❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
❯♠❛ ❈❧❛ss❡ ❞❡ Pr♦❜❧❡♠❛s ❊❧í♣t✐❝♦s ❆ss✐♥t♦t✐❝❛♠❡♥t❡
▲✐♥❡❛r❡s ❡♠
R
N
♣♦r
❲❡s❧❡② ❞❡ ❋r❡✐t❛s ▼❡♥❞❡s
❇r❛sí❧✐❛
Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)
MM538c
Mendes, Wesley de Freitas
Uma Classe de Problemas Elípticos Assintoticamente Lineares em R^N. / Wesley de Freitas Mendes;
orientador Ricardo Ruviaro. -- Brasília, 2016. 77 p.
Dissertação (Mestrado - Mestrado em Matemática) --Universidade de Brasília, 2016.
❉❡❞✐❝❛tór✐❛
❆♦s ♠❡✉s ♣❛✐s
❏♦sé ❈❛r❧♦s ❡ ▼❛r❧✐ ▼❛r✐❛
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ q✉❡ ♠❡ ♣❡r♠✐t✐✉ ❡st❛r ❛q✉✐ ❡ r❡❛❧✐③❛r ♠❡✉ s♦♥❤♦ ❞❡ ❝♦♥❤❡❝❡r ❡ss❡ ♠✉♥❞♦ ❞❛ ▼❛t❡♠át✐❝❛✳ P♦r ♠❡ ♠♦str❛r ❝♦♠♦ s❡r ♠❛✐s ❝❛❧♠♦ ❡ ♣♦r r❡✈❡❧❛r s✉❛ ♣r❡s❡♥ç❛ ❡♠ ♠✐♥❤❛ ✈✐❞❛✳ ❆❣r❛❞❡ç♦ ♣♦r ❡ss❛ ❣r❛♥❞❡ ❝♦♥q✉✐st❛✳
❆♦ ♠❡✉ ♣❛✐✱ ❏♦sé ❈❛r❧♦s✱ ♣♦r s❡♠♣r❡ ❡st❛r ♣r❡s❡♥t❡ ❡♠ ♠✐♥❤❛ ❡❞✉❝❛çã♦✱ ♣❡❧♦s ❝♦♥s❡❧❤♦s ✈❛❧✐♦s♦s ❡ ♣❡❧♦ ❡①❡♠♣❧♦ ❞❡ ❤♦♠❡♠ q✉❡ q✉❡r♦ s❡r ✉♠ ❞✐❛✳ ➚ ♠✐♥❤❛ ♠ã❡✱ ▼❛r❧✐ ▼❛r✐❛✱ ♣❡❧♦ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡ ♣❡❧♦ ❝❛r✐♥❤♦✳ ➚ ♠✐♥❤❛ ❛✈ó✱ ❩❡❧♠❛✱ q✉❡ ♣❛r❛ ♠✐♠ é s✐♥ô♥✐♠♦ ❞❡ ❛♠♦r✳ ❆❣r❛❞❡ç♦ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ♣❡❧❛ ❝♦♥✜❛♥ç❛ q✉❡ ❞❡♣♦s✐t❛r❛♠ ❡♠ ♠✐♠✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ✐r♠ã♦s ❲✐❧❧✐❛♠✱ ❲❡♥❞❡❧ ❡ ❇ár❜❛r❛ ♣♦r ✐❧✉♠✐♥❛r❡♠ ♠✐♥❤❛ ✈✐❞❛✳
➚ ♠✐♥❤❛ ♥❛♠♦r❛❞❛✱ ▼✐❧❡♥❡ ❙♦❛r❡s✱ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ♠♦t✐✈❛♥❞♦ ❝♦♠ ✉♠ ❜❡❧♦ s♦rr✐s♦✱ ♣♦r s❡r ♣❛❝✐❡♥t❡ ❡ ♠❡ ❛❣✉❡♥t❛r ❢❛❧❛r ❞❡ ▼❛t❡♠át✐❝❛ ♦ t❡♠♣♦ t♦❞♦✳
❆♦s ♠❡✉s ♠❡❧❤♦r❡s ❛♠✐❣♦s✱ ❏♦sé ▼❛r✐❛ ❡ ❘♦♥② ▲✐♥s✱ q✉❡ ✜③❡r❛♠ ❞❡ ♠✐♥❤❛ ✈✐❞❛ ✉♠❛ ❢❡st❛✳ ➚ ♠✐♥❤❛ ❝♦❧❡❣❛ ▼❛②r❛ ❙♦❛r❡s✱ q✉❡ ❡st❡✈❡ ❝♦♠✐❣♦ ❞❡s❞❡ ♦ ✐♥í❝✐♦ ❞♦ ▼❡str❛❞♦ ♠❡ ❛♥✐♠❛♥❞♦ ❝♦♠ s❡✉ ❥❡✐t♦ ú♥✐❝♦ ❞❡ s❡r✳ ❆❣r❛❞❡ç♦ ❛♦s ❞❡♠❛✐s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡ss❛ ❥♦r♥❛❞❛✱ ❛ ❡❧❡s ♠❡✉ s✐♥❝❡r♦ ♦❜r✐❣❛❞♦✳ ❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦✱ ❛❣r❛❞❡ç♦ ♣❡❧♦s ❝♦♥❤❡❝✐♠❡♥t♦s tr❛♥s♠✐t✐❞♦s ❡ ♣❡❧♦ t❡♠♣♦ ❞✐s♣♦♥✐✲ ❜✐❧✐③❛❞♦✳ ❊♠ ❡s♣❡❝✐❛❧ ❛❣r❛❞❡ç♦ à ♣r♦❢❡ss♦r❛ ▲✐❧✐❛♥❡ ❞❡ ❆❧♠❡✐❞❛✱ ❛♦ ♣r♦❢❡ss♦r ▼❛✉r♦ ❘❛❜❡❧♦ ❡ à ♣r♦❢❡ss♦r❛ ❈át✐❛ ❘❡❣✐♥❛✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ à ♣r♦❢❡ss♦r❛ ❏❛q✉❡❧✐♥❡ ●♦❞♦② ❡ ❛♦ ♣r♦❢❡ss♦r ❊❞❝❛r❧♦s ❉♦♠✐♥❣♦s ♣♦r ❢♦r♠❛r❡♠ ♠✐♥❤❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❘✐❝❛r❞♦ ❘✉✈✐❛r♦✱ ❛❣r❛❞❡ç♦ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ✈❛❧✐♦s♦s ❡ ♣♦r ♠❡ ❝♦❧♦❝❛r ♥♦ ❝❛♠✐♥❤♦ ❝❡rt♦ s❡♠♣r❡ q✉❡ ♠❡ ❞❡s✈✐❛✈❛✳ ❙❡r❡✐ ❡t❡r♥❛♠❡♥t❡ ❣r❛t♦ ♣♦r t❡r ♠❡ ❛❝♦❧❤✐❞♦ ❝♦♠♦ ♦r✐❡♥t❛♥❞♦✳ ❆❣r❛❞❡ç♦ ♣♦r s❡r ❡ss❡ ❡①❡♠♣❧♦ ❞❡ ♣r♦✜ss✐♦♥❛❧ ❞❡❞✐❝❛❞♦ ❡ ♠❛✐s ❞♦ q✉❡ ✉♠ ♦r✐❡♥t❛❞♦r s✐♥t♦ q✉❡ ❣❛♥❤❡✐ ✉♠ ❛♠✐❣♦✳
❆❣r❛❞❡ç♦ à ❈◆Pq ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ à ❡st❡ tr❛❜❛❧❤♦✳
❘❡s✉♠♦
❇✉s❝❛r❡♠♦s ♥❡st❡ tr❛❜❛❧❤♦ ❡st❛❜❡❧❡❝❡r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r
(Pλ) −∆u+λu=f(x, u)u, x∈RN,
♦♥❞❡ λ > ✵ é ✉♠ ♣❛râ♠❡tr♦ ❡ f ∈C(RN ×R+,R+) s❛t✐s❢❛③ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s ❡s♣❡❝í✜❝❛s✳ P❛r❛ ✐ss♦✱
✉s❛♠♦s ❛ té❝♥✐❝❛ ✈❛r✐❛❝✐♦♥❛❧ ❡ ♥♦ss❛ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ s❡rá ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❈❡r❛♠✐✳ ❊st❛❜❡❧❡❝❡r❡♠♦s t❛♠❜é♠ r❡s✉❧t❛❞♦s ❞❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ (Pλ)❝♦♠
✉♠❛ ❝♦♥❞✐çã♦ ❡①tr❛ ❞❡ s✐♠❡tr✐❛ ♥❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡✳
P❛❧❛✈r❛s✲❈❤❛✈❡✿ Pr♦❜❧❡♠❛ ❙❡♠✐❧✐♥❡❛r❀ ❙♦❧✉çã♦ P♦s✐t✐✈❛❀ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛❀ ❈♦♥❞✐çã♦ ❞❡ ❈❡r❛♠✐❀ ❚é❝♥✐❝❛ ❱❛r✐❛❝✐♦♥❛❧❀ ❘❡s✉❧t❛❞♦s ❞❡ ▼✉❧t✐♣❧✐❝✐❞❛❞❡✳
❆❜str❛❝t
❲❡ s❡❡❦ ✐♥ t❤✐s ✇♦r❦ t♦ ❡st❛❜❧✐s❤ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢♦r t❤❡ s❡♠✐❧✐♥❡❛r ♣r♦❜❧❡♠
(Pλ) −∆u+λu=f(x, u)u, x∈RN
✇❤❡r❡λ >0✐s ❛ ♣❛r❛♠❡t❡r ❛♥❞f ∈C(RN×R+,R+)s❛t✐s✜❡s s♦♠❡ s♣❡❝✐✜❝s ❤②♣♦t❤❡s❡s✳ ❋♦r t❤✐s✱ ✇❡ ✉s❡
t❤❡ ✈❛r✐❛t✐♦♥❛❧ t❡❝❤♥✐q✉❡ ❛♥❞ ♦✉r ♠❛✐♥ t♦♦❧ ✇✐❧❧ ❜❡ t❤❡ ▼♦✉♥t❛✐♥✲P❛ss ❚❤❡♦r❡♠ ✇✐t❤ ❈❡r❛♠✐ ❝♦♥❞✐t✐♦♥✳ ❲❡ ❡st❛❜❧✐s❤✱ ❛s ✇❡❧❧✱ ♠✉❧t✐♣❧✐❝✐t② r❡s✉❧ts ❢♦r t❤❡ ♣r♦❜❧❡♠ (Pλ)✇✐t❤ ❛♥ ❡①tr❛ s②♠♠❡tr② ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡
♥♦♥❧✐♥❡❛r✐t②✳
❑❡②✲❲♦r❞s✿ ❙❡♠✐❧✐♥❡❛r Pr♦❜❧❡♠❀ P♦s✐t✐✈❡ ❙♦❧✉t✐♦♥❀ ▼♦✉♥t❛✐♥✲P❛ss❀ ❈❡r❛♠✐ ❈♦♥❞✐t✐♦♥❀ ❱❛r✐❛t✐♦♥❛❧ ❚❡❝❤♥✐q✉❡❀ ▼✉❧t✐♣❧✐❝✐t② ❘❡s✉❧ts✳
◆♦t❛çõ❡s
❆♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ✈❛♠♦s ✉t✐❧✐③❛r ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿
BR, BR(0)✱ ❜♦❧❛ ❛❜❡rt❛ ❝❡♥tr❛❞❛ ❡♠ ③❡r♦ ❡ ❝♦♠ r❛✐♦ R✳
BR(y), BR+y✱ ❜♦❧❛ ❝❡♥tr❛❞❛ ❡♠y ❡ ❝♦♠ r❛✐♦R✳
p∗= N p
N−p✱ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ❞❡ ❙♦❜♦❧❡✈✳
(P S)c✱ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧c✳
(Ce)c✱ ❝♦♥❞✐çã♦ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧c✳
un→u✱ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ✭❡♠ ♥♦r♠❛✮✳
un⇀ u✱ ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛✳
suppf✱ s✉♣♦rt❡ ❞❛ ❢✉♥çã♦f✳
h·,·i✱ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳
C, Ci✱ ❞❡♥♦t❛♠ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s✳
o(1)✱ ♦r❞❡♠ ♣❡q✉❡♥❛✳
R+✱ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥❡❣❛t✐✈♦s✳
D(A)✱ ❞♦♠í♥✐♦ ❞♦ ♦♣❡r❛❞♦rA✳
