Um estudo sobre a equação de Schrödinger biharmônica

❯♠ ❡st✉❞♦ s♦❜r❡ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r

❜✐❤❛r♠ô♥✐❝❛

❍❡❧♦ís❛ ▲♦♣❡s ❞❡ ❙♦✉s❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❚❛❞❡✉ ❞❡ ❖❧✐✈❡✐r❛ P✐♠❡♥t❛ ❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❙✉❡tô♥✐♦ ❞❡ ❆❧♠❡✐❞❛ ▼❡✐r❛

Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧

❯◆■❱❊❘❙■❉❆❉❊ ❊❙❚❆❉❯❆▲ P❆❯▲■❙❚❆

❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❡ Pr❡s✐❞❡♥t❡ Pr✉❞❡♥t❡

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧

❯♠ ❡st✉❞♦ s♦❜r❡ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r

❜✐❤❛r♠ô♥✐❝❛

❍❡❧♦ís❛ ▲♦♣❡s ❞❡ ❙♦✉s❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❚❛❞❡✉ ❞❡ ❖❧✐✈❡✐r❛ P✐♠❡♥t❛ ❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❙✉❡tô♥✐♦ ❞❡ ❆❧♠❡✐❞❛ ▼❡✐r❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✧❏✉❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✧ ❝❛♠♣✉s ❞❡ Pr❡s✐❞❡♥t❡ Pr✉❞❡♥t❡ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ♠❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧✳

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2

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7' "#8 9,('6 ':-'$ ';($'$()'

($''') <'

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r t❡r ❢❡✐t♦ ♣♦ssí✈❡❧ ❡st❡ tr❛❜❛❧❤♦✱ ♣♦✐s ❡♠ ♠✐♥❤❛ tr❛❥❡tór✐❛ ❡❧❡ ❝♦❧♦❝♦✉ ♣❡ss♦❛s ❡s♣❡❝✐❛✐s q✉❡ ❢♦r❛♠ ✐♠♣r❡s❝✐♥❞í✈❡✐s ♣❛r❛ r❡❛❧✐③á✲❧♦✳

❆❣r❛❞❡ç♦ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ♠ã❡ ❋át✐♠❛ q✉❡ ❡♠ ♠❡✐♦ ❞❡ t♦❞♦s ♦s ✐♠♣r❡✈✐st♦s q✉❡ ❛ ✈✐❞❛ ♥♦s ♦❢❡r❡❝❡ ❡❧❛ s❡♠♣r❡ ♠❡ ❞❡✉ ❡①❡♠♣❧♦s ❞❡ ❢♦rç❛ ❞❡ ✈♦♥t❛❞❡✱ ❣❡♥❡r♦s✐❞❛❞❡ ❡ ❞❡ ❞❡❞✐❝❛çã♦✳ ❆♦ ♠❡✉ ♣❛✐ ❏♦sé ♣❡❧❛s ❝♦♥✈❡rs❛s s♦❜r❡ ❛ ✈✐❞❛✱ s❡♠♣r❡ ❜r❡✈❡s ♠❛s ✈❛❧✐♦s❛s✱ s✐♥t♦ ❢❛❧t❛ ❞❡❧❛s✳ P♦r ❡❧❡s t❡r❡♠ ♠❡ ❞❛❞♦ ♦ ♠❡❧❤♦r q✉❡ ♣✉❞❡r❛♠ ♦❢❡r❡❝❡r✳

❆♦s ♠❡✉s ✐r♠ã♦s ❆❧✐♥❡ ❡ ❏✉♥✐♦r ♣❡❧❛s ❜r✐♥❝❛❞❡✐r❛s ❞❡ ❝r✐❛♥ç❛✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ❡ ♣♦r ❝✉✐❞❛r❡♠ ❞❛ ♠ã❡ ❡ ❞♦ ♣❛✐ q✉❛♥❞♦ ❡✉ ♥ã♦ ❡st❛✈❛ ♣♦r ♣❡rt♦✳

❆♦ ♠❡✉ ♥❛♠♦r❛❞♦ ❡ ❛♠✐❣♦ ❏✉♥✐♦r ♣❡❧♦ s❡✉ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ♣♦r s❡♠♣r❡ ♠❡ ❞❛r ❢♦rç❛s✱ ♣♦r t♦❞♦s ♦s ❞♦♠✐♥❣♦s ❡ ❢❡r✐❛❞♦s q✉❡ ♣❛ss❛♠♦s ❡st✉❞❛♥❞♦ ❆♥á❧✐s❡ ❡ ♣♦r ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳

