❡r♠✐♥❛❞♦ ❢✉♥❝✐♦♥❛❧✳ ❊ss❡ ❡st✉❞♦ s❡rá ❜❛s❡❛❞♦ ♥♦s tr❛❜❛❧❤♦s ❬✶✵❪✱ ❬✾❪ ❡ ❬✶✷❪✳
✶✳✶ ❙❡q✉ê♥❝✐❛s P❛❧❛✐s✲❙♠❛❧❡ ❡ ❙❡q✉ê♥❝✐❛s ❞❡ ❈❡r❛♠✐
❆ ❛♣❧✐❝❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦✱ s♦❜ ❝❡rt❛s ❤✐♣ót❡s❡s✱ ❛ ♦❜✲ t❡♥çã♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ P❛❧❛✐s✲❙♠❛❧❡ ♦✉ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐✱ ♣❛r❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❢✉♥❝✐♦♥❛❧✳ ◆❡ss❛ s❡çã♦ ✈❛♠♦s ❞❡✜♥✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ s❡q✉ê♥❝✐❛ P❛❧❛✐s✲❙♠❛❧❡ ❡ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡I :X →R✱ ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡ ❝❧❛ss❡ C1✳ ❙✉♣♦♥❤❛
q✉❡ ❡①✐st❛♠ c∈R❡(un)⊂X t❛✐s q✉❡
I(un)→c ❡ kI′(un)k →0,
❡♥tã♦ ❞✐③❡♠♦s q✉❡ (un) é ✉♠❛ s❡q✉ê♥❝✐❛ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧ ❝ ♣❛r❛ I✱ ♦✉ ❞❡ ❢♦r♠❛ ❛❜r❡✈✐❛❞❛✱ (un)
é ✉♠❛ s❡q✉ê♥❝✐❛ (P S)c ♣❛r❛ I✳ ❆❧é♠ ❞✐ss♦✱ ❞✐③❡♠♦s q✉❡ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧ c✱
q✉❛♥❞♦ t♦❞❛ s❡q✉ê♥❝✐❛ (P S)c ♣❛r❛I✱ ♣♦ss✉✐ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳
❉❡✜♥✐çã♦ ✶✳✷✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡I :X →R✱ ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡ ❝❧❛ss❡ C1✳ ❙✉♣♦♥❤❛
q✉❡ ❡①✐st❛♠ c∈R❡(un)⊂X t❛✐s q✉❡
I(un)→c ❡ kI′(un)kH−1 1 +kunk→0,
❡♥tã♦ ❞✐③❡♠♦s q✉❡(un)é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ ❝ ♣❛r❛I✱ ♦✉ ❞❡ ❢♦r♠❛ ❛❜r❡✈✐❛❞❛✱(un)é ✉♠❛
s❡q✉ê♥❝✐❛ (Ce)c ♣❛r❛I✳ ❆❧é♠ ❞✐ss♦✱ ❞✐③❡♠♦s q✉❡I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧c✱ q✉❛♥❞♦ t♦❞❛
✶✳✷ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞❡ ❊♥❡r❣✐❛ ▼í♥✐♠❛ ✺
❖❜s❡r✈❡ q✉❡ t♦❞❛ s❡q✉ê♥❝✐❛ (Ce)c ♣❛r❛ I✱ é t❛♠❜é♠ ✉♠❛ s❡q✉ê♥❝✐❛ (P S)c ♣❛r❛ I✳ ❆❧é♠ ❞✐ss♦✱
♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❞❡ss❛s s❡q✉ê♥❝✐❛s✱ ✉t✐❧✐③❛♥❞♦ ❛❧❣✉♠ ❛r❣✉♠❡♥t♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡✱ é ♣♦ssí✈❡❧ ❝♦♥❝❧✉✐r q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ ❡❧❛ ❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❞♦ ❢✉♥❝✐♦♥❛❧ ❡♠ q✉❡stã♦✱ ♦ q✉❡ ❢♦r♥❡❝❡ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❛ss♦❝✐❛❞♦ ❛♦ ❢✉♥❝✐♦♥❛❧✳
✶✳✷ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞❡ ❊♥❡r❣✐❛ ▼í♥✐♠❛
❖ ♦❜❥❡t✐✈♦ ❞❡ss❛ s❡çã♦ é ❡♥✉♥❝✐❛r ❡ ♣r♦✈❛r ♦ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✱ ❡ ♠❛✐s ❛✐♥❞❛ ❣❛r❛♥t✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ♣❛r❛ ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❛❞♦✳
❚❡♦r❡♠❛ ✶✳✶✳ ❙❡❥❛♠ X, d
✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ ❡ I : x → (−∞,+∞] ✉♠ ❢✉♥❝✐♦♥❛❧ s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❙✉♣♦♥❤❛ q✉❡Is❡❥❛ ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✱ ♦✉ s❡❥❛✱ inf
u∈XI(u)>−∞✳ ❊♥tã♦
❞❛❞♦ε >0 ❡v0∈X t❛✐s q✉❡
I(v0)≤ inf
u∈XI(u) +ε, ✭✶✳✶✮
❡①✐st❡ uε∈X✱ t❛❧ q✉❡
✭❛✮I(uε)≤I(v0)≤ inf
u∈XI(u) +ε;
✭❜✮d(v0, uε)≤√ε;
✭❝✮ P❛r❛ ❝❛❞❛w∈X, w6=uε✱ ✈❛❧❡ q✉❡
I(uε)< I(w) +√ε d(uε, w).