σ(A)✱ ❡s♣❡❝tr♦ ❞♦ ♦♣❡r❛❞♦rA✳
δij =
1, s❡i=j
0, s❡i6=j ✱ ❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r✳
X⊥✱ ❝♦♠♣❧❡♠❡♥t♦ ♦rt♦❣♦♥❛❧ ❛X✳
σess(A)✱ ❡s♣❡❝tr♦ ❡ss❡♥❝✐❛❧ ❞♦ ♦♣❡r❛❞♦rA✳
֒→✱ ✐♠❡rsã♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❡♠ ♦✉tr♦✳
C(X, Y)✱ ❡s♣❛ç♦ ❞❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s ❞❡X ❡♠Y✳
C1(X, Y)✱ ❡s♣❛ç♦ ❞♦s ❢✉♥❝✐♦♥❛✐s ❝♦♥t✐♥✉❛♠❡♥t❡
❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ X ❡♠Y✳
∂u ∂xi ♦✉
uxi✱ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❞❡ u❡♠ r❡❧❛çã♦ ❛xi✳
∂u
∂η =η.∇u✱ ❞❡r✐✈❛❞❛ ♥♦r♠❛❧ ❡①t❡r✐♦r✳
∆u=
N
X
i=1
∂2u
∂x2
i
✱ ▲❛♣❧❛❝✐❛♥♦ ❞❡u✳
∇u=∂u
∂x1
, ∂u ∂x2
,· · · , ∂u ∂xN
✱ ❣r❛❞✐❡♥t❡ ❞❡ u✳
X∗=nf :X→R
f é ❧✐♠✐t❛❞❛ o
✱ ❡s♣❛ç♦ ❞✉❛❧ ❞❡X✳
kukp=
hZ
Ω|
u|pdxi1/p✱ ♥♦r♠❛ ❞♦ ❡s♣❛ç♦Lp(Ω)✳
kukλ=
hZ
|∇u|2+λu2dxi1/2✱ ♥♦r♠❛ ❞❡ H1(RN)✳
kuk∞= inf
n C >0
|f(x)| ≤C ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ o
✱ ♥♦r♠❛ ❞♦ ❡s♣❛ç♦L∞(Ω)✳
Hk(Ω)✱ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈Wk,2(Ω)✳
H1
0(Ω)✱ ❢❡❝❤♦ ❞❡ C0∞(Ω)❝♦♠ ❛ ♥♦r♠❛k · kH1✳
D1,2(Ω)✱ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ C∞
0 (Ω)✳
Lp(Ω) =nu: Ω→R ♠❡♥s✉rá✈❡❧
Z
Ω|
u|pdx <∞o✳ L∞(Ω) =nu: Ω→R ♠❡♥s✉rá✈❡❧
|f(x)| ≤C ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ o
✳
Lploc(Ω) =nu: Ω→R ♠❡♥s✉rá✈❡❧
u|K ∈L
p(Ω),∀K⊂⊂Ω ❝♦♠♣❛❝t♦o✳
Wk,p(Ω) =nu∈Lp(Ω) D
αu∈Lp(Ω),∀α ♠✉❧t✐ í♥❞✐❝❡✱ t❛❧ q✉❡ |α| ≤ko✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ Pr❡❧✐♠✐♥❛r❡s ✸
✶✳✶ ❖♣❡r❛❞♦r❡s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✹ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✺ ❈❛r❛❝t❡r✐③❛çã♦ ❞❡λ1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷ ❊str✉t✉r❛ ❱❛r✐❛❝✐♦♥❛❧ ✲ ❆ ❈♦♥❞✐çã♦ ❞❡ ❈❡r❛♠✐ ✷✸
✷✳✶ ❈♦♥❞✐çõ❡s ♣❛r❛f ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷✳✷ ❘❡❣✉❧❛r✐❞❛❞❡ ❞♦ ❋✉♥❝✐♦♥❛❧Iλ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✷✳✸ ❈♦♥❞✐çã♦ ❞❡ ❈❡r❛♠✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸ ❊①✐stê♥❝✐❛ ❞❡ ❙♦❧✉çã♦ P♦s✐t✐✈❛ ✺✻
✸✳✶ ●❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✷ ❊①✐stê♥❝✐❛ ❞❡ ❙♦❧✉çã♦ P♦s✐t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✸ ❊①❡♠♣❧♦ ❞❡f ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✹ ❊①✐stê♥❝✐❛ ❞❡ ▼ú❧t✐♣❧❛s ❙♦❧✉çõ❡s ✻✽
✹✳✶ ❙✐♠❡tr✐❛ ♥❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✳✷ ❘❡s✉❧t❛❞♦ ❞❡ ▼✉❧t✐♣❧✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
■♥tr♦❞✉çã♦
❊st✉❞❛r❡♠♦s ♥❡st❡ tr❛❜❛❧❤♦ ❛ q✉❡stã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡✈✐❞♦ ❛ ❈♦st❛ ❡ ❚❡❤r❛♥✐[✻]✱
❜❡♠ ❝♦♠♦ r❡s✉❧t❛❞♦s ❞❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r✿
(Pλ) −∆u+λu=f(x, u)u, x∈RN,
♦♥❞❡ λ >0 é ✉♠ ♣❛râ♠❡tr♦ ❡ f ∈C(RN ×R+,R+)s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ✭❝♦♥❞✐çõ❡s ♣r❡❝✐s❛s
s❡rã♦ ✐♥❞✐❝❛❞❛s ♥♦ ❈❛♣ít✉❧♦ ✷✮✿
lim
s→0f(x, s) = 0, ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x, ✭✶✮
f(x, s)é ✉♠❛ ❢✉♥çã♦ ♥ã♦ ❞❡❝r❡s❝❡♥t❡ ❞❡s∈[0,∞)✱ ♣❛r❛ t♦❞♦x∈RN,❡ ❡①✐st❡♠ ❢✉♥çõ❡sg∈C(RN,R+)
❡h∈C(R+,R+)❝♦♠✿
lim
s→∞f(x, s) =g(x), |xlim|→∞f(x, s) =h(s) ❡ |x|→∞lim, s→∞f(x, s) =l∞∈(0,∞). ✭✷✮
◆❡ss❛s ❝♦♥❞✐çõ❡s✱(Pλ)é ✉♠ ♣r♦❜❧❡♠❛ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r✳ ◗✉❛♥❞♦ ❡ss❛ ❡q✉❛çã♦ é ❝♦♥s✐❞❡r❛❞❛
❡♠ ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ Ω ⊂ RN ✭❝♦♠✱ ❞✐❣❛♠♦s✱ ❛ ❝♦♥❞✐çã♦ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t✮✱ ❡①✐st❡ ✉♠❛
❡①t❡♥s❛ ❧✐t❡r❛t✉r❛ q✉❡ ❛❜♦r❞❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦✱ ❜❡♠ ❝♦♠♦ r❡s✉❧t❛❞♦s ❞❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡✳ ❉❡ ♣❛rt✐❝✉❧❛r ✐♥t❡r❡ss❡ é ❡♥tã♦ ♦ ❝❛s♦ r❡ss♦♥❛♥t❡✱ ♦♥❞❡ −λ∈ σ(S)❡S é ❛ ❧✐♥❡❛r✐③❛çã♦ ❛ss✐♥tót✐❝❛ ❞♦ ♣r♦❜❧❡♠❛✳ ❊♠
♦✉tr❛s ♣❛❧❛✈r❛s✱S :D(S)⊂L2(Ω)→L2(Ω)é ♦ ♦♣❡r❛❞♦r ❞❛❞♦ ♣♦r✿
Su(x) =−∆u(x)−g(x)u(x) ❡ D(S) =H01(Ω)∩H2(Ω). ✭✸✮
◆❡ss❡ ❝❛s♦✱ ❛ q✉❡stã♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s é ♠❛✐s ❞❡❧✐❝❛❞❛✳ ➱ ❝❧❛r♦ q✉❡ ❞❡s❞❡ q✉❡ Ω é ❧✐♠✐✲
t❛❞♦✱ σ(S) ❝♦♥s✐st❡ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧ ❞❡ ❛✉t♦✈❛❧♦r❡s ❝♦♠ ♠✉❧t✐♣❧✐❝✐❞❛❞❡s ✜♥✐t❛s ❡✱ ♣♦rt❛♥t♦✱
r❡ss♦♥â♥❝✐❛ é ✉♠ ❢❡♥ô♠❡♥♦ r❛r♦✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ ♦ ♥♦ss♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♠❡♥♦s s❡ t❡♠ ❢❡✐t♦ q✉❛♥❞♦Ω =RN ♥♦ ❝❛s♦ ❞♦ ♣r♦❜❧❡♠❛ (Pλ)✳ ❯♠❛ ❞❛s ❞✐✜❝✉❧❞❛❞❡s ♥❡ss❡ ❝❛s♦ é ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ❡s♣❡❝tr♦ ❞♦ ♦♣❡r❛❞♦r S ✐♥❝❧✉✐ ✉♠❛ ♣❛rt❡
❡ss❡♥❝✐❛❧✱ ❛ s❛❜❡r [−l∞,∞)✱ ❞❡ ♠♦❞♦ q✉❡ ♣r❡❝✐s❛♠♦s ❧✐❞❛r ❝♦♠ ✉♠ ♣r♦❜❧❡♠❛ r❡ss♦♥❛♥t❡ ♠✉✐t♦ ♠❛✐s
❝♦♠♣❧✐❝❛❞♦✳ ❆ ♦✉tr❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❧✐❞❛r ❝♦♠ t❛✐s ♣r♦❜❧❡♠❛s ❡♠ RN é ❛ ❢❛❧t❛ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❡①✐❜✐❞❛
■♥tr♦❞✉çã♦ ✷
◆♦ ❈❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ❝♦♥❝❡✐t♦s ♣r❡❧✐♠✐♥❛r❡s ❡ss❡♥❝✐❛✐s ❛♦ ❜♦♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ♥♦ss♦s ♣r✐♥❝✐♣❛✐s t❡♦r❡♠❛s ❡ ❧❡♠❛s✳ ◆♦ ❈❛♣ít✉❧♦ ✷✱ ✐♥tr♦❞✉③✐♠♦s ❛ ❡str✉t✉r❛ ✈❛r✐❛❝✐♦♥❛❧ ❡ ❡st✉❞❛♠♦s ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ ❈❡r❛♠✐✳ Pr✐♠❡✐r♦✱ ❞❡✜♥✐r❡♠♦s
Λ= inf
Z h
|∇u|2−g(x)u2idx u∈H
1(RN
), Z
u2dx= 1
.