➚ ♠✐♥❤❛ ❡①✲❝♦❧❡❣❛ ❞❡ r❡♣ú❜❧✐❝❛ ❡ ❛♠✐❣❛ ❏✉❧✐❛♥❛ ●♦r✐ ✭❏✉❤✮ ♣❡❧♦s três ❛♥♦s ❞❡ ❝♦♥✲ ✈✐✈ê♥❝✐❛✱ ♣♦r s❡❣✉r❛r ❛ ❜❛rr❛ q✉❛♥❞♦ ❛ ❜♦❧s❛ ❛tr❛s❛✈❛ ❡ ♣♦r t♦❞❛s ❛s ❝♦♥✈❡rs❛s q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ❛ ❛♠❛❞✉r❡❝❡r✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ❯❊▼ q✉❡ ❞✉r❛♥t❡ ♦s q✉❛tr♦ ❛♥♦s ❞❡ ❣r❛❞✉❛çã♦ ♠❡ ♣r♦♣♦r❝✐♦♥❛✲ r❛♠ ♠♦♠❡♥t♦s ♠❡♠♦rá✈❡✐s✳ ❆♦s ♣r♦❢❡ss♦r❡s ❞❡ ❧á✱ ❛❣r❛❞❡ç♦ ♣♦r t♦❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♦❢❡✲ r❡❝✐❞♦✱ ♣♦r t❡r❡♠ ♠❡ ❞❛❞♦ ❛ ❜❛s❡ ❞❡ ♠✐♥❤❛ ❢♦r♠❛çã♦✳ ❊♠ ❡s♣❡❝✐❛❧ à ♣r♦❢❡ss♦r❛ ❈❧❛✉❞❡t❡ ▼❛t✐❧❞❡ ❲❡❜❧❡r ▼❛rt✐♥s q✉❡ ♣♦r ❞♦✐s ❛♥♦s ♠❡ ♦r✐❡♥t♦✉ ♥♦ ♣r♦❥❡t♦ ❞❡ ✐♥✐❝✐❛çã♦ ❝✐❡♥tí✜❝❛✱ ♣♦r t❡r ♠❡ ✐♥❝❡♥t✐✈❛❞♦ ❛ ❡♥tr❛r ♥♦ ♠❡str❛❞♦✱ ♠❛✐s ❞♦ q✉❡ ✐st♦✱ ♣♦r t❡r ♠❡ ❢❡✐t♦ ❛❝r❡❞✐t❛r q✉❡ ❡r❛ ♣♦ssí✈❡❧ ❡ ♣♦r t❡r ♠❡ ❞❛❞♦ t♦t❛❧ ❛♣♦✐♦ ♣❛r❛ q✉❡ r❡❛❧♠❡♥t❡ ❛❝♦♥t❡❝❡ss❡✳

❆♦s ♣r♦❢❡ss♦r❡s ❡ ❛♠✐❣♦s ❞♦ Pós✲▼❛❝ ♣❡❧❛ ❛♣r❡♥❞✐③❛❣❡♠ ❡ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥✲ tr❛çã♦ ❡ ❛♠✐③❛❞❡✳