Pr♦✈❛✳ Pr✐♠❡✐r❛♠❡♥t❡ ❝♦♥s✐❞❡r❡ ❡♠ X, d❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ♣❛r❝✐❛❧✿
w≺v⇔I(w)≤I(v)−√ε d(w, v).
❱❛♠♦s ♠♦str❛r q✉❡≺é r❡✢❡①✐✈❛✱ ❛♥t✐ss✐♠étr✐❝❛ ❡ tr❛♥s✐t✐✈❛✳ ❉❡ ❢❛t♦✱ ❞❛❞♦w∈X✱ ✈❛❧❡ q✉❡
I(w) =I(w)−√ε d(w, w),
♦✉ s❡❥❛✱w≺w, ❡ ❝♦♠ ✐ss♦✱ ✈❡♠♦s q✉❡≺é r❡✢❡①✐✈❛✳ ❚❛♠❜é♠✱ ❞❛❞♦sw, v ∈X t❛✐s q✉❡w≺v ❡v≺w✱
s❡❣✉❡ q✉❡
I(w)≤I(v)−√ε d(w, v) ❡ I(v)≤I(w)−√ε d(v, w),
❧♦❣♦✱ s♦♠❛♥❞♦ t❛✐s ❞❡s✐❣✉❛❧❞❛❞❡s✱ ♦❜t❡♠♦s q✉❡
0≤ −2√ε d(w, v)≤0,
♦✉ s❡❥❛✱ d(w, v) = 0❡w =v✱ ❛ss✐♠≺é ❛♥t✐ss✐♠étr✐❝❛✳ P♦r ✜♠✱ ❞❛❞♦s w, v ❡u∈X t❛✐s q✉❡w ≺v ❡ v≺u✱ ✈❛❧❡ q✉❡
✶✳✷ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞❡ ❊♥❡r❣✐❛ ▼í♥✐♠❛ ✻
❊♥tã♦ s♦♠❛♥❞♦ t❛✐s ❞❡s✐❣✉❛❧❞❛❞❡s ❡ ❛♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ ♦❜t❡♠♦s
I(w) ≤ I(u)−√ε d(w, v)−√ε d(v, u)
≤ I(u)−√ε d(w, u),
✐st♦ é✱ w≺u❡≺é tr❛♥s✐t✐✈❛✳ ❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ ≺é r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ♣❛r❝✐❛❧✳
❆❣♦r❛ ❝♦♥s✐❞❡r❡ ❛ s❡q✉ê♥❝✐❛(An)❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡X✱ ❞❡ ♠♦❞♦ q✉❡
A0={w∈X:w≺v0},
❧♦❣♦v0∈A0 ❡ ❝♦♠ ✐ss♦A06=∅✳ ❊♥tã♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ í♥✜♠♦✱ t♦♠❡v1∈A0✱ t❛❧ q✉❡ I(v1)≤ inf
u∈A0I(u) + 1,
❡ ❛ss✐♠ ❞❡✜♥❛
A1={w∈X:w≺v1}.
❆♥❛❧♦❣❛♠❡♥t❡✱ ♥♦t❡ q✉❡ v1∈A1 ❡A16=∅✳ ❆ss✐♠✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ í♥✜♠♦ t♦♠❡v2∈A1✱ t❛❧ q✉❡ I(v2)≤ inf
u∈A1I(u) +
1 2.
❈♦♠ ✐ss♦✱ ❞❡✜♥❛
A2={w∈X:w≺v2},
❡ ♣r♦❝❡❞❡♥❞♦ r❡❝✉rs✐✈❛♠❡♥t❡✱ ♣❛r❛ t♦❞♦ n∈ N, ❝♦♠♦ vn−1 ∈An−1✱ ❡♥tã♦ An−1 6= ∅✱ ❛ss✐♠ t♦♠❛♥❞♦ vn∈An−1 t❛❧ q✉❡
I(vn)≤ inf u∈An−1
I(u) + 1
n, ✭✶✳✷✮
❞❡✜♥❛
An={w∈X:w≺vn}.
❊ ♦❜s❡r✈❡ q✉❡An ⊃An+1✱ ♣❛r❛ t♦❞♦n∈N✱ ♣♦✐s ❞❛❞♦w∈An+1s❡❣✉❡ q✉❡w≺vn+1❡ ❝♦♠♦vn+1∈An✱
❡♥tã♦ vn+1≺vn ❡ ♣❡❧❛ tr❛♥s✐t✐✈✐❞❛❞❡w≺vn✱ ❡w∈An✳
❆❧é♠ ❞✐ss♦✱Ané ❢❡❝❤❛❞♦✱ ♣♦✐s ❞❛❞❛ (wk)⊂An t❛❧ q✉❡wk →w∈X✱ q✉❛♥❞♦k→ ∞✱ s❡❣✉❡ q✉❡
wk ≺vn ❡ ❛ss✐♠
I(wk)≤I(vn)−√ε d(wk, vn).