❙❡ (Pλ) ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ❡♥tã♦✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ❞❡✈❡♠♦s t❡r λ < |Λ|✱ ❞❡ ❢♦r♠❛ q✉❡ ❛ss✉♠✐♠♦s
0 < λ < |Λ| ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳ ❆❞❡♠❛✐s✱ ❡①♣❧♦r❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ♣❛r❛ ♣r♦❜❧❡♠❛s ❧✐♥❡❛r❡s ❞❡
❛✉t♦✈❛❧♦r ❡♠RN✱ ❡ s✐st❡♠❛t✐❝❛♠❡♥t❡ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡ ❈♦♠♣❛❝✐❞❛❞❡ ❞❡ ▲✐♦♥s✱ s♦♠♦s
❝❛♣❛③❡s ❞❡ ♠♦str❛r q✉❡ ❛ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ ❈❡r❛♠✐ ✈❛❧❡ ♣❛r❛ ✉♠ ❝❡rt♦ ✐♥t❡r✈❛❧♦ ❞❡ ✈❛❧♦r❡s ❞❡ ❡♥❡r❣✐❛ ❞♦ ❢✉♥❝✐♦♥❛❧ ❝♦rr❡s♣♦♥❞❡♥t❡✳
◆♦ ❈❛♣ít✉❧♦ ✸✱ ♣r♦✈❛♠♦s ♥♦ss♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡ ❡①✐stê♥❝✐❛ ❡ ❡st❛❜❡❧❡❝❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ (Pλ)✱ ♣❛r❛ t♦❞♦0< λ <|Λ|✳ ■st♦ é ❢❡✐t♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛❝❤❛♥❞♦ ✉♠❛ ❝❛♥❞✐❞❛t❛ ♣❛r❛
✉♠ ♥í✈❡❧ ❝rít✐❝♦ ❛tr❛✈és ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳ ❊♥tã♦✱ s♦❜ ✉♠❛ ❝♦♥❞✐çã♦ ❛❞✐❝✐♦♥❛❧ ♣❛r❛
f(x, s)✱ ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ♥♦ ✐♥✜♥✐t♦ é ✉s❛❞♦ ♣❛r❛ ♠♦str❛r q✉❡ ♥♦ss♦ ♥í✈❡❧
❝❛♥❞✐❞❛t♦ é ❞❡ ❢❛t♦ ♦ ♥í✈❡❧ ♦♥❞❡ ❛ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ ❈❡r❛♠✐ ✈❛❧❡✱ ♣❡r♠✐t✐♥❞♦ ❛ss✐♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡ t❡♦r❡♠❛s ❞❡ ♣♦♥t♦s ❝rít✐❝♦s✳ ❱❛❧❡ ♥♦t❛r q✉❡✱ ❝♦♠♦Λ é ♦ ♠❡♥♦r ♣♦♥t♦ ❡s♣❡❝tr❛❧ ❞❡S ❡σess(S) = [−l∞,∞)✱ t❡♠♦s
Λ ≤ −l∞ ❡ ❡♥tã♦✱ s❡ 0 < λ < |Λ|✱ ♣♦❞❡ ♠✉✐t♦ ❜❡♠ ❛❝♦♥t❡❝❡r q✉❡ −λ ∈ σ(S)✳ ◆ã♦ ♦❜st❛♥t❡✱ ♥♦ss♦
r❡s✉❧t❛❞♦ ❞❡ ❡①✐stê♥❝✐❛✱ q✉❡ ✈❡r❡♠♦s ♥♦ ❚❡♦r❡♠❛ ✸✳✶✱ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ♣r♦❜❧❡♠❛ s❡r ♦✉ ♥ã♦ r❡ss♦♥❛♥t❡✳ ❋✐♥❛❧♠❡♥t❡✱ ♥♦ ❈❛♣ít✉❧♦ ✹✱ ❝♦♥s✐❞❡r❛♠♦s ❛ q✉❡stã♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ ♠ú❧t✐♣❧❛s s♦❧✉çõ❡s q✉❛♥❞♦f(x, s)
é ✉♠❛ ❢✉♥çã♦ ♣❛r ❞❡ s✳ ◆♦ss❛ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ♥❡ss❛ s❡çã♦ é ✉♠❛ ✈❛r✐❛♥t❡ ❞♦ t❡♦r❡♠❛ ❞♦ ♣♦♥t♦
❝rít✐❝♦ ❛❜str❛t♦ ♣❛r❛ ❢✉♥❝✐♦♥❛✐s ♣❛r❡s✳
❙❛❧✈♦ ♠❡♥çã♦ ❡♠ ❝♦♥trár✐♦✱ t♦❞❛s ❛s ✐♥t❡❣r❛✐s s❡rã♦ t♦♠❛❞❛s s♦❜r❡ t♦❞♦ ♦RN ❡C, Ci r❡♣r❡s❡♥t❛rã♦
❈❛♣ít✉❧♦
1
Pr❡❧✐♠✐♥❛r❡s
❆♥t❡s ❞❡ ❝♦♠❡ç❛r♠♦s ♥♦ss♦ tr❛❜❛❧❤♦ ♣r✐♥❝✐♣❛❧✱ ♣r❡❝✐s❛♠♦s ❝♦♥❤❡❝❡r ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ út❡✐s ♥♦ ❞❡❝♦rr❡r ❞♦ ♠❡s♠♦✳
✶✳✶ ❖♣❡r❛❞♦r❡s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠
❱❛♠♦s ❝♦♥❤❡❝❡r ♣r✐♠❡✐r❛♠❡♥t❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❞❡ ✷❛ ♦r❞❡♠✱ ♦ ❧❡♠❛ ❞❡ ❍♦♣❢ ❡ ♦ ❧❡♠❛ ❞❡ ▲✐♦♥s✱ q✉❡ s❡rã♦ ✉s❛❞♦s ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✳ ◆♦ q✉❡ s❡❣✉❡✱ ❡st❛♠♦s ♥♦s ❜❛s❡❛♥❞♦ ❡♠ ❊✈❛♥s ❬✽❪ ❡ ❆❞❛♠s ❬✶❪✳
❈♦♥s✐❞❡r❡ ♦ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❞❡ ✷❛ ♦r❞❡♠ ❞❛❞♦ ♣❡❧❛ ❡①♣r❡ssã♦
Lu:=
N
X
i,j=1
aij(x)uxixj +
N
X
i=1
bi(x)uxi+c(x)u, ✭✶✳✶✮ ♦♥❞❡ u ∈C2(Ω)✱ ♦s ❝♦❡✜❝✐❡♥t❡s aij, bi, c : Ω → Rsã♦ ❢✉♥çõ❡s ❞❛❞❛s ❡ Ω ⊂RN é ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦✳
❖❜s❡r✈❡ q✉❡ ❝♦♠♦ u∈ C2(Ω)✱ ❡♥tã♦ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✇❛r③ t❡♠♦s u
xixj = uxjxi✱ ♣❛r❛ t♦❞♦ i, j ∈
1, ..., N✳ ▲♦❣♦✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡✱ ♣❛r❛ ❝❛❞❛x∈Ω✱ ❛ ♠❛tr✐③A(x) = [aij(x)]
N×N é s✐♠étr✐❝❛✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❉✐③❡♠♦s q✉❡ ♦ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ ❡♠(1.1)é ❡❧í♣t✐❝♦ ♥♦ ♣♦♥t♦x∈Ωs❡ ❛ ❢♦r♠❛ q✉❛❞rát✐❝❛
❛ss♦❝✐❛❞❛ à ♠❛tr✐③ A(x)é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✱ ♦✉ s❡❥❛✱ s❡λ(x)❢♦r ♦ ♠❡♥♦r ❛✉t♦✈❛❧♦r ❞❡ A(x)✱ ❡♥tã♦
N
X
i,j=1
aij(x)ξ
iξj ≥λ(x)|ξ|2>0
♣❛r❛ t♦❞♦ ξ= (ξ1, ..., ξN)∈RN \ {0}✳ ❖ ♦♣❡r❛❞♦r é ❞✐t♦ ❡❧í♣t✐❝♦ ❡♠Ω s❡ ❢♦r ❡❧í♣t✐❝♦ ❡♠ ❝❛❞❛ ♣♦♥t♦ ❞❡
Ω✳ ❋✐♥❛❧♠❡♥t❡✱ ❞✐③❡♠♦s q✉❡ L é ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦ ❡♠ Ω s❡ ❡①✐st❡ θ0 >0 t❛❧ q✉❡ λ(x)≥θ0 ♣❛r❛
t♦❞♦ x∈Ω✳ ❉✐③❡♠♦s q✉❡L ❡stá ♥❛ ❢♦r♠❛ ❞✐✈❡r❣❡♥t❡ q✉❛♥❞♦
Lu:=−
N
X
i,j
(aij(x)uxi)xj +
N
X
i=1
✶✳✶ ❖♣❡r❛❞♦r❡s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✹
❖❜s❡r✈❡ q✉❡ q✉❛♥❞♦Lé ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦✱ ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡
ξA(x)ξ=
N
X
i,j=1
aij(x)ξ
iξj≥θ0|ξ|2, ξ∈RN.