✻ ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ▼❛r❝♦s ❚❛❞❡✉ ❞❡ ❖❧✐✈❡✐r❛ P✐♠❡♥t❛ ♣♦r s✉❛ ✐♠❡♥s❛ ♣❛❝✐✲ ê♥❝✐❛✱ s❡♠♣r❡ ❞✐s♣♦st♦ ❛ ♠❡ ❛t❡♥❞❡r✳ P❡❧❛ ♠❛♥❡✐r❛ ❞❡❞✐❝❛❞❛ ❡ ♣r♦✜ss✐♦♥❛❧ q✉❡ ❣✉✐♦✉ ❡st❡ tr❛❜❛❧❤♦✱ s❡♥❞♦ ❛ss✐♠ ✉♠ ❡①❡♠♣❧♦ ❞❡ ♣r♦❢❡ss♦r q✉❡ ♣r❡t❡♥❞♦ s❡❣✉✐r✳ P❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ♣♦r t♦❞❛s ❛s ❝♦♥✈❡rs❛s ❞❡ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ t❡ór✐❝♦ ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s ❊❧í♣t✐❝❛s✱ ❡st✉❞❛♠♦s ✉♠❛ ✈❡rsã♦ ❡st❛❝✐♦♥ár✐❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ♥ã♦✲❧✐♥❡❛r ❜✐❤❛r♠ô♥✐❝❛✳ ❖ ♦❜❥❡t✐✈♦ ♣r✐♥✲ ❝✐♣❛❧ ✈❡rs❛ s♦❜r❡ r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ♥ã♦✲tr✐✈✐❛✐s✱ q✉❛♥❞♦ ✉♠ ♣❛râ♠❡tr♦ ǫt❡♥❞❡ ❛ ③❡r♦✳ ❙ã♦ ✉t✐❧✐③❛❞♦s ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ♣❛r❛ ❡st✉❞❛r ❡①✐stê♥❝✐❛

❞❛s s♦❧✉çõ❡s ❢r❛❝❛s ♥ã♦✲tr✐✈✐❛✐s ❝♦♠ ❤✐♣ót❡s❡s s♦❜r❡ ♦ ♣♦♥t❡❝✐❛❧ ❡ s♦❜r❡ ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡✳

❆❜str❛❝t

■♥ t❤✐s t❤❡♦r❡t✐❝❛❧ ✇♦r❦ ✐♥ ❊❧❧✐♣t✐❝ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✇❡ st✉❞② ❛ st❛t✐♦✲ ♥❛r② ✈❡rs✐♦♥ ♦❢ t❤❡ ❜✐❤❛r♠♦♥✐❝ ♥♦♥❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥✳ ❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ❛✐♠s ❡①✐st❡♥❝❡ r❡s✉❧ts ❛♥❞ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥s ✇❤❡♥ ❛ ♣❛r❛♠❡t❡r ǫ t❡♥❞s t♦

③❡r♦✳ ❱❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞s ❛r❡ ✉s❡❞ t♦ st✉❞② t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ✇❡❛❦ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥s ✉♥❞❡r ❝❡rt❛✐♥ ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ♣♦t❡♥t✐❛❧ ❛♥❞ t❤❡ ♥♦♥❧✐♥❡❛r✐t②✳

❮♥❞✐❝❡ ❞❡ ◆♦t❛çõ❡s

|A| é ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A⊂RN❀

Ck_{(Ω) ={u: Ω→R;u é ❝♦♥t✐♥✉❛♠❡♥t❡ k ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧}}

Ck

0(Ω) ={u∈Ck(Ω);supp(u)é ❝♦♠♣❛❝t♦}❀

Ck,α_{(Ω) ={u∈C}k_(Ω);Dk_{u é α−❍ö❧❞❡r ❝♦♥tí♥✉❛}❀}

uLp_(Ω) =

Ω

|u|p_dx

1_p

❀

Lp_{(Ω) ={u: Ω→R; u é ♠❡♥s✉rá✈❡❧ ❡ u}

Lp_(Ω) <∞}❀

uL∞

(Ω) = inf{a≥0;{x∈Ω;|u(x)|> a}= 0}❀

L∞_{(Ω) ={u: Ω→R;u é ♠❡♥s✉rá✈❡❧ ❡ u}

L∞_(Ω) <∞}❀

Wm,p_{(Ω) ={u ∈ L}p_(Ω);Dα_{u ∈L}p_{(Ω)∀|α| ≤m}✱ ♦♥❞❡ α = (α}

1, . . . , αN) é ✉♠ ♠✉❧t✐✲

í♥❞✐❝❡❀

uWm,p_(Ω) =

_m

i=1

Di_u Lp_(Ω)