❚❛♠❜é♠✱ ❝♦♠♦I é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡✱ ❡♥tã♦ s❡k→ ∞✱ ✈❛❧❡ q✉❡
I(w) ≤ lim inf
k→∞ I(wk)
≤ lim inf
k→∞
h
I(vn)−√ε d(wk, vn)
i
= I(vn)−√εlim sup k→∞
d(wk, vn)
✶✳✷ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞❡ ❊♥❡r❣✐❛ ▼í♥✐♠❛ ✼
✐♠♣❧✐❝❛♥❞♦ q✉❡ w≺vn✱ ❡ ♣♦r ❞❡✜♥✐çã♦w∈An✱ ♣♦rt❛♥t♦An é ❢❡❝❤❛❞♦✳ ❆❣♦r❛ ❛✜r♠❛♠♦s q✉❡
∞
\
n=0
An6=∅.
❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡vk∈An ♣❛r❛k≥n✱ ❛❧é♠ ❞✐ss♦✱ ❞❛❞♦sk > l≥ns❡❣✉❡ q✉❡An⊃Al⊃Ak✱ ❧♦❣♦vk≺vl
❡ ♣♦r ✭✶✳✷✮✱ ♦❜t❡♠♦s q✉❡
d(vk, vl) ≤
1
√
ε
h
I(vl)−I(vk)
i
≤ √1
ε
h
I(vl)− inf u∈Al−1
I(u)i
≤ √1
ε l,
♣♦rt❛♥t♦ (vn)é s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ An✱ ♣❛r❛ t♦❞♦n∈N✳ ❆ss✐♠(vn)❝♦♥✈❡r❣❡ ♣❛r❛ uε∈An✱ ♣❛r❛
t♦❞♦ n ∈ N✱ ♦✉ s❡❥❛✱ uε ∈
∞
\
n=0
An 6=∅. ❊ ❛✐♥❞❛ ❛✜r♠❛♠♦s q✉❡ ❞✐❛♠(An) →0✱ q✉❛♥❞♦ n → ∞✱ ❧♦❣♦
❝♦♥❝❧✉í♠♦s q✉❡
∞
\
n=0
An={uε}.
❉❡ ❢❛t♦✱ ❞❛❞♦w∈An ✈❛❧❡ q✉❡w≺vn≺vn−1❡ ❛ss✐♠
√
ε d(w, vn)≤I(vn)−I(w).
❈♦♠♦ w∈An ⊂An−1✱ ❡♥tã♦−I(w)≤ − inf
u∈An−1
I(u)❡ ♣♦r ✭✶✳✷✮✱ ✈❛❧❡ q✉❡
d(w, vn) ≤
1
√
ε
inf
u∈An−1
I(u) +1
n−u∈infAn−1
I(u)
= √1
ε n.
❈♦♠ ✐ss♦✱ ♣❛r❛ q✉❛✐sq✉❡rw, v∈An✱ s❡❣✉❡ q✉❡
d(w, v)≤d(w, vn) +d(vn, v)≤
2
√
ε n.
❖✉ s❡❥❛✱
0≤ lim
n→∞w,vsup∈A
n
d(w, v)≤ lim
n→∞ 2
√ε n = 0.
❉❡ss❛ ❢♦r♠❛✱ ❝♦♠♦ ❞✐❛♠(An) = sup w,v∈An
d(w, v)✱ ❡♥tã♦ ❞✐❛♠(An)→0✱ q✉❛♥❞♦n→ ∞.P♦rt❛♥t♦✱ ❝♦♥❝❧✉í✲
♠♦s q✉❡ ∞
\
n=0
An={uε}✱ ❝❛s♦ ❝♦♥trár✐♦✱ ❤❛✈❡r✐❛v∈
∞
\
n=0
An✱ ❝♦♠v6=uε✱ ❡ ❛ss✐♠ ❞✐❛♠(An)≥d(v, uε)>0✱
♣❛r❛ t♦❞♦n∈N✱ ❝♦♥tr❛❞✐③❡♥❞♦ q✉❡ ❞✐❛♠(An)→0✱ q✉❛♥❞♦n→ ∞.