❊ ❛ss✐♠ t♦♠❛♥❞♦ ξ=ei✱ ✈❡t♦r ❞❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ❞❡RN✱ ♦❜t❡♠♦s
eiA(x)ei=aii(x)≥θ0|ei|2=θ0, i= 1, ..., N ❡ x∈Ω. ✭✶✳✷✮
❚❡♦r❡♠❛ ✶✳✶✳ ❙❡❥❛L✉♠ ♦♣❡r❛❞♦r ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦ ❡♠Ω❝♦♠c≡0❡♠Ω✳ ❙❡u∈C2(Ω)∩C(Ω)
❡ Lu≥0 ❡♠ Ω✱ ❡♥tã♦ max
Ω u= max∂Ω u.
❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡Lu >0 ❡♠Ω❡ q✉❡ ❡①✐st❡x˜∈Ωt❛❧ q✉❡u(˜x) = max
Ω
u✳ ❈♦♠♦
L é ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦✱ ❛ ♠❛tr✐③ ❞♦s ❝♦❡✜❝✐❡♥t❡sA=A(˜x)é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✳ P♦r As❡r s✐♠étr✐❝❛✱
❡①✐st❡ ✉♠❛ ♠❛tr✐③ ♦rt♦❣♦♥❛❧O=ON×N✱ ♦✉ s❡❥❛✱O−1=OT✱ t❛❧ q✉❡
OAOT =
λ1 0 ... 0
0 λ2 ... 0
: : ✳✳✳ 0 0 0 ... λN
❡ ♣♦r L s❡r ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦✱ t❡♠♦s λi ≥θ0 >0, i= 1, ..., N✳ ❖ t❡r♠♦ ❣❡r❛❧ ❞❛ ♠❛tr✐③ ❛❝✐♠❛ é
❞❛❞♦ ♣♦r
δklλk= N
X
j=1
okj N
X
i=1
aijoTil = N
X
i,j=1
okjaijoli. ✭✶✳✸✮
❈♦♥s✐❞❡r❛♥❞♦ ❛❣♦r❛ ❛ ♥♦✈❛ ✈❛r✐á✈❡❧y(x) := ˜x+O(x−x˜)✱ ♥♦t❡ q✉❡ y(˜x) = ˜x❡
y−x˜ = O(x−x˜) ⇒
OT(y−x˜) =OTO(x−x˜) = O−1O(x−x˜) =x−x˜ ⇒
˜
x+OT(y−x˜) = x
❛ss✐♠
u(x) =u(˜x+OT(y−x˜)) :=v(y(x)).
❖❜s❡r✈❡ q✉❡y(˜x)é ♦ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❞❛ ❢✉♥çã♦v✱ ♣♦✐sx˜é ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❞❡u✱ ❡ ♣♦rt❛♥t♦
∇u(˜x) =∇v(y(˜x)) =∇v(˜x) = 0 ❡ D2v(˜x)≤0,
❝♦♠ ❛ s❡❣✉♥❞❛ ✐♥❡q✉❛çã♦ ❛❝✐♠❛ s✐❣♥✐✜❝❛♥❞♦ q✉❡ ❛ ♠❛tr✐③ ❍❡ss✐❛♥❛ ❞♦ v ♥♦ ♣♦♥t♦x˜ é ♥ã♦ ♣♦s✐t✐✈❛✳ ❙❡
y = (y1, ..., yN)✱ ❡♥tã♦
yk = ˜xk+ N
X
j=1
okj(xj−x˜j) ⇒
∂yk
∂xi
✶✳✶ ❖♣❡r❛❞♦r❡s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✺
♣❛r❛ ❝❛❞❛k= 1, ..., N✱ ❧♦❣♦
uxi=
N
X
k=1
∂v ∂yk
∂yk
∂xi
=
N
X
k=1
vykoki, ❡ ❞♦ ♠❡s♠♦ ♠♦❞♦
uxixj =
N
X
k, l=1
vykylokiolj, ✭✶✳✹✮ ♣❛r❛ i, j= 1,· · · , N✳
❈♦♠♦∇u(˜x) = 0✱ ♦❜t❡♠♦s
Lu(˜x) =
N
X
i,j=1
aij(˜x)uxixj(˜x) +
N
X
i=1
bi(˜x)uxi(˜x), ♣♦✐s c≡0, ❡♠ Ω
=
N
X
i,j=1
aij(˜x)uxixj(˜x), ♣♦✐s
N
X
i=1
bi(˜x)uxi(˜x) = (b.∇u)(˜x) = 0
=
N
X
i,j=1
aij(˜x)
N
X
k,l=1
vykylokiolj, ♣♦r (1.4)
=
N
X
k,l=1
vykyl
N
X
i,j=1
aij(˜x)okiolj
=
N
X
k,l=1
vykylδklλk, ♣♦r (1.3)
=
N
X
k=1
vykykλk.
❯♠❛ ✈❡③ q✉❡D2v(˜x)≤0✱ t❡♠♦s q✉❡e
kD2v(˜x)ek≤0 ❡ ✐st♦ ✐♠♣❧✐❝❛ q✉❡vykyk(˜x)≤0✱ ♣❛r❛k= 1, ..., N✳ ❈♦♠♦ ♦s ♥ú♠❡r♦sλi′ssã♦ ♣♦s✐t✐✈♦s✱ ❝♦♥❝❧✉í♠♦s ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ q✉❡
Lu(˜x) =
N
X
k=1
vykyk(˜x)λk≤0,
♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ▲♦❣♦✱ s❡ Lu > 0 ❡♠ Ω✱ ❛ ❢✉♥çã♦u ♥ã♦ ♣♦❞❡ ❛ss✉♠✐r s❡✉ ♠á①✐♠♦ ❡♠Ω✱ ✐st♦ é✱ max
Ω
u= max
∂Ω u✳
❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ♦ ❝❛s♦ ❣❡r❛❧Lu≥0✳ ❙❡❥❛γ∈R❛r❜✐trár✐♦✱ε >0❡ ❝♦♥s✐❞❡r❡
uε(x) :=u(x) +εeγx1, x= (x1, ..., xN)∈Ω.
❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡L✱ ❛ ❡q✉❛çã♦ ✭✶✳✷✮✱ ❛ r❡❣✉❧❛r✐❞❛❞❡ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❡Lu≥0✱ ♦❜t❡♠♦s
Luε = Lu+εL(eγx1)
= Lu+εeγx1(a11(x)γ2+b1(x)γ)
≥ εeγx1(θ
0γ2− kb1k∞γ).
✶✳✶ ❖♣❡r❛❞♦r❡s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✻
❞❡♠♦♥str❛çã♦ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡
max
Ω
uε= max ∂Ω uε.
▼❛s u≤uε✱ ❡ ♣♦rt❛♥t♦
max
Ω
u≤max
Ω
uε= max
∂Ω uε≤max∂Ω u+εmax∂Ω e
γx1.
❋❛③❡♥❞♦ε→0+✱ ❝♦♥❝❧✉í♠♦s q✉❡max
Ω u≤max∂Ω u✳ ❯♠❛ ✈❡③ q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❝♦♥trár✐❛ é tr✐✈✐❛❧♠❡♥t❡
s❛t✐s❢❡✐t❛✱ ❝♦♥❝❧✉í♠♦s q✉❡
max
Ω u= max∂Ω u.
❚❡♦r❡♠❛ ✶✳✷✳ ✭Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ❋r❛❝♦✮ ❙❡❥❛ L ✉♠ ♦♣❡r❛❞♦r ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦ ❡♠ Ω ❝♦♠
c≤0 ❡♠Ω✳ ❙❡u∈C2(Ω)∩C(Ω) ❡Lu≥0❡♠ Ω✱ ❡♥tã♦ max Ω
u≤max
∂Ω u + ✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ Ω+ := {x∈ Ω| u(x)>0}✳ ❙❡ Ω+ ❢♦r ✈❛③✐♦✱ ❡♥tã♦ u≤0 ❡♠ Ω✳ ❚♦♠❡ x∈ ∂Ω❡
(xn) ⊂Ω t❛❧ q✉❡ xn → x✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡u ❛té ❛ ❢r♦♥t❡✐r❛✱ t❡♠♦s q✉❡ u(x) = lim
n→∞u(xn) ≤0✱
❛ss✐♠u≤0❡♠Ω❡ ♣♦rt❛♥t♦
max
Ω
u≤0 = max
∂Ω u +,
♦♥❞❡u+(x) := max{u(x),0}.
▲♦❣♦✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ Ω+ 6=∅✳ ❚♦♠❡ ❡♥tã♦ x∈Ω+✱ ❧♦❣♦ u(x)>0 ❡ ❝♦♠♦ ué ❝♦♥tí♥✉❛✱ ❡♥tã♦
❡①✐st❡r >0t❛❧ q✉❡u >0❡♠Br(x)❡ ❛ss✐♠ Ω+é ❛❜❡rt♦ ❡♠Ω❡✱ ♣♦rt❛♥t♦✱ ❛❜❡rt♦ ❡♠RN✳ ❙❡❥❛
Ku:=Lu−c(x)u=
N
X
i,j=1
aij(x)u xixj +
N
X
i=1
bi(x)u xi,
❞❡ss❡ ♠♦❞♦✱ ❝♦♠♦ c≤0 ❡♠ Ω✱ ♦❜t❡♠♦s Ku≥0✱ ♣❛r❛ u∈C2(Ω+)∩C(Ω+)✳ ❙❡❣✉❡ ❡♥tã♦ ❞♦ ❚❡♦r❡♠❛
1.1✱ ❛♣❧✐❝❛❞♦ ❛♦ ♦♣❡r❛❞♦rK✱ q✉❡
max
Ω+
u= max
∂Ω+u.
❯♠❛ ✈❡③ q✉❡Ω = Ω+∩Ω\Ω+ ❡u≤0 ❡♠Ω\Ω+✱ s❡❣✉❡ q✉❡
max
Ω u= maxΩ+ u= max∂Ω+u.
❆ss✐♠✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡
max
∂Ω+u≤max∂Ω u
+.