1_p

❀

W₀m,p(Ω) =C∞

0 (Ω)✱ ♦♥❞❡ ♦ ❢❡❝❤♦ é t♦♠❛❞♦ ❝♦♠ r❡s♣❡✐t♦ ❛ ♥♦r♠❛ .Wm,p_(Ω)❀ Hm_{(Ω) =W}m,2_(Ω)❀

Hm

0 (Ω) =C0∞(Ω)✱ ♦♥❞❡ ♦ ❢❡❝❤♦ é t♦♠❛❞♦ ❝♦♠ r❡s♣❡✐t♦ ❛ ♥♦r♠❛ .Hm_(Ω)❀

∆u=

N

i=1

∂2_u

∂x2

i

❀ ∆2_{u= ∆(∆u)❀}

2∗ =

2N N −4❀ 2∗ = 2N

N −2✳

❙✉♠ár✐♦

❈❛♣ít✉❧♦s

✶ ■♥tr♦❞✉çã♦ ✶✼

✷ Pr❡❧✐♠✐♥❛r❡s ✷✶

✷✳✶ ❉✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✳✶ ❚❡♦r❡♠❛s ❞❡ ❞❡♥s✐❞❛❞❡ ❡ ✐♠❡rsã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸ ❊①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ✸✶

✸✳✶ ❖♣r♦❜❧❡♠❛ ♠♦❞✐✜❝❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✹ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✺✺

❈❛♣ít✉❧♦

✶

■♥tr♦❞✉çã♦

◆♦s ú❧t✐♠♦s ❛♥♦s✱ ✈ár✐♦s ❛✉t♦r❡s tê♠ ❡st✉❞❛❞♦ ❞✐✈❡rs❛s q✉❡stõ❡s r❡❧❛t✐✈❛s à ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❡st❛❝✐♦♥ár✐❛ ♥ã♦✲❧✐♥❡❛r

⎧ ⎨ ⎩

−ǫ2_{∆u+V(x)u=f(u) ❡♠ Ω}

u∈H1_(Ω), ✭✶✳✶✮

❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ◆❡✉♠❛♥♥ ♦✉ ❉✐r✐❝❤❧❡t✱ ♦♥❞❡ Ω ⊂ RN _{é ✉♠ ❞♦♠í♥✐♦ ♥ã♦}

♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧✐♠✐t❛❞♦✳▼♦t✐✈❛❞♦ ♣♦r ❋❧♦❡r ❡ ❲❡✐♥st❡✐♥ ❬✼❪✱ ❘❛❜✐♥♦✇✐t③ ❡♠ ❬✶✸❪ ✉s♦✉ ❛r❣✉♠❡♥t♦s ❞♦ t✐♣♦ ♣❛ss♦ ❞❛ ♠♦♥t❛♥❤❛ ♣❛r❛ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞❡ (1.1) ♣❛r❛ ǫ > 0 s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ♦♥❞❡ N ≥ 3✱ Ω = RN ❡ _f é ✉♠❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ s✉♣❡r❧✐♥❡❛r ❡ s✉❜❝rít✐❝❛✳❙♦❜r❡ ♦ ♣♦t❡♥❝✐❛❧ V ❢♦✐ ❛ss✉♠✐❞❛ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦

0< V0 = inf

x∈RNV(x)<lim inf_|x|→∞ V(x).

❊♠ ❬✶✺❪✱ ❲❛♥❣ ♣r♦✈♦✉ q✉❡ ❛s s♦❧✉çõ❡s ❞♦ ♣❛ss♦ ❞❛ ♠♦♥t❛♥❤❛ ❡♥❝♦♥tr❛❞❛s ♣♦r ❘❛❜✐✲ ♥♦✇✐t③ ❡♠ ❬✶✸❪ ❛♣r❡s❡♥t❛♠ ✉♠ ❢❡♥ô♠❡♥♦ ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡♠ t♦r♥♦ ❞❡ ✉♠ ♠í♥✐♠♦ ❣❧♦❜❛❧ ❞❡ V ✱ s❡ ǫ→0✳❊♠ ❬✺❪✱ ❉❡❧ P✐♥♦ ❡ ❋❡❧♠❡r ✉s❛r❛♠ ✉♠ ♠ét♦❞♦ ❞❡ ♣❡♥❛❧✐③❛çã♦ ♣❛r❛ ♣r♦✲