P♦r ✜♠✱ ✈❛♠♦s ♠♦str❛r q✉❡uεs❛t✐s❢❛③ ✭❛✮✱ ✭❜✮ ❡ ✭❝✮✳ ❉❡ ❢❛t♦✱ ❝♦♠♦uε∈A0✱ ♣♦r ❞❡✜♥✐çã♦ s❡❣✉❡
q✉❡uε≺v0 ❡ ❛ss✐♠
✶✳✷ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞❡ ❊♥❡r❣✐❛ ▼í♥✐♠❛ ✽
♦ q✉❡ ♣r♦✈❛ ✭❛✮✳ ❚❛♠❜é♠ ♦❜s❡r✈❡ q✉❡−I(uε)≤ − inf
u∈XI(u)❡ ♣♦r ✭✶✳✶✮✱ ✈❛❧❡ q✉❡
d(v0, uε) ≤ √1
ε
h
I(v0)−I(uε)
i
≤ √1
ε
inf
u∈XI(u) +ε−uinf∈XI(u)
≤ √ε,
♦ q✉❡ ♣r♦✈❛ ✭❜✮✳ ❊ ♣♦r ú❧t✐♠♦✱ ❞❛❞♦ w6=uε✱ ♥♦t❡ q✉❡ w ♥ã♦ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠uε✱ ❝❛s♦ ❝♦♥trár✐♦✱
✈❛❧❡r✐❛ q✉❡
w≺uε≺vn, ∀ n∈N,
❡ ❝♦♠ ✐ss♦✱w∈
∞
\
n=0
An={uε},♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❆ss✐♠✱ ❝♦♠♦w⊀uε✱ ✈❛❧❡ q✉❡
I(w)> I(uε)−√ε d(w, uε),
♦ q✉❡ ♣r♦✈❛ ✭❝✮✱ ❡ ❝♦♠♣❧❡t❛ ♦ r❡s✉❧t❛❞♦✳
P❛r❛ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ♥♦ q✉❛❧ X,k · k
é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ❛ ✜♠ ❞❡ ✉t✐❧✐③❛r ♦ ❚❡♦r❡♠❛ ✶✳✶ ♣❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♣❛r❛ I ♥♦ ♥í✈❡❧ m= inf
u∈XI(u)✱ ✈❛♠♦s ❞❡✜♥✐r
✉♠❛ ♥♦✈❛ ♠étr✐❝❛δ:X×X →R+ s♦❜r❡X ❡ ♠♦str❛r q✉❡ ❛ ♠étr✐❝❛ ❞❛❞❛ ♣❡❧❛ ♥♦r♠❛ ❡ ❛ ♠étr✐❝❛ δs❡
r❡❧❛❝✐♦♥❛♠✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❙❡❥❛ c∈C([0,1];X) ✉♠❛ ❝✉r✈❛ q✉❛❧q✉❡r✱ ❞❡✜♥✐♠♦s ♦ ❝♦♠♣r✐♠❡♥t♦ ❣❡♦❞és✐❝♦ℓ(c) ❞❛ ❝✉r✈❛ c ❝♦♠♦ s❡♥❞♦ ❞❛❞♦ ♣♦r✿
ℓ(c) =
Z 1
0
kc′(t)k 1 +kc(t)kdt.
❈♦♠ ✐ss♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r t❛♠❜é♠δ:X×X →R+✱ ❛ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦su❡v∈X✱
❝♦♠♦ s❡♥❞♦ ❞❛❞❛ ♣♦r
δ(u, v) := inf
ℓ(c) :c∈C1([0,1], X), c(0) =u, c(1) =v . ✭✶✳✸✮
❈❧❛r❛♠❡♥t❡ δ(u, v) ≤ ku−vk, ♣♦✐s ❞❛❞♦ ˜c(t) = (1−t)u−tv ∈ C1([0,1];X)✱ s❡❣✉❡ q✉❡
k˜c′(t)k=ku−vk✱ ❧♦❣♦
ℓ(˜c) =
Z 1
0
ku−vk
1 +k(1−t)u−tvkdt≤ ku−vk,
❡ ❛♣❧✐❝❛♥❞♦ ♦ í♥✜♠♦✱
δ(u, v)≤ ku−vk.
P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ B⊂X ❧✐♠✐t❛❞♦ ♥❛ ♥♦r♠❛ ❞❡X✱ ❡①✐st❡R >0✱ t❛❧ q✉❡kxk ≤R✱
✶✳✷ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞❡ ❊♥❡r❣✐❛ ▼í♥✐♠❛ ✾
s❡❣✉❡ q✉❡kc(t)k ≤R, ∀t∈[0,1]✱ ❡ ❢❛③❡♥❞♦ β= 1
1 +R✱ ♦❜t❡♠♦s q✉❡
ℓ(c) =
Z 1
0
kc′(t)k 1 +kc(t)kdt
≥ 1 +1R
Z 1
0 k
c′(t)kdt
≥ 1 +1R
Z 1
0
c′(t)dt
= 1
1 +R
c(1)−c(0)
= βku−vk.
❈♦♠ ✐ss♦✱ ❛♣❧✐❝❛♥❞♦ ♦ í♥✜♠♦ s♦❜r❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s c∈C1([0,1], B
R[0])✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❡①✐st❡β >0✱
t❛❧ q✉❡
δ(u, v)≥βku−vk, ∀ u, v∈B.
❉❡ss❛ ❢♦r♠❛✱ B ⊂X é ❧✐♠✐t❛❞♦ ♥❛ ♠étr✐❝❛ ❞❛ ♥♦r♠❛ ❞❡ X s❡✱ ❡ s♦♠❡♥t❡ s❡✱ B é ❧✐♠✐t❛❞♦ ♥❛
♠étr✐❝❛δ✳ ▼❛✐s ❛✐♥❞❛✱ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠δ✱ é s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ♥❛ ♠étr✐❝❛ ❞❛ ♥♦r♠❛✱
❧♦❣♦ é ❝♦♥✈❡r❣❡♥t❡ ♥❛ ♠étr✐❝❛ ❞❛ ♥♦r♠❛✱ ♣♦✐s X,k · k é ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ❡ ♣❡❧❛ r❡❧❛çã♦ ❛❝✐♠❛ ❡♥tr❡
❛s ♠étr✐❝❛s✱ ❝♦♥✈❡r❣❡ t❛♠❜é♠ ❡♠ δ✳ P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ X, δ
é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❡ ♣♦❞❡♠♦s ❛♣❧✐❝❛r s♦❜r❡ ❡❧❡ ♦ ❚❡♦r❡♠❛ ✶✳✶✳
❈♦r♦❧ár✐♦ ✶✳✶✳ ❙❡❥❛♠ X,k · k
✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ❡I:X →R✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡ ❝❧❛ss❡C1✱ ❡
❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ (un)⊂X t❛❧ q✉❡
I(un)→ inf
u∈XI(u) =m
❡
1 +kunkkI′(un)kX′ →0.