❈♦♥s✐❞❡r❡x0∈∂Ω+ t❛❧ q✉❡u(x0) = max
∂Ω+u✳ ❆ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡u❡ ❛ ❞❡✜♥✐çã♦ ❞❡ Ω
+ ✐♠♣❧✐❝❛♠ q✉❡
u(x0)≥0✳ ❚❡♠♦s ❞♦✐s ❝❛s♦s ❛ ❝♦♥s✐❞❡r❛r✿
❈❛s♦ ✶✮u(x0) = 0 :
◆❡st❡ ❝❛s♦ ❞❡✈❡♠♦s t❡ru≤0❡♠Ω♣♦✐su(x)≤max
Ω u= max∂Ω+u=u(x0) = 0✳ ▲♦❣♦✱u
+= 0❡♠ ∂Ω❡
♣♦rt❛♥t♦
u(x0) = max
∂Ω+u= 0 = max∂Ω u
✶✳✶ ❖♣❡r❛❞♦r❡s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✼
❈❛s♦ ✷✮u(x0)>0 :
◆❡st❡ ❝❛s♦✱ ❝♦♠♦ Ω+ é ❛❜❡rt♦ ❡♠ Ω✱ ❞❡✈❡♠♦s t❡r x
0 ∈ ∂Ω✳ ❉❡ ❢❛t♦✱ s❡ ♥ã♦ ❢♦ss❡ ❛ss✐♠✱ t❡rí❛♠♦s
x0∈Ω❡ ❝♦♠♦ué ❝♦♥tí♥✉❛✱ ❡♥tã♦us❡r✐❛ ♣♦s✐t✐✈❛ ❡♠ t♦❞❛ ✉♠❛ ❜♦❧❛Bε(x0)⊂Ω+✱ ❝♦♥tr❛r✐❛♥❞♦ ♦ ❢❛t♦
❞❡ q✉❡x0∈∂Ω+✳ ❉❛í
max
∂Ω+u=u(x0) =u
+(x
0)≤max
∂Ω u +,
❡ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳
❚❡♦r❡♠❛ ✶✳✸✳ ✭Pr✐♥❝í♣✐♦ ❞❛ ❈♦♠♣❛r❛çã♦✮✳ ❙❡❥❛ ▲ ✉♠ ♦♣❡r❛❞♦r ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦ ❡♠Ω❝♦♠c≤0
❡ u∈C2(Ω)∩C(Ω)✳ ❙❡Lu≥0 ❡♠Ω❡ u≤0 ❡♠∂Ω✱ ❡♥tã♦ u≤0 ❡♠Ω✳
❉❡♠♦♥str❛çã♦✳ P❡❧♦ ❚❡♦r❡♠❛1.2✱ t❡♠♦s q✉❡
u(x)≤max
Ω u≤max∂Ω u += 0,
♣♦✐s u+= 0 ❡♠∂Ω✱ ❧♦❣♦u≤0 ❡♠Ω✳
▲❡♠❛ ✶✳✶✳ ✭▲❡♠❛ ❞❡ ❍♦♣❢✮✳ ❙✉♣♦♥❤❛ q✉❡B ⊂RN é ✉♠❛ ❜♦❧❛ ❛❜❡rt❛✱Lé ✉♠ ♦♣❡r❛❞♦r ✉♥✐❢♦r♠❡♠❡♥t❡
❡❧í♣t✐❝♦ ❡♠ B✱ u∈C2(B) ❡Lu≥0 ❡♠ B✳ ❙✉♣♦♥❤❛ ❛✐♥❞❛ q✉❡ ❡①✐st❡ x
0∈∂B t❛❧ q✉❡u é ❝♦♥tí♥✉❛ ❡♠
x0 ❡u(x)< u(x0)✱ ♣❛r❛ t♦❞♦ x∈B✳ ❊♥tã♦✱
✐✮ s❡c= 0❡♠ B ❡ ❡①✐st❡ ❛ ❞❡r✐✈❛❞❛ ♥♦r♠❛❧ ∂u
∂η(x0)✱ ❡♥tã♦ ∂u
∂η(x0)>0❀
✐✐✮ s❡c≤0 ❡♠ Ω❡u(x0)≥0✱ ❡♥tã♦ ✈❛❧❡ ♦ ♠❡s♠♦ r❡s✉❧t❛❞♦ ❞♦ ✐t❡♠ ❛❝✐♠❛✳
❆♥t❡s ❞❡ ♣r♦✈❛r ♦ ❧❡♠❛ ❞❡ ❍♦♣❢ ✈❛❧❡ ♦❜s❡r✈❛r q✉❡ s❡x0∈∂B é ✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❧♦❝❛❧ ❡ ❡①✐st❡
∂u
∂η(x0)✱ ❡♥tã♦ é s❡♠♣r❡ ✈❡r❞❛❞❡ q✉❡
∂u
∂η(x0) = limh→0−
u(x0+hη)−u(x0)
h ≥0
✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ s✐♥❛❧ ❞❡Lu✳ ❆ ✐♥❢♦r♠❛çã♦ ❛❞✐❝✐♦♥❛❧ ❞❛❞❛ ♣❡❧♦ ❧❡♠❛ é q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ❡str✐t❛✳
❉❡♠♦♥str❛çã♦✳ P♦❞❡♠♦s s✉♣♦r✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ u∈C(B)❡ q✉❡u(x)< u(x0)♣❛r❛ t♦❞♦
x∈B\{x0}✳ ❉❡ ❢❛t♦✱ s❡ ♥ã♦ ❢♦r ❡ss❡ ♦ ❝❛s♦✱ é s✉✜❝✐❡♥t❡ t♦♠❛r ✉♠❛ ♥♦✈❛ ❜♦❧❛B′ ⊂B q✉❡ é ✐♥t❡r♥❛♠❡♥t❡
t❛♥❣❡♥t❡ à B ♥♦ ♣♦♥t♦x0✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♥❢♦r♠❡ ✈❡r❡♠♦s ♣♦st❡r✐♦r♠❡♥t❡✱ ♣♦❞❡♠♦s t❛♠❜é♠ s✉♣♦r q✉❡
B =Br(0)✳
❋❡✐t❛s ❛s ❝♦♥s✐❞❡r❛çõ❡s ❛❝✐♠❛✱ ✈❛♠♦s ❛ss✉♠✐r ✐♥✐❝✐❛❧♠❡♥t❡ ❛s ❤✐♣ót❡s❡s ❞♦ ✐t❡♠ ✭✐✐✮ ❡ ❝♦♥s✐❞❡r❛r✱ ♣❛r❛
γ >0❛ s❡r ❞❡t❡r♠✐♥❛❞♦✱ ❛ ❢✉♥çã♦
v(x) :=e−γ|x|2−e−γr2, x∈B.
P❛r❛ ❝❛❞❛i, j= 1, ..., N✱ t❡♠♦s q✉❡
vxi=−2γxie
−γ|x|2
❡
vxixj =
(
4γ2x
ixje−γ|x|
2
, s❡ i6=j,
4γ2x2
ie−γ|x|
2
−2γe−γ|x|2
✶✳✶ ❖♣❡r❛❞♦r❡s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✽
♦✉ s❡❥❛✱
vxixj = (4γ
2x
ixj−2γδij)e−γ|x|
2
,
❞❡ ♠♦❞♦ q✉❡
Lv(x) =
N
X
i,j=1
aij(x)v xixj +
N
X
i=1
bi(x)v
xi+c(x)v
= e−γ|x|2
N
X
i,j=1
(4γ2aij(x)xixj−2γδijaij(x))−2γ N
X
i=1
(bi(x)xi) +c(x)
−c(x)e−γr2.
❯s❛♥❞♦ ❛s ❤✐♣ót❡s❡ s♦❜r❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡L✱ t❡♠♦s q✉❡
N
X
i,j=1
aij(x)xixj ≥θ0|x|2,
N
X
i=1
bi(x)xi≤ |x| N
X
i=1
kbik∞=C1,
N
X
i,j=1
δijaij(x)≤ N
X
i=1
kaijk∞=C2
❡
−c(x)e−γr2 ≥0, ♣♦✐s c≤0
❝♦♠C1, C2≥0✳ ❆s ❡st✐♠❛t✐✈❛s ❛❝✐♠❛ ✐♠♣❧✐❝❛♠ q✉❡
Lv(x)≥e−γ|x|24γ2θ0|x|2−2γ(C1+C2)− kck∞
.
❉❡ss❡ ♠♦❞♦✱ ❢❛③❡♥❞♦C3:=C1+C2❡ ❞❡♥♦t❛♥❞♦Ar:=Br(0)\Br/2(0)✱ t❡♠♦s q✉❡✱ ♣❛r❛ t♦❞♦x∈Ar✱
✈❛❧❡
Lv(x)≥e−γ|x|24γ2θ0
r
2
2
−2γC3− kck∞
.
❊s❝♦❧❤❡♥❞♦ γ > 0 ❣r❛♥❞❡ ♦ s✉✜❝✐❡♥t❡ ❞❡ ♠♦❞♦ q✉❡ ♦ t❡r♠♦ ❡♥tr❡ ♣❛rê♥t❡s❡s ❛❝✐♠❛ s❡❥❛ ♣♦s✐t✐✈♦✱
❝♦♥❝❧✉í♠♦s q✉❡
Lv≥0, ❡♠ Ar.
◆♦t❡ q✉❡ s❡x∈B=Br(0)✱ ❡♥tã♦
|x|2 < r2 ⇒
−γ|x|2 > −γr2 ⇒ e−γ|x|2 > e−γr2,
❧♦❣♦v(x) =e−γ|x|2−e−γr2>0 ❡♠B ❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱v é ♣♦s✐t✐✈❛ ❡♠∂B
r/2(0)❡ ✉♠❛ ✈❡③ q✉❡x0é ✉♠
♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❡str✐t♦ ❞❡u❡ ❛ ❢✉♥çã♦ vé ❝♦♥tí♥✉❛ ♥♦ ❝♦♠♣❛❝t♦∂Br/2(0)✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡rε >0❞❡
t❛❧ ♠♦❞♦ q✉❡
u(x0)≥u(x) +εv(x), x∈∂Br/2(0).
◆♦t❡ ❛✐♥❞❛ q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣❡r♠❛♥❡❝❡ ✈á❧✐❞❛ ❡♠∂Br(0)♣♦✐s✱ ♥❡ss❡ ❝♦♥❥✉♥t♦ ❛ ❢✉♥çã♦vs❡
❛♥✉❧❛✳ ❉❡ss❡ ♠♦❞♦✱ ❛ ❢✉♥çã♦
✶✳✶ ❖♣❡r❛❞♦r❡s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✾
é t❛❧ q✉❡
(
Lw=Lu+εLv−c(x)u(x0)≥0, em Ar,
w≤0, em ∂Ar.