✈❛r ❛ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ (1.1)✱ ❝♦♠ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ s✉♣❡r❧✐♥❡❛r ❡ s✉❜❝rít✐❝❛ ❡ ❝♦♠ ♦ ♣♦t❡♥❝✐❛❧ V s❛t✐s❢❛③❡♥❞♦ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦

inf

x∈ΛV(x)<xinf∈∂ΛV(x),

♦♥❞❡Λé ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♥t✐❞♦ ❡♠Ω✳❊ss❡s ❛r❣✉♠❡♥t♦s ✐♥s♣✐r❛r❛♠ ♠✉✐t♦s ❛✉t♦r❡s ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ❡♥tr❡ ❡❧❡s ❆❧✈❡s ❡ ❋✐❣✉❡✐r❡❞♦ ❡♠ ❬✷✱ ✸❪✱ q✉❡ ❝♦♥s✐❞❡r❛r❛♠ ♦ ♣r♦❜❧❡♠❛

✶✳ ■♥tr♦❞✉çã♦ ✶✽ (1.1) ❝♦♠ ♦ ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡ s✉❜st✐t✉í❞♦ ♣❡❧♦ p✲▲❛♣❧❛❝✐❛♥♦ ❡ ♦❜t✐✈❡r❛♠ ❡①✐stê♥❝✐❛✱

♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s✳

❊♠❜♦r❛ ♠✉✐t♦s ❛✉t♦r❡s t❡♥❤❛♠ ❡st✉❞❛❞♦ ♦ ♣r♦❜❧❡♠❛ (1.1) ♣❛r❛ ♦s ♦♣❡r❛❞♦r❡s ▲❛♣❧❛✲ ❝✐❛♥♦ ❡ p✲▲❛♣❧❛❝✐❛♥♦✱ ♣♦✉❝♦s tr❛❜❛❧❤♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s tr❛t❛♥❞♦ ❞❡ ❡q✉❛çõ❡s ❞❡

❙❝❤rö❞✐♥❣❡r ❜✐❤❛r♠ô♥✐❝❛s✱ ❞❛❞❛s ♣♦r

⎧ ⎨ ⎩

ǫ4_∆2_{u+V(x)u=f(u) ❡♠ R}N

u∈H2_(RN_), ✭✶✳✷✮

❊♠ ❬✶✶✱ ✶✷❪✱ P✐♠❡♥t❛ ❡ ❙♦❛r❡s ♣r♦✈❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ (1.2)✱ ♦♥❞❡ ❡st❛s ❛♣r❡s❡♥t❛♠ ✉♠ ❢❡♥ô♠❡♥♦ ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡♠ t♦r♥♦ ❞❡ ✉♠ ♣♦♥t♦ ❞❡

♠í♥✐♠♦ ❣❧♦❜❛❧ ❞❡ V✱ t❛❧ ❝♦♠♦ ❢❡✐t♦ ❡♠ ❬✺❪✳ ❏á ♥♦ tr❛❜❛❧❤♦ ❬✻❪✱ ❋✐❣✉❡✐r❡❞♦ ❡ P✐♠❡♥t❛

♣r♦✈❛♠ ❛ ❡①✐stê♥❝✐❛ ❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s q✉❡ s❡ ❝♦♥❝❡♥tr❛♠ ❡♠ t♦r♥♦ ❞❡ ✉♠ ♠í♥✐♠♦ ❣❧♦❜❛❧ ❞❡ V, ❧❡✈❛♥❞♦✲s❡ ❡♠ ❝♦♥t❛ ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞♦ ❝♦♥❥✉♥t♦ ♦♥❞❡ V

❛t✐♥❣❡ ♦ s❡✉ ♠í♥✐♠♦✱ ♦♥❞❡ ❛ ❡q✉❛çã♦ ❝♦♥s✐❞❡r❛❞❛ é s❡♠❡❧❤❛♥t❡ ❛ (1.2)✳

◆❡st❡ tr❛❜❛❧❤♦✱ r❡❛❧✐③❛♠♦s ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞♦ tr❛❜❛❧❤♦ ❬✶✶❪✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ✉♠❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ f s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s✿

(f1) f ∈C1(R)✱

(f2) f(ξ) = o(ξ)✱ q✉❛♥❞♦ ξ→0✱

(f3) ❡①✐st❡♠ ❝♦♥st❛♥t❡s a1, a2 >0t❛✐s q✉❡

|f(ξ)| ≤a1|ξ|+a2|ξ|p, ∀ξ ∈R

♦♥❞❡ 1≤p < N_N+4₋₄✱

(f4) ♣❛r❛ ❛❧❣✉♠ 2< θ≤p+ 1✱ t❡♠♦s ♣❛r❛ t♦❞♦ ξ = 0

0≤θF(ξ)< f(ξ)ξ, ♦♥❞❡ F(ξ) =

ξ

0

f(t)dt✱

(f5) ❛ ❢✉♥çã♦ ξ→

f(ξ)