■st♦ é✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♣❛r❛ I ♥♦ ♥í✈❡❧m.
Pr♦✈❛✳ ❈♦♠♦ X,k·k
é ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ♣❡❧❛s ❝♦♥s✐❞❡r❛çõ❡s ❛❝✐♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ X, δ
é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳ ❚❛♠❜é♠ ♣♦r ❤✐♣ót❡s❡Ié s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡ é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✳
❈♦♠ ✐ss♦✱ ❞❛❞♦n∈N,t♦♠❡ε= 1
n2✱ ❡ ❡♥tã♦ ♦ ❚❡♦r❡♠❛ ✶✳✶ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡un∈X✱ t❛❧ q✉❡ ♣♦r
✭❛✮✱ ✈❛❧❡ q✉❡
I(un)≤ inf u∈XI(u) +
1
n2,
❡ ♣♦r ✭❝✮✱ ✈❛❧❡ q✉❡
I(w)≥I(un)−
1
nδ(w, un), ∀ w∈X.
❉❡ss❡ ♠♦❞♦✱ ♦❜t❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛(un)⊂X✱ ❝♦♠n∈N✱ t❛❧ q✉❡
inf
u∈XI(u)≤I(un)≤uinf∈XI(u) +
1
✶✳✸ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✶✵
♣♦rt❛♥t♦✱ s❡❣✉❡ q✉❡
lim
n→∞I(un) = infu∈XI(u).
❆❧é♠ ❞✐ss♦✱ ❢❛③❡♥❞♦w=un+tu✱ ♣❛r❛ t >0❡u∈X ❛r❜✐trár✐♦s✱ s❡❣✉❡ q✉❡
I(un+tu)−I(un)≥ −1
nδ(un, un+tu).
❆ss✐♠ ❞✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣♦r t✱ ❡ r❡❧❡♠❜r❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❛ ❞✐stâ♥❝✐❛
❣❡♦❞és✐❝❛ δ✱ ♦❜t❡♠♦s
1
t
h
I(un+tu)−I(un)
i
≥ −nt1 δ(un, un+tu)
≥ −1
nkuk
Z t
0
ds
1 +kun+suk
,
❡♥tã♦✱ ❢❛③❡♥❞♦t→0✱ ❝♦♠♦ Ié ❞❡ ❝❧❛ss❡C1✱ ❝♦♥❝❧✉í♠♦s q✉❡
I′(un)u≥ −1
n 1 +kunk
−1
kuk.
❆❣♦r❛ ❝♦♠♦ ué ❛r❜✐trár✐♦✱ tr♦❝❛♥❞♦u♣♦r−u✱ ♦❜t❡♠♦s t❛♠❜é♠
I′(un)u≤
1
n 1 +kunk
−1
kuk,
❡ ❛ss✐♠✱
|I′(u
n)u| kuk ≤
1
n 1 +kunk
−1
,
♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠
0≤ 1 +kunk
kI′(u
n)kX′ ≤
1
n.
P♦rt❛♥t♦
lim
n→∞ 1 +kunk
kI′(un)kX′ = 0.
❊ ❞❡ss❛ ❢♦r♠❛✱ (un)é ❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♣r♦❝✉r❛❞❛✳
✶✳✸ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛
◆❡ss❛ s❡çã♦✱ ✈❛♠♦s ❞❡✜♥✐r ❛ ❣❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❡ ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ ✉♠ ❞❛❞♦ ❢✉♥❝✐♦♥❛❧I:X →R✱ ❡♠ q✉❡(X,k · k)é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❊♠ s❡❣✉✐❞❛✱ ✈❛♠♦s ✉s❛r ♦ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞ ♣❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ ✉♠ ❢✉♥❝✐♦♥❛❧ Iq✉❡ ♣♦ss✉❛ ❛ ❣❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳
❉❡✜♥✐çã♦ ✶✳✹✳ ❈♦♥s✐❞❡r❡ (X,k · k) ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ❡ I : X → R ✉♠ ❢✉♥❝✐♦♥❛❧ t❛❧ q✉❡
I(0) = 0✱ ❞✐③❡♠♦s q✉❡I♣♦ss✉✐ ❛ ❣❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ♦✉ ❛❜r❡✈✐❛❞❛♠❡♥t❡✱ ❛ ❣❡♦♠❡tr✐❛ P▼✱