❙❡❣✉❡ ❡♥tã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❈♦♠♣❛r❛çã♦ ✭❚❡♦r❡♠❛1.3✮ q✉❡w≤0❡♠Ar✳ ❖❜s❡r✈❡ ❛❣♦r❛ q✉❡✱ ❝♦♠♦
x0 ∈∂B✱ t❡♠♦s q✉❡ v(x0) = 0✳ ▲♦❣♦✱ w(x0) =u(x0) +εv(x0)−u(x0) = 0❡✱ ♣♦rt❛♥t♦✱ x0 é ✉♠ ♣♦♥t♦
❞❡ ♠á①✐♠♦ ❞❡ w ❡♠ Ar✳ ❉❡ss❡ ♠♦❞♦✱ ♣❡❧❛ ♦❜s❡r✈❛çã♦ ❛♥t❡s ❞❛ ❞❡♠♦♥str❛çã♦✱ s✉♣♦♥❞♦ q✉❡ ❡①✐st❡ ❛
❞❡r✐✈❛❞❛ ♥♦r♠❛❧ ❞❡u♥♦ ♣♦♥t♦x0✱ ❞❡✈❡♠♦s t❡r ∂w
∂η(x0)≥0✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠
∂w
∂η(x0) = ∇w(x0).η
= ∇u+εv−u(x0)
(x0).η
= ∇u(x0).η+ε∇v(x0).η≥0,
❧♦❣♦✱ ♥♦t❛♥❞♦ q✉❡ x0
r é ♦ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ ❞❡Br(0)✱ t❡♠♦s ∇u(x0).η = ∂u
∂η(x0) ≥ −ε∇v(x0).
x0 r
= −ε−2γx0e−γ|x0|
2x0 r
= 2γε|x0|
2
r e
−γ|x0|2 >0.
■ss♦ ❡st❛❜❡❧❡❝❡ ❛ ✈❡r❛❝✐❞❛❞❡ ❞❡ (ii)♥♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❜♦❧❛ B ❡stá ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠✳ P❛r❛ ♦ ❝❛s♦
❣❡r❛❧ ❡♠ q✉❡ B =Br(y)✱ ❜❛st❛ ❝♦♥s✐❞❡r❛r v(x) = e−γ|x−y|
2
−e−γr2✱ ♣❛r❛ x∈ B
r(y) ❡ ♣r♦❝❡❞❡r ❝♦♠♦
❛❝✐♠❛✳ ❆ ♣r♦✈❛ ❞♦ ✐t❡♠(i)t❛♠❜é♠ ♣♦❞❡ s❡r ❢❡✐t❛ r❡♣❡t✐♥❞♦ ♦s ♠❡s♠♦s ♣❛ss♦s✳
Pr❡❝✐s❛♠♦s t❛♠❜é♠ ❝♦♥❤❡❝❡r ♦ ❢❛♠♦s♦ ▲❡♠❛ ❞❡ ▲✐♦♥s✳
▲❡♠❛ ✶✳✷✳ ✭▲❡♠❛ ❞❡ ▲✐♦♥s✮ ❙❡❥❛♠ R >0 ❡2≤q <2∗✳ ❙❡(u
n) é ❧✐♠✐t❛❞❛ ❡♠H1(RN)❡ s❡
sup
y∈RN
Z
BR(y)
|un|qdx→0, q✉❛♥❞♦ n→+∞
❡♥tã♦
un→0 ❡♠ Lp(RN), ♣❛r❛ t♦❞♦ 2< p <2∗.
❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ❝❛s♦N ≥3✳ ❈♦♥s✐❞❡r❡q < s <2∗❡u∈H1(RN)✳ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡
❞❛ ✐♥t❡r♣♦❧❛çã♦ ❡ ❛ ✐♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈✱ t❡♠♦s q✉❡
hZ
BR(y)
|u|sdxi1/s=kuk Ls(B
R(y)) ≤ kuk
1−λ Lq(B
R(y))kuk
λ L2∗
(BR(y))
≤ Ckuk1L−q(λB
R(y))kuk
λ H1(B
R(y))
= Ckuk1L−q(λBR(y))
hZ
BR(y)
|u|2+|∇u|2dxi
λ
2
✶✳✶ ❖♣❡r❛❞♦r❡s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✶✵
❧♦❣♦
Z
BR(y)
|u|sdx≤Cskuk(1Lq−(Bλ)s R(y))
hZ
BR(y)
|u|2+|∇u|2dxi
λ
2s
♦♥❞❡ λ := s−q 2∗−q
2∗ s
✱ ♥♦t❡ q✉❡ 0 < λ < 1✱ ❥á q✉❡ q < s < 2∗ ✳ ❊s❝♦❧❤❡♥❞♦ λ = 2
s ♦❜t❡♠♦s
(1−λ)s=s−2❡
Z
BR(y)
|u|sdx≤CskukLs−q(2BR(y))
Z
BR(y)
|u|2+|∇u|2dx.
❈♦♥s✐❞❡r❡ ❛❣♦r❛ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❜♦❧❛s{BR(yi)}i∈Nq✉❡ ❝♦❜r❡♠ RN✱ ❞❡ ♠♦❞♦ q✉❡ ❝❛❞❛ ♣♦♥t♦ ❞❡RN
❡st❡❥❛ ❝♦♥t✐❞♦ ❡♠ ♥♦ ♠á①✐♠♦ N+ 1❜♦❧❛s✱ ❧♦❣♦ t❡♠♦s q✉❡
Z
|u|sdx =
Z
S∞
i=1BR(yi)
|u|sdx ≤
∞
X
i=1
Z
BR(yi)
|u|sdx ≤
∞
X
i=1
CskukLs−q(2BR(yi))
Z
BR(yi)
|u|2+|∇u|2dx
≤ Cssup i∈Nk
uksL−q(2BR(yi))
∞
X
i=1
Z
|u|2+|∇u|2.χ
BR(yi)
dx
≤ Cs sup
y∈RNk
uksL−q(2B R(y))
Z
|u|2+|∇u|2
∞
X
i=1
χ
BR(yi)
dx
≤ (N+ 1)Cs sup
y∈RNk
uksL−q(2BR(y))
Z
|u|2+|∇u|2dx,
♦♥❞❡ ✉s❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ▼♦♥ót♦♥❛ ❞❡ ▲❡❜❡s❣✉❡ ♥❛ ♣❡♥ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡
χ
BR(yi)
(x) =
(
1, se x∈BR(yi),
0, se x /∈BR(yi).
❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣❛r❛(un)❡ ✉s❛♥❞♦ ❛s ❤✐♣ót❡s❡s✱ ❝❤❡❣❛♠♦s ❡♠ un→0✱ ❡♠Ls(RN)✳
❈♦♠♦2< s <2∗✱ ❡♥tã♦ ♣❡❧❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❞❛ ✐♥t❡r♣♦❧❛çã♦ ❡ ❛ ✐♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈H1(RN)֒→Lr(RN)✱
♣❛r❛ 2≤r≤2∗✱ t❡♠♦s q✉❡
✭❛✮ s❡2< p≤s✱ ❡♥tã♦
kunkp≤ kunkβ2kunks1−β ≤Ckunk1s−β, ♦♥❞❡ β =
s−p
s−2
2
p
;
✭❜✮ s❡s≤p <2∗✱ ❡♥tã♦
kunkp≤ kunkµskunk12−∗µ≤Ckunk1s−µ, ♦♥❞❡ µ=
s−p
s−2∗
2∗ p
.
✶✳✷ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ✶✶
✶✳✷ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧
❆s ❞❡♠♦♥str❛çõ❡s q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛ s❡❣✉✐r ♣♦❞❡♠ s❡r ✈✐st❛s ❝♦♠ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ❡♠ ❇r❡③✐s ❬✺❪ ❡ ❑r❡②s③✐❣ ❬✶✸❪✳
❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠ ❡s♣❛ç♦H ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ é ❞✐t♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt s❡H é ❝♦♠♣❧❡t♦ ❝♦♠ ❛ ♥♦r♠❛
✐♥❞✉③✐❞❛ ♣❡❧♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳
❚❡♦r❡♠❛ ✶✳✹✳ ✭❚❡♦r❡♠❛ ❞❡ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❘✐❡s③✮ ❙❡❥❛ H ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h·,·iH✳ ❉❛❞♦g∈H∗✱ ❡①✐st❡ ✉♠ ú♥✐❝♦u∈H t❛❧ q✉❡
hu, xiH =g(x), ♣❛r❛ t♦❞♦ x∈H. ✭✶✳✺✮
❉❡♠♦♥str❛çã♦✳ ❙❡ g = 0✱ ❡♥tã♦ (1.5) é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ u = 0✳ ❙❡❥❛ ❡♥tã♦ g 6= 0 ❡ ❝♦♥s✐❞❡r❡ ♦ ♥ú❝❧❡♦
❞❡ g✱ q✉❡ é ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❢❡❝❤❛❞♦ ❞❡♥♦t❛❞♦ ♣♦rN(g)✳ ❈♦♠♦g = 06 ❡♥tã♦ N(g)6=H✱ ❡ s❡❣✉❡ q✉❡ ♦
❝♦♠♣❧❡♠❡♥t♦ ♦rt♦❣♦♥❛❧ ❞❡ N(g)♥ã♦ é ♥✉❧♦✱ ♦✉ s❡❥❛✱N⊥(g)6= 0✳ ❚♦♠❡ ❡♥tã♦06=u
0∈N⊥(g)❡ ❞❡✜♥❛
v=g(x)u0−g(u0)x,
♦♥❞❡x∈H é ❛r❜✐trár✐♦✳ ❆♣❧✐❝❛♥❞♦g✱ ♦❜t❡♠♦s
g(v) =g(x)g(u0)−g(u0)g(x) = 0.
■st♦ ♥♦s ♠♦str❛ q✉❡v∈N(g)✳ ❈♦♠♦u0⊥N(g)✱ t❡♠♦s
0 = hv, u0iH
= hg(x)u0−g(u0)x, u0iH
= g(x)hu0, u0iH−g(u0)hx, u0iH.
❈♦♠♦hu0, u0iH=ku0k2H 6= 0✱ ♦❜t❡♠♦s
g(x) = g(u0)
hu0, u0iHh
x, u0iH=
D
x, g(u0) hu0, u0iH
u0
E
H.