ξ ✱ é ❝r❡s❝❡♥t❡ ♣❛r❛ ξ >0✱ ❞❡❝r❡s❝❡♥t❡ ♣❛r❛ ξ <0✳

◗✉❛♥t♦ ❛♦ ♣♦t❡♥❝✐❛❧ V✱ ✈❛♠♦s s✉♣♦r ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

(V1) V ∈C0(RN)∩L∞(RN)✱

✶✳ ■♥tr♦❞✉çã♦ ✶✾

0< V(x0) =V0 = inf

RN V <inf_∂Ω V.

❖ ♦❜❥❡t✐✈♦ é ♠♦str❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦

❚❡♦r❡♠❛ ✶ ❙❡❥❛♠f ❡ V ❢✉♥çõ❡s q✉❡ s❛t✐s❢❛③❡♠(f1)−(f5)❡(V1)−(V2)r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❊♥tã♦ ♣❛r❛ t♦❞❛ s❡q✉ê♥❝✐❛ ǫn →0 ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥t✐♥✉❛r❡♠♦s ❛ ❞❡♥♦t❛r

♣♦r {ǫn} t❛❧ q✉❡ (1.2) ✭❝♦♠ ǫn =ǫ) ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ♥ã♦✲tr✐✈✐❛❧ un ∈H2(RN)✳ ❆✐♥❞❛

♠❛✐s✱s❡♥❞♦ xn ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❞❡ |un|✱❡♥tã♦ xn∈Ω ❡ ❛✐♥❞❛

lim

n→∞V(xn) = infRN V.

◆♦ tr❛❜❛❧❤♦ ❬✶✶❪✱ ♦s ❛✉t♦r❡s ♣r♦✈❛♠ ♦s r❡s✉❧t❛❞♦s s❡♠ ✉t✐❧✐③❛r ❛ ❝♦♥❞✐çã♦ (f4) s♦❜r❡

❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡✱ ❝❤❛♠❛❞❛ ❞❡ ❝♦♥❞✐çã♦ ❞❡ ❆♠❜r♦s❡tt✐✲❘❛❜✐♥♦✇✐t③✳❆♦ ✐♥✈és ❞❡ss❛✱ ❝♦♥✲ s✐❞❡r❛r❛♠ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ s✉♣❡r❧✐♥❡❛r✐❞❛❞❡ s♦❜r❡ f ♠❛✐s ❢r❛❝❛✱ ❞❡ ♠♦❞♦ ❛ ❛❜r❛♥❣❡r ✉♠

♥ú♠❡r♦ ♠❛✐♦r ❞❡ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡s✱ ♦ q✉❡ ❡①✐❣✐✉ ❛ ❛❞✐çã♦ ❞❡ ♠✉✐t♦s r❡s✉❧t❛❞♦s té❝♥✐❝♦s ❛♦ tr❛❜❛❧❤♦✳

◆♦ss❛ ♣r♦♣♦st❛✱ ❛♦ ❛♣r❡s❡♥t❛r ♦ t❡①t♦ ❝♦♠ ✉♠❛ ❝♦♥❞✐çã♦ ♠❛✐s r❡str✐t✐✈❛✱ ❢♦✐ ♣❛r❛ q✉❡ ❡st❡ s❡ t♦r♥❡ ♠❛✐s ❛❝❡ssí✈❡❧✱ ♣♦r ❝♦♥t❡r ♠❡♥♦s ❞❡t❛❧❤❡s té❝♥✐❝♦s✱ ❛ ✉♠ ✐♥✐❝✐❛♥t❡ ♥❛ ár❡❛ q✉❡ s❡ ✐♥t❡r❡ss❡ ♣❡❧♦ tr❛❜❛❧❤♦ ❬✶✶❪✱ ♦♥❞❡ t♦❞❛s ❛s ❞❡♠♦♥str❛çõ❡s ❛♣r❡s❡♥t❡♠ s♦♠❡♥t❡ ♦s ❞❡t❛❧❤❡s ✐♥❡r❡♥t❡s à té❝♥✐❝❛ ❞❡ ♣❡♥❛❧✐③❛çã♦ ✉t✐❧✐③❛❞❛ ❡ ♥ã♦ ❛♦s q✉❡ sã♦ té❝♥✐❝♦s ♦r✐✉♥❞♦s ❞❛ ❢❛❧t❛ ❞❛ ❝♦♥❞✐çã♦ ❞❡ ❆♠❜r♦s❡tt✐✲❘❛❜✐♥♦✇✐t③✳