✶✳✸ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✶✶
✭P▼✶✮ ❊①✐st❡♠ρ, α >0 t❛✐s q✉❡I(u)≥α >0✱ ♣❛r❛ t♦❞♦ u∈X,❝♦♠kuk=ρ;
✭P▼✷✮ ❊①✐st❡ e∈X✱ ❝♦♠kek> ρ✱ t❛❧ q✉❡I(e)<0✳
❉❡✜♥✐çã♦ ✶✳✺✳ ❈♦♥s✐❞❡r❡ X,k · k
✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ❡ I: X →R ✉♠ ❢✉♥❝✐♦♥❛❧ q✉❡ ♣♦ss✉✐
❛ ❣❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ❡♥tã♦ ✜❝❛ ❜❡♠ ❞❡✜♥✐❞♦ ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ I✱ ♦✉
❛❜r❡✈✐❛❞❛♠❡♥t❡✱ ♦ ♥í✈❡❧ P▼ ♣❛r❛ I✱ ❞❛❞♦ ♣♦r
c= inf
˜
γ∈Γ˜tmax∈[0,1]I(˜γ(t)), ✭✶✳✹✮
❡♠ q✉❡
˜
Γ ={˜γ∈C([0,1], X) ; ˜γ(0) = 0, γ˜(1) =e}. ✭✶✳✺✮
◆♦t❡ q✉❡✱ ❞❡ ❢❛t♦✱ Γ =˜ {γ˜ ∈ C([0,1], X) ; ˜γ(0) = 0, ˜γ(1) = e} 6=∅✱ ♣♦✐s s❡ γ˜(t) = te✱ ❡♥tã♦ ˜
γ ∈Γ˜. ❆❧é♠ ❞✐ss♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ c ❝♦♥❝❧✉í♠♦s q✉❡ c ≥α > 0✳ ❆ s❡❣✉✐r ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ r❡s✉❧✲
t❛❞♦ q✉❡ é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ ✐♥ú♠❡r♦s ♣r♦❜❧❡♠❛s✱ ♣♦✐s ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ P▼ ♣❛r❛ ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡ ❝❧❛ss❡C1✱ ❞❡s❞❡ q✉❡ ❡st❡ s❛t✐s❢❛ç❛ ❛ ❣❡♦♠❡tr✐❛ P▼✳
❚❡♦r❡♠❛ ✶✳✷✳ ❙❡❥❛♠ X,k · k ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ❡
I : X → R ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡ ❝❧❛ss❡ C1
q✉❡ ♣♦ss✉✐ ❛ ❣❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♣❛r❛I ♥♦ ♥í✈❡❧ ❞♦
P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ✐st♦ é✱ ❡①✐st❡ (un)⊂X✱ t❛❧ q✉❡
I(un)→c ❡ kI′(un)k 1 +kunk
→0,
q✉❛♥❞♦ n→ ∞✱ ❡♠ q✉❡ cé ❞❛❞♦ ♣♦r ✭✶✳✹✮✳
Pr♦✈❛✳ ❆ ✐❞❡✐❛ é s❡❣✉✐r ♦s ♣❛ss♦s ❞♦ ❈♦r♦❧ár✐♦ ✶✳✶ ♣❛r❛ ♦❜t❡r ❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♣r♦❝✉r❛❞❛✳ P❛r❛ ✐ss♦✱ ❞❛❞♦ Γ˜ ⊂C([0,1], X)✱ ❞❡✜♥✐❞♦ ❡♠ ✭✶✳✺✮✱ ❝♦♥s✐❞❡r❡✲♦ ❝♦♠♦ s✉❜❡s♣❛ç♦ ♠étr✐❝♦ ❞❡C([0,1], X) ❝♦♠ ❛ ♠étr✐❝❛ ❞❛❞❛ ♣❡❧❛ ♥♦r♠❛k · k∞✱ ♦✉ s❡❥❛✱ Γ˜, d
é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ t❛❧ q✉❡
d(˜γ1,γ˜2) =k˜γ1−˜γ2k∞= max
t∈[0,1]k˜γ1(t)−˜γ2(t)k, ∀ γ˜1,γ˜2∈
˜ Γ.
❆ss✐♠✱ ❝♦♠♦ C([0,1], X) é ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❜❛st❛ ♠♦str❛r q✉❡ Γ˜ é ❢❡❝❤❛❞♦✱ ❝♦♠ r❡s♣❡✐t♦ ❛ ♠étr✐❝❛ d✱ ♣❛r❛ ❣❛r❛♥t✐r q✉❡ Γ˜, d
é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳ ❉❡ss❛ ❢♦r♠❛✱ ❞❛❞❛ (˜γn)⊂ Γ˜ ✉♠❛
s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣✐♥❞♦ ♣❛r❛ γ˜ ❡♠ (C([0,1], X)✱ q✉❛♥❞♦n → ∞✱ ♥♦t❡ q✉❡γ˜n(0) = 0❡ ˜γn(1) =e✱ ♣❛r❛