❙❡ ❡s❝r❡✈❡r♠♦s
u= g(u0)
hu0, u0iHu0,
♦❜t❡♠♦s
g(x) =hx, uiH,
❡ ❝♦♠♦x∈H é ❛r❜✐trár✐♦✱ ✜❝❛ ♣r♦✈❛❞♦(1.5)✳
P❛r❛ ♣r♦✈❛r ❛ ✉♥✐❝✐❞❛❞❡✱ s✉♣♦♥❤❛ q✉❡✱ ♣❛r❛ t♦❞♦x∈H✱ t❡♥❤❛♠♦s
g(x) =hx, u1iH =hx, u2iH,
❡♥tã♦ hx, u1−u2iH = 0♣❛r❛ t♦❞♦x✳ ❊♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ x=u1−u2✱ t❡♠♦s
✶✳✷ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ✶✷
♣♦rt❛♥t♦ u1−u2= 0✱ ❞❡ ♠♦❞♦ q✉❡ ✈❛❧❡ ❛ ✉♥✐❝✐❞❛❞❡✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❯♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ (X,k · kX) é ❞✐t♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ s❡X é ❝♦♠♣❧❡t♦ ❝♦♠
❛ ♥♦r♠❛ kukX✳
❉❡✜♥✐çã♦ ✶✳✹✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ ❡ (xn) ⊂ X ✉♠❛ s❡q✉ê♥❝✐❛✳ ❉✐③❡♠♦s q✉❡ (xn)
❝♦♥✈❡r❣❡ ❢r❛❝❛♠❡♥t❡ ❡♠ X✱ s❡ ❡①✐st❡ x ∈ X t❛❧ q✉❡✱ ♣❛r❛ t♦❞❛ f ∈ X∗✱ t❡♥❤❛♠♦s hf, xni → hf, xi✳
❉❡♥♦t❛♠♦s ❡st❡ ❢❛t♦ ♣♦r xn⇀ x✳
❚❡♦r❡♠❛ ✶✳✺✳ ❙❡❥❛ (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦X✳
(i)s❡xn→x✱ ❡♥tã♦ xn⇀ x❡♠ X❀
(ii)s❡ xn⇀ x❡♠ X✱ ❡♥tã♦ (xn)é ❧✐♠✐t❛❞❛ ❡ kxkX ≤lim inf
n→∞ kxnkX❀
(iii)s❡ xn⇀ x❡♠ X ❡fn →f ❡♠X∗✱ ❡♥tã♦ hfn, xni → hf, xi✳
❉❡♠♦♥str❛çã♦✳ Pr♦♣♦s✐çã♦ ✸✳✺✱ ♣á❣✐♥❛ ✺✽ ❞❡ ❇r❡③✐s [✺]✳
❚❡♦r❡♠❛ ✶✳✻✳ ❙❡❥❛ X ❡♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡✢❡①✐✈♦ ❡(xn)⊂X ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✳ ❊♥tã♦ ❡①✐st❡
✉♠❛ s✉❜s❡q✉ê♥❝✐❛ (xnk)⊂(xn)t❛❧ q✉❡xnk⇀ x❡♠ X✳
❉❡♠♦♥str❛çã♦✳ P❡❧♦ t❡♦r❡♠❛ ❞❡ ❑❛❦✉t❛♥✐ ✭❚❡♦r❡♠❛ ✸✳✶✼✱ ♣á❣✐♥❛ ✻✼ ❞❡ ❇r❡③✐s [✺]✮✱ t❡♠♦s q✉❡ ❛ ❜♦❧❛
✉♥✐tár✐❛ ❞❡Xé ❢r❛❝❛♠❡♥t❡ ❝♦♠♣❛❝t❛✳ ❚♦♠❡(xn)❧✐♠✐t❛❞❛✱ ❧♦❣♦(xn)⊂BR✱ ♣❛r❛ ❛❧❣✉♠R >0✱ ❡ ♣♦rBR
s❡r ❢r❛❝❛♠❡♥t❡ ❝♦♠♣❛❝t❛✱ t❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛(xj)⊂(xn)t❛❧ q✉❡xj⇀ x, ❡♠ X✳
❚❡♦r❡♠❛ ✶✳✼✳ ❙❡❥❛♠ H ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt s❡♣❛rá✈❡❧ ❡ T :H →H ✉♠ ♦♣❡r❛❞♦r ❝♦♠♣❛❝t♦ ❡ ❛✉t♦❛❞✲
❥✉♥t♦✳ ❊♥tã♦ H ❛❞♠✐t❡ ✉♠❛ ❜❛s❡ ❤✐❧❜❡rt✐❛♥❛ ❢♦r♠❛❞❛ ♣♦r ❛✉t♦❢✉♥çõ❡s ❞❡T✱ ♦✉ s❡❥❛✱ ❛❞♠✐t❡ ✉♠❛ ❜❛s❡
(uj) t❛❧ q✉❡T uj =µjuj✱hui, ujiH = 0 ♣❛r❛i6=j ❡huj, ujiH = 1✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❞✐♠❡♥sã♦ ❞❡ q✉❛❧q✉❡r
❛✉t♦❡s♣❛ç♦ é ✜♥✐t❛✳
❉❡♠♦♥str❛çã♦✳ ❚❡♦r❡♠❛ ✻✳✶✶✱ ♣á❣✐♥❛ ✶✻✼ ❞❡ ❇r❡③✐s[✺]✳
❙❡❥❛ ❛❣♦r❛ ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ r❡❛❧E ❡T :D(T)⊂E→E ❧✐♥❡❛r✳ P❛r❛ ❝❛❞❛λ∈R❞❡✜♥❛
Tλ:D(T) → E
u 7→ T u−λu,
♦✉ s❡❥❛✱ Tλ=T −λI✳
❉❡✜♥✐çã♦ ✶✳✺✳ ❖ ♦♣❡r❛❞♦r
Rλ:Tλ(D(T)) → D(T)
Tλu 7→ u,
q✉❛♥❞♦ ❡①✐st✐r✱ é ❝❤❛♠❛❞♦ ❞❡ ♦♣❡r❛❞♦r r❡s♦❧✈❡♥t❡ ❞❡ T✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s Rλ=Tλ−1✳
❉❡✜♥✐çã♦ ✶✳✻✳ ❉✐③❡♠♦s q✉❡ λ∈Ré ✉♠ ✈❛❧♦r r❡❣✉❧❛r ❞❡T s❡✿ (i)Rλ ❡①✐st✐r❀
(ii)Rλ ❢♦r ❝♦♥tí♥✉♦❀
✶✳✸ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ ✶✸
❉❡✜♥✐çã♦ ✶✳✼✳ ❈♦♠ r❡❧❛çã♦ ❛ ✉♠ ♦♣❡r❛❞♦r T✱ t❡♠♦s✿
a) ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s q✉❡ sã♦ ✈❛❧♦r❡s r❡❣✉❧❛r❡s ❞❡ T✱ ❞❡♥♦t❛❞♦ ♣♦r ρ(T)✱ é ❝❤❛♠❛❞♦ ❞❡
r❡s♦❧✈❡♥t❡ ❞❡ T✳
b)❖ ❝♦♠♣❧❡♠❡♥t❛r ❡♠ R❞♦ r❡s♦❧✈❡♥t❡ ❞❡ T✱ ❞❡♥♦t❛❞♦ ♣♦r σ(T)✱ é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❡❝tr♦ ❞❡ T✳
❚❡♦r❡♠❛ ✶✳✽✳ ❙❡❥❛♠ E ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ ❝♦♠ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ❡ T : E →E ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r
❝♦♠♣❛❝t♦✳ ❊♥tã♦✿
(1) 0∈σ(T)❀
(2)σ(T)\ {0}=A(T)\ {0}✱ ♦♥❞❡ A(T)é ♦ ❝♦♥❥✉♥t♦s ❞♦s ❛✉t♦✈❛❧♦r❡s ❞❡T❀
(3)❖❝♦rr❡ ❛♣❡♥❛s ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❛❧t❡r♥❛t✐✈❛s✿ (i)σ(T) ={0}❀
(ii)σ(T)\ {0} é ✜♥✐t♦ ❡✱ ♣♦rt❛♥t♦✱ ❞✐s❝r❡t♦❀
(iii)σ(T)\ {0}=µn →0✳
❉❡♠♦♥str❛çã♦✳ ❚❡♦r❡♠❛ ✻✳✽✱ ♣á❣✐♥❛ ✶✻✹ ❞❡ ❇r❡③✐s [✺]✳
✶✳✸ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛
▼♦str❛r❡♠♦s ❛q✉✐ r❡s✉❧t❛❞♦s ❞❡ ▼❡❞✐❞❛ ❡ ■♥t❡❣r❛çã♦ q✉❡ s❡rã♦ ✉s❛❞♦s t❛♥t♦ ❡①♣❧í❝✐t❛ q✉❛♥t♦ ✐♠♣❧✐✲ ❝✐t❛♠❡♥t❡ ♥❡st❡ tr❛❜❛❧❤♦✳ ❖s ♣ró①✐♠♦s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❝♦♥s✉❧t❛❞♦s ❡♠ ❇❛rt❧❡ ❬✸❪ ❡ ❇r❡③✐s [✺]✳
❙❡❥❛(A,❆, µ)✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ ♦♥❞❡ A é ✉♠ ❝♦♥❥✉♥t♦✱ ❆ é ✉♠❛ σ✲á❧❣❡❜r❛ ❡µé ✉♠❛ ♠❡❞✐❞❛✳
❉❡♥♦t❛r❡♠♦s ♣♦r ▼+(A,❆)❛s ❢✉♥çõ❡s ❆✲♠❡♥s✉rá✈❡✐s ♥ã♦ ♥❡❣❛t✐✈❛s ❞❡A♣❛r❛Re=R∪ {∞}✳
❚❡♦r❡♠❛ ✶✳✾✳ ✭❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ▼♦♥ót♦♥❛ ❞❡ ▲❡❜❡s❣✉❡✮ ❙❡❥❛♠(A,❆, µ)✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛
❡ (fn)✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s ❡♠A✱ ❡ s✉♣♦♥❤❛ q✉❡✿
(i) 0≤f1≤f2≤...✱ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡A❀
(ii)fn →f✱ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡A✳
❊♥tã♦f é ♠❡♥s✉rá✈❡❧✱ ❡
Z
A
fndµ→
Z
A
f dµ,
q✉❛♥❞♦ n→ ∞✳
❉❡♠♦♥str❛çã♦✳ ❙❛❜❡♠♦s q✉❡ ♦ ❧✐♠✐t❡ ❞❡ ✉♠ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s é ♠❡♥s✉rá✈❡❧✱ ❧♦❣♦ f é
♠❡♥s✉rá✈❡❧✳ ❈♦♠♦fn≤fn+1≤f, ∀n∈N✱ s❡❣✉❡ q✉❡
Z
A
fndµ≤
Z
A
fn+1dµ≤
Z
A
f dµ.
P♦rt❛♥t♦
lim
n→∞
Z
A
fndµ≤
Z
A
f dµ. ✭✶✳✻✮
P❛r❛ ♦❜t❡r♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♦♣♦st❛✱ s❡❥❛ α∈(0,1) ❡ s❡❥❛ ϕ ✉♠❛ ❢✉♥çã♦ s✐♠♣❧❡s ❝♦♠ 0≤ ϕ≤f✳
❉❡✜♥❛
An=
n x∈A
fn(x)≥αϕ(x) o
✶✳✸ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ ✶✹
❧♦❣♦An ∈❆, An ⊂An+1✱ ❡A=
∞
[
n=1
An✳ ❚❡r❡♠♦s ❡♥tã♦
Z
An
αϕ dµ≤ Z
An
fndµ≤
Z
A
fndµ. ✭✶✳✼✮
❈♦♠♦ ❛ s❡q✉ê♥❝✐❛(An)é ♠♦♥ót♦♥❛ ❝r❡s❝❡♥t❡ ❡A=
∞
[
n=1
An✱ s❡❣✉❡ q✉❡
Z
A
ϕ dµ= lim
n→∞
Z
An
ϕ dµ.
❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ ❡♠(1.7)✱ ♦❜t❡♠♦s
α Z
A
ϕ dµ≤ lim
n→∞
Z
A
fndµ,
❡ ❝♦♠♦ ❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✈❛❧❡ ♣❛r❛ t♦❞♦ α∈(0,1)✱ ❝♦♥❝❧✉í♠♦s q✉❡
Z
A
ϕ dµ≤ lim
n→∞
Z
A
fndµ.
❈♦♠♦ ϕé ✉♠❛ ❢✉♥çã♦ s✐♠♣❧❡s ❛r❜✐trár✐❛ s❛t✐s❢❛③❡♥❞♦0≤ϕ≤f✱ ❝❤❡❣❛♠♦s ❡♠
Z
A
f dµ= sup
ϕ
Z
A
ϕ dµ≤ lim
n→∞
Z
A
fndµ. ✭✶✳✽✮
❈♦♠❜✐♥❛♥❞♦(1.6) ❡(1.8)✱ ♦❜t❡♠♦s ♦ r❡s✉❧t❛❞♦✳
▲❡♠❛ ✶✳✸✳ ✭▲❡♠❛ ❞❡ ❋❛t♦✉✮ ❙❡ (fn)⊂▼+(A,❆)✱ ❡♥tã♦
Z
A
(lim inf
n→∞ fn)dµ≤lim infn→∞
Z
A
fndµ.
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛gm= inf{fm, fm+1,...}❞❡ ♠♦❞♦ q✉❡gm≤fn s❡♠♣r❡ q✉❡m≤n✳ ❆ss✐♠ t❡♠♦s
Z
A
gmdµ≤
Z
A
fndµ,
❧♦❣♦
Z
A
gmdµ≤lim inf n→∞
Z
A
fndµ.
❈♦♠♦ ❛ s❡q✉ê♥❝✐❛(gm)é ♥ã♦ ❞❡❝r❡s❝❡♥t❡ ❡ ❝♦♥✈❡r❣❡ ♣❛r❛lim inf
n→∞ fn✱ ♦ ❚❡♦r❡♠❛ ✶✳✾ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛
▼♦♥ót♦♥❛ ✐♠♣❧✐❝❛ q✉❡
Z
A
lim inf
n→∞ fndµ= limn→∞
Z
A
gmdµ≤lim inf n→∞
Z
A
fndµ.
✶✳✸ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ ✶✺
✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s ❡♠A✱ t❛❧ q✉❡
fn→f, ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ A.
❙❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ✐♥t❡❣rá✈❡❧g t❛❧ q✉❡
|fn| ≤g, ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ A,
❡♥tã♦ f é ✐♥t❡❣rá✈❡❧ ❡
Z
A
f dµ= lim
n→∞
Z
A
fndµ.
❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡f é ✐♥t❡❣rá✈❡❧ ❡ ❝♦♠♦g+fn≥0✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ▲❡♠❛ ✶✳✸ ❞❡ ❋❛t♦✉ ♣❛r❛
♦❜t❡r
Z
A
g dµ+
Z
A
f dµ =
Z
A
(g+f)dµ
≤ lim inf
n→∞
Z
A
(g+fn)dµ
= lim inf
n→∞
Z
A
g dµ+
Z
A
fndµ
=
Z
A
g dµ+ lim inf
n→∞
Z
A
fndµ.
❙❡❣✉❡ q✉❡
Z
A
f dµ≤lim inf
n→∞
Z
A
fndµ.
❈♦♠♦g−fn ≥0✱ ♠❛✐s ✉♠❛ ❛♣❧✐❝❛çã♦ ❞♦ ▲❡♠❛ ✶✳✸ ❞❡ ❋❛t♦✉ ♥♦s ❞á
Z
A
g dµ− Z
A
f dµ =
Z
A
(g−f)dµ
≤ lim inf
n→∞
Z
A
(g−fn)dµ
=
Z
A
g dµ−lim sup
n→∞
Z
A
fndµ,
♦✉ s❡❥❛✱
lim sup
n→∞
Z
A
fndµ≤
Z
A
f dµ.
❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡
Z
A
f dµ= lim
n→∞
Z
A
fndµ.
❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡❥❛♠ (A,❆, µ)✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛✱f :A→R✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❡1≤p <∞✳
❉❡✜♥✐♠♦s
kfkp=
hZ
A|
✶✳✸ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ ✶✻
❡ Lp(A)❛ ❝♦❧❡çã♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s ❡♠ At❛✐s q✉❡
kfkp<∞.
❉❡✜♥✐♠♦s t❛♠❜é♠L∞(A)❝♦♠♦ ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐sf t❛✐s q✉❡|f(x)| ≤M✱
❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ A✱ ♣❛r❛ ❛❧❣✉♠ M >0✳ ❉❡✜♥✐♠♦s ❛ ♥♦r♠❛kfk∞ ❡♠ L∞(A)♣♦r
kfk∞= inf
n
M >0
|f|< M, ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ A o
.
❚❡♦r❡♠❛ ✶✳✶✶✳ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✮ ❈♦♥s✐❞❡r❡♠♦s (A,❆, µ)✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ ❡1≤p, q≤ ∞
❝♦♠ 1
p+
1
q = 1✳ ❙❡f ∈L
p(A), g∈Lq(A)✱ ❡♥tã♦ f g∈L1(A)❡
kf gk1≤ kfkpkgkq.
❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛p= 1❡q=∞✱ ❧♦❣♦
Z
A
f g dµ
≤
Z
A|
f g|dµ
≤ kgk∞
Z
A|
f|dµ
= kgk∞kfk1≤ ∞,
❡ t❡♠♦s ♦ r❡s✉❧t❛❞♦ ♥♦ ❝❛s♦ p= 1✳
P❛r❛ ♦ ❝❛s♦ p > 1✱ s❡❥❛α ∈ (0,1) ❡ϕ(t) := αt−tα ♣❛r❛ t ≥ 0✳ ▲♦❣♦✱ ϕ′(t) = α−αtα−1 ❡ ❛ss✐♠
ϕ′(t)<0♣❛r❛0< t <1❡ϕ′(t)>0♣❛r❛t >1✳ ▲♦❣♦✱t= 1é ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❞❡ϕ✱ ♦✉ s❡❥❛✱ϕ(t)≥ϕ(1)
❡ϕ(t) =ϕ(1)s❡✱ ❡ s♦♠❡♥t❡ s❡✱t= 1✳
❚❡♠♦s ❡♥tã♦ q✉❡ϕ(t)≥ϕ(1) ✐♠♣❧✐❝❛ ❡♠
tα≤αt+ (1−α), t≥0.
❙❡❥❛♠ a, b♥ã♦ ♥❡❣❛t✐✈♦s ❡t=a
b✱ ❧♦❣♦ t❡r❡♠♦s
aαb−α≤αab−1+ (1−α),
❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦rb✱ t❡♠♦s
aαb1−α≤αa+ (1−α)b,
♦♥❞❡ ❛ ✐❣✉❛❧❞❛❞❡ ✈❛❧❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a=b✳
❙❡❥❛♠ ❛❣♦r❛ p ❡ q s❛t✐s❢❛③❡♥❞♦ 1 < p < ∞ ❡ 1p+ 1
q = 1 ❡ t♦♠❡ α=
1
p✳ ❙❡❣✉❡ q✉❡ s❡ A ❡ B sã♦
♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥❡❣❛t✐✈♦s✱ ❡♥tã♦
AB≤ A
p
p + Bq
q ,
❡ ❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Ap=Bq.
❙✉♣♦♥❤❛ q✉❡f ∈Lp✱g∈Lq ❡kfk
p,kgkq 6= 0✱ ❡♥tã♦ ♦ ♣r♦❞✉t♦f gé ♠❡♥s✉rá✈❡❧ ❡ t♦♠❛♥❞♦A= |
✶✳✸ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ ✶✼
❡B= |g(x)|
kgkq ✱ ♦❜t❡♠♦s
|f(x)g(x)| kfkpkgkq ≤
|f(x)|p
pkfkpp +
|g(x)|q
qkgkqq .
❈♦♠♦ ♦s ❞♦✐s t❡r♠♦s à ❞✐r❡✐t❛ sã♦ ✐♥t❡❣rá✈❡✐s✱ s❡❣✉❡ q✉❡f gé ✐♥t❡❣rá✈❡❧✳ ❆❧é♠ ❞✐ss♦✱ ✐♥t❡❣r❛♥❞♦✱ ♦❜t❡♠♦s
kf gk1
kfkpkgkq ≤
1
p+
1
q = 1,
♦ q✉❡ ♣r♦✈❛ ♥♦ss♦ r❡s✉❧t❛❞♦✳
❚❡♦r❡♠❛ ✶✳✶✷✳ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ■♥t❡r♣♦❧❛çã♦✮ ❙❡❥❛♠ 1≤s≤r≤t≤ ∞❡θ∈(0,1)t❛❧ q✉❡
1
r = θ s+
1−θ t .
❙✉♣♦♥❤❛♠♦s t❛♠❜é♠ q✉❡ u∈Ls(Ω)∩Lt(Ω)✱ ♦♥❞❡ Ωé ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦✳ ❊♥tã♦✱u∈Lr(Ω)❡
kukr≤ kuksθkuk1t−θ.
❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ❝♦♠ ♦s ❡①♣♦❡♥t❡s ❝♦♥❥✉❣❛❞♦s θr
s +
(1−θ)r t = 1✱
♦❜t❡♠♦s
kukrr ≤
Z
Ω|
u|rdx
=
Z
Ω|
u|θr|u|(1−θ)rdx
≤
Z
Ω|
u|θrθrs dx
θrsZ
Ω|
u|(1−θ)r(1−tθ)rdx
(1
−θ)r t
=
Z
Ω|
u|sdx
θr sZ
Ω|
u|tdx
(1−θ)r t
= kukθrs kuk
(1−θ)r t ,
❧♦❣♦
kukr≤ kukθskuk
(1−θ)
t ,
❡ ❝♦♠♦u∈Ls(Ω)∩Lt(Ω)✱ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳
❚❡♦r❡♠❛ ✶✳✶✸✳ ❈♦♥s✐❞❡r❡ ✉♠❛ s❡q✉ê♥❝✐❛ (fn) ⊂ Lp(Ω) ❡ f ∈ Lp(Ω)✱ ❞❡ ♠♦❞♦ q✉❡ kfn −fkp → 0✱
q✉❛♥❞♦ n→ ∞✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛(fnk)t❛❧ q✉❡✿
(i)fnk(x)→f(x)✱ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡Ω❀
(ii)|fnk(x)| ≤g(x)✱ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ Ω✱ ❝♦♠g∈L
p(Ω)✳