t♦❞♦ n∈N✳ ▲♦❣♦ ❝♦♠♦k˜γn−γ˜k∞→0✱ q✉❛♥❞♦n→ ∞✱ ❡♥tã♦˜γ(0) = 0 ❡˜γ(1) =e✱ ❛ss✐♠˜γ∈˜Γ,❡ s❡❣✉❡ q✉❡ Γ˜ é ❢❡❝❤❛❞♦✱ ♣♦rt❛♥t♦ ˜Γ, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳ ❈♦♠ ✐ss♦✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ❛ ♠étr✐❝❛
d✱ q✉❡ ❛❞✈é♠ ❞❛ ♥♦r♠❛✱ ❡ ❝♦♥str✉✐r ❛ ♠étr✐❝❛ ❣❡♦❞és✐❝❛δΓ˜✱ ♣❛r❛ ˜Γ✳ ❈♦♠♦ ✈✐st♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ ✉♠❛
✈❡③ q✉❡ Γ˜, d
é ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ s❡❣✉❡ q✉❡ Γ˜, δ˜
Γ
é t❛♠❜é♠ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❧♦❣♦ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ❚❡♦r❡♠❛ ✶✳✶ ♣❛r❛ ❡ss❡ ❡s♣❛ç♦✳
❆❣♦r❛ ❞❡✜♥❛ ♦ ❢✉♥❝✐♦♥❛❧Ψ : ˜Γ→R✱ ❞❛❞♦ ♣♦r
Ψ(˜γ) = max
✶✳✸ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✶✷
P❛r❛ ❝❛❞❛t∈[0,1]✜①❛❞♦✱ ❝♦♥s✐❞❡r❡Xt=
n
˜
γ(t) ; ˜γ∈Γ˜o⊂X✳ ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡I:X →Ré ❞❡ ❝❧❛ss❡
C1✱ ❡♥tã♦ é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡✱ ❡ ❞❡♥♦t❛♥❞♦ ♣♦r I
t ❛ r❡str✐çã♦ I|Xt✱ s❡❣✉❡ q✉❡ It : Xt →R é
s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡✳ ▼❛s ♦❜s❡r✈❡ q✉❡Ψ = max
t∈[0,1]It✱ ❡ ❝♦♠♦Ité s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡✱ ♣❛r❛
t♦❞♦ t∈[0,1]✱ s❡❣✉❡ q✉❡Ψé s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦γ˜ ∈Γ✱ ❝♦♠♦˜ I s❛t✐s❢❛③ ❛ ❣❡♦♠❡tr✐❛ P▼✱ ❡ ✈❛❧❡˜γ(0) = 0✱ kγ˜(1)k> ρ
❡ γ˜∈C([0,1], X)✱ ❡♥tã♦ ❡①✐st❡ t0∈(0,1)❝♦♠kγ˜(t0)k=ρ✱ ❡ ♣♦r(P M1)✱ s❡❣✉❡ q✉❡
Ψ(˜γ)≥I(˜γ(t0))≥α >0,
❛ss✐♠Ψé ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❈♦♠ ✐ss♦ ❡st❛♠♦s ♥❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ✶✳✶✱ ❡ ♣r♦❝❡❞❡♥❞♦ ♣❛r❛ ♦ ❡s♣❛ç♦ Γ˜, δ˜
Γ
❞❡ ❢♦r♠❛ s❡♠❡❧❤❛♥t❡ ❛ ♣r♦✈❛ ❞♦ ❈♦r♦❧ár✐♦ ✶✳✶ ✱ ❞❛❞♦
n∈N ❡ ε= 1
n2✱ ❡①✐st❡γ˜n ∈Γ✱ t❛❧˜
q✉❡
Ψ(˜γn)≤ inf
˜
γ∈Γ˜Ψ(˜γ) +
1
n2 =c+
1
n2, ✭✶✳✼✮
❡
Ψ(γ)≥Ψ(˜γn)−
1
nδΓ˜(γ,γ˜n), ∀ γ∈Γ˜. ✭✶✳✽✮
P♦rt❛♥t♦ ❞❡✜♥✐♥❞♦
Mn=
n
t∈[0,1] ; I(˜γn(t)) = max
s∈[0,1]I(˜γn(s)) = Ψ(˜γn) o
,
❡ t♦♠❛♥❞♦t˜n∈Mn✱ ♣♦r ✭✶✳✼✮ s❡❣✉❡ q✉❡
c≤I(˜γn(˜tn))≤c+ 1
n2, ∀ n∈N,
✐st♦ é✱
lim
n→∞I(˜γn(˜tn)) =c. ✭✶✳✾✮
❆❧é♠ ❞✐ss♦✱ ✜①❛❞♦ n∈ N✱ ❝♦♥s✐❞❡r❡ γ˜ ∈ C([0,1], X)❛r❜✐trár✐♦✱ t❛❧ q✉❡ kγ˜k˜Γ =k˜γ(˜tn)k✱ ❡
˜
γ(0) = ˜γ(1) = 0✱ ❡♥tã♦ ❢❛③❡♥❞♦γ(s) = ˜γn(s) +tγ˜(s)✱ ❝♦♠ t >0✱ ❝♦♠♦˜γn ∈Γ✱ s❡❣✉❡ q✉❡˜ γ∈Γ✱ ❡ ♣❛r❛˜
t s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ✈❛❧❡ q✉❡ max
s∈[0,1]I(γ(s)) = I(γ(˜tn)) = I ˜γn(˜tn) +t˜γ(˜tn)
,
❛ss✐♠ ♣♦r ✭✶✳✽✮ s❡❣✉❡ q✉❡
I γ˜n(˜tn) +t˜γ(˜tn)
−I(˜γn(˜tn))≥ −
1
nδ˜Γ(˜γn+t˜γ,˜γn).
❈♦♠ ✐ss♦✱ ❞✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣♦r t✱ ❡ r❡❧❡♠❜r❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❛ ❞✐stâ♥❝✐❛
❣❡♦❞és✐❝❛ δγ˜✱ ❡ ❛ ❞❡✜♥✐çã♦ ❞❡k · kΓ˜✱ ♦❜t❡♠♦s
1
t
h
I γ˜n(˜tn) +t˜γ(˜tn)
−I(˜γn(˜tn))
i
≥ −nt1 δ˜Γ(˜γn+t˜γ,γ˜n)
≥ −nt1
Z t
0
kγ˜kΓ˜
1 +kγ˜n+sγ˜k˜Γ ds
≥ −nt1
Z t
0
k˜γ(˜tn)k
1 +kγ˜n(˜tn) +s˜γ(˜tn)k
ds
≥ −nt1 k˜γ(˜tn)k
Z t
0
ds
1 +kγ˜n(˜tn) +sγ˜(˜tn)k
✶✳✸ ❯♠❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ◆í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✶✸
❊ ❝♦♠♦ Ié ❞❡ ❝❧❛ss❡ C1✱ ❛♣❧✐❝❛♥❞♦ ♦ ❧✐♠✐t❡ ❝♦♠ t→0✱ ♦❜t❡♠♦s q✉❡
I′(˜γn(˜tn))˜γ(˜tn)≥ −
1
n
1 +k˜γn(˜tn)k
−1
k˜γ(˜tn)k.
❆❣♦r❛ ❝♦♠♦ γ˜ é ❛r❜✐trár✐♦✱ tr♦❝❛♥❞♦γ˜ ♣♦r−˜γ✱ ♦❜t❡♠♦s
I′(˜γn(˜tn))˜γ(˜tn)≤ 1
n
1 +kγ˜n(˜tn)k
−1
k˜γ(˜tn)k,
❡ ❛ss✐♠✱
I
′(˜γ
n(˜tn))˜γ(˜tn)
k˜γ(˜tn)k ≤
1
n
1 +kγ˜n(˜tn)k
−1
,
♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠
0≤1 +kγ˜n(˜tn)k
kI′(˜γn(˜tn))kX′ ≤
1
n.
❈♦♠♦ n∈N✱ ✜①❛❞♦ ❛❝✐♠❛✱ é q✉❛❧q✉❡r✱ ❝♦♥❝❧✉í♠♦s q✉❡
lim
n→∞
1 +kγ˜n(˜tn)k
kI′(˜γn(˜tn))kX′ = 0. ✭✶✳✶✵✮
P♦r ✜♠✱ ❝♦♥s✐❞❡r❛♥❞♦(un)⊂X✱ t❛❧ q✉❡ un = ˜γn(˜tn)✱ ♣❛r❛ t♦❞♦n∈N✱ ♣♦r ✭✶✳✾✮ ❡ ✭✶✳✶✵✮ s❡❣✉❡ q✉❡
lim
n→∞I(un) =c ❡ nlim→∞ 1 +kunk
kI′(un)kX′ = 0.
P♦rt❛♥t♦ ♦❜t❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♣❛r❛I ♥♦ ♥í✈❡❧ P▼✳
❈❛♣ít✉❧♦
2
❯♠ Pr♦❜❧❡♠❛ ❆✉tô♥♦♠♦ ♥♦
R
N
❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ ❝❛♣ít✉❧♦ é ❛♥❛❧✐s❛r ❞❡t❡r♠✐♥❛❞❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❛✉tô♥♦♠♦s✱ ❛ ✜♠ ❞❡ r❡❧❛❝✐✲ ♦♥❛r ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❝♦♠ ♦ ♥í✈❡❧ ❞❡ ♠í♥✐♠♦ ❞♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛✳ P❛r❛ ✐ss♦ ❢❛r❡♠♦s ✉s♦ ❞❡ r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s ❞❡✈✐❞♦s ❛ ❇❡r❡st②❝❦✐ ❡ ▲✐♦♥s ❬✻❪ ♣❛r❛N ≥3✱ ❡ ❛ ❇❡r❡st②❝❦✐✱ ●❛❧❧♦✉ët
❡ ❑❛✈✐❛♥ ❬✺❪✱ ♣❛r❛N = 2✳ ❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛ ❛♣r❡s❡♥t❛❞♦ ❡♠ ✭✸✮✱ ♥❛ ✐♥tr♦❞✉çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ ✐st♦
é✱
−∆u=h(u), ❡♠ RN,
♣❛r❛ ♦ q✉❛❧ ✈❛♠♦s ❛ss✉♠✐r ❛s s❡❣✉✐♥t❡s ❤✐♣ót❡s❡s s♦❜r❡ h✿
✭❤✵✮ h:R→Ré ❝♦♥tí♥✉❛ ❡ í♠♣❛r❀
✭❤✶✮ s❡N ≥3✱−∞<lim inf
s→0 h(s)
s <lim sups→0 h(s)
s =−L <0,
s❡N= 2✱ lim
s→0 h(s)
s =−L∈(−∞,0)❀
✭❤✷✮ s❡N ≥3✱ lim
s→+∞|h(s)|s
−(N+2)/(N−2)= 0✱
s❡N= 2✱ ♣❛r❛ ❝❛❞❛α >0 ❡①✐st❡ ✉♠Cα>0✱ t❛❧ q✉❡
|h(s)| ≤Cαeαs 2
, ♣❛r❛ t♦❞♦ s∈R.
❆ss♦❝✐❛♠♦s ❛♦ ♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦ ♦ ❢✉♥❝✐♦♥❛❧ ♥❛t✉r❛❧J :H →R✱ ❞❛❞♦ ♣♦r
J(u) =
Z
RN
1
2|∇u|
2
−H(u)dx, ✭✷✳✶✮
❡♠ q✉❡ H(u) =
Z u
0
h(s)ds✳ ❙♦❜ ❛s ❤✐♣ót❡s❡s ✭❤✵✮✲✭❤✷✮ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ J ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ❡