❈❛♣ít✉❧♦

✷

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦ ✈❡r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s✱ ♥♦t❛çõ❡s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ❞❡s❝r❡✈❡♠♦s ✉♠❛ ♥♦✈❛ ❝❧❛ss❡ ❞❡ ♦❜❥❡t♦s ❡♠ q✉❡ é ❝♦❡r❡♥t❡ ❞❡✜♥✐r ✉♠❛ ❞❡r✐✈❛❞❛ ✭❣❡♥❡r❛❧✐③❛❞❛✮♦♥❞❡ ❛s r❡❣r❛s ❞♦ ❝á❧❝✉❧♦ s❡ ♠❛♥t❡♥❤❛♠ ❡ ❛♦ ♠❡s♠♦ t❡♠♣♦ ♣♦ss❛✲s❡ ✐♥❝❧✉✐r ❢✉♥çõ❡s ♥ã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s ♥♦ s❡♥t✐❞♦ ❝❧áss✐❝♦✳ ❚❛✐s ♦❜❥❡t♦s sã♦ ❛s ❝❤❛♠❛❞❛s ❞✐str✐❜✉✐çõ❡s✳

❆♣ós ✈❡r❡♠♦s ✉♠❛ ✐♠♣♦rt❛♥t❡ ♦♣❡r❛çã♦ ❝♦♠ ❞✐str✐❜✉✐çõ❡s✱ ❞❡✜♥✐♠♦s ♦s ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❡ ❛❞♠✐t✐♠♦s ♦s ❚❡♦r❡♠❛s ❞❡ ■♠❡rsõ❡s✳ P♦r ✜♠ ❞❡♠♦str❛♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳

✷✳✶ ❉✐str✐❜✉✐çõ❡s

❙❡❥❛φ : Ω⊂RN _→R₍C₎ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥♦ ❛❜❡rt♦ Ω✳ ❊♥tã♦ ❞❡✜♥✐♠♦s✿

❉❡✜♥✐çã♦ ✶ ❖ ❝♦♥❥✉♥t♦ supp(φ) = {x∈Ω;φ(x)= 0} é ❝❤❛♠❛❞♦ ❞❡ s✉♣♦rt❡ ❞❡ φ✳ ❙❡

❡st❡ ❝♦♥❥✉♥t♦ ❛❧é♠ ❞❡ ❢❡❝❤❛❞♦ ❢♦r ❝♦♠♣❛❝t♦ ❞✐③❡♠♦s q✉❡ φ t❡♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳

❉❡✜♥✐çã♦ ✷ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❛s ❢✉♥çõ❡s C∞(RN₎ ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱ ♦ q✉❛❧ ❝❤❛✲ ♠❛♠♦s ❞❡ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡✱ s❡rá ❞❡♥♦t❛❞♦ ♣♦r D(RN_{)✳ ❙❡ Ω é ✉♠ ❛❜❡rt♦ ❞♦ R}N_✱

❛✐♥❞❛ ♣♦❞❡♠♦s ❢❛❧❛r ❞♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s C∞ ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ❝♦♥t✐❞♦ ❡♠ Ω✳ ❊st❡

❡s♣❛ç♦ s❡rá ❞❡♥♦t❛❞♦ ♣♦r D(Ω)✳

❱❛♠♦s ❢♦r♥❡❝❡r à D(Ω) ✉♠❛ t♦♣♦❧♦❣✐❛ t❛❧ q✉❡ ❢❛ç❛ ❞❡ D(Ω) ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t♦♣♦✲ ❧ó❣✐❝♦✳ Pr❡❝✐s❛♠♦s ❡♥tã♦ s❛❜❡r ♦ q✉❡ sã♦ s❡q✉ê♥❝✐❛s ❝♦♥✈❡r❣❡♥t❡s ❡♠ D(Ω)✳

✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✷ ❉❡✜♥✐çã♦ ✸ ❯♠❛ s❡q✉ê♥❝✐❛ {φm} ❞❡ ❢✉♥çõ❡s ❡♠ D(Ω) é ❞✐t❛ ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ③❡r♦ s❡

❡①✐st✐r ✉♠ ❝♦♠♣❛❝t♦ K ⊂Ωt❛❧ q✉❡ supp(φm)⊂K✱ ♣❛r❛ t♦❞♦ m ❡ t♦❞❛s ❛s s✉❛s ❞❡r✐✈❛❞❛s

❝♦♥✈❡r❣❡♠ ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ ③❡r♦ ❡♠ K✳

❉❡✜♥✐çã♦ ✹ ❯♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❛r T ❞❡✜♥✐❞♦ ❡♠ D(Ω) é ❞✐t♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ Ωs❡♠✲ ♣r❡ q✉❡✱ s❡ φm →0 ❡♠ D(Ω) ❡♥tã♦ T(φm)→0✱ q✉❛♥❞♦ m→ ∞✳

❖ ❡s♣❛ç♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✱ ♦ q✉❛❧ é ♦ ❞✉❛❧ ❞♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡✱ é ❞❡♥♦t❛❞♦ ♣♦r D′_(Ω)✳

❊①❡♠♣❧♦ ✶ ✭❆ ❞✐tr✐❜✉✐çã♦ ❞❡ ❉✐r❛❝✮ ❙❡❥❛ x ∈ RN✳ ❉❡✜♥❛ _δx ♣♦r _{δx(φ) = φ(x})✱ ♣❛r❛ t♦❞❛ φ ∈ D(RN_{)✳ ➱ ❝❧❛r♦ q✉❡ δ}

x ❞❡✜♥❡ ✉♠❛ ❞✐str✐❜✉✐çã♦✳ ❊♠ ♣❛rt✐❝✉❧❛r s❡ x = 0

❡s❝r❡✈❡♠♦s ❛♣❡♥❛s δ ❡ ❡st❡ é ♦ ❝♦♥❤❡❝✐❞♦ δ ❞❡ ❉✐r❛❝✳

❉❡✜♥✐çã♦ ✺ ❯♠❛ ❢✉♥çã♦ f : Ω ⊂ RN _{→ R(C) é ❞✐t❛ ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s❡ ♣❛r❛}

q✉❛❧q✉❡r ❝♦♠♣❛❝t♦ K ⊂ Ω t✐✈❡r♠♦s q✉❡

K

|f|<∞. ❉❡♥♦t❛♠♦s ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s

❞❡✜♥✐❞❛s ❡♠ Ω q✉❡ sã♦ ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ♣♦r L1

loc(Ω)✳

❊①❡♠♣❧♦ ✷ ❉❛❞❛ ✉♠❛ ❢✉♥çã♦ ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ f ❞❡✜♥❛ ♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r Tf ❡♠

D(Ω) ♣♦r Tf(φ) =

Ω

f φdx. ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ Tf é ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ D′(Ω)✳

❱❛♠♦s ❛❣♦r❛ ✈❡r ✉♠❛ ♦♣❡r❛çã♦ ❢❛♠✐❧✐❛r ❛♦ ❈á❧❝✉❧♦ ❛♣❧✐❝❛❞❛ à ❞✐str✐❜✉✐çõ❡s✱ q✉❡ é ❛ ❞❡ ❞✐❢❡r❡♥❝✐❛çã♦✳ ❈♦♠❡ç❛r❡♠♦s ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ♠✉❧t✐✲í♥❞✐❝❡ ❞❡ ❙❝❤✇❛rt③✱ ❛ q✉❛❧ s❡rá ♠✉✐t♦ út✐❧✳

❉❡✜♥✐çã♦ ✻ ❯♠ ♠✉❧t✐✲í♥❞✐❝❡ α é ✉♠❛ n✲✉♣❧❛ α = (α1,· · · , αn)✱ αi ≥0 ✐♥t❡✐r♦s✳ ❆ss♦✲

❝✐❛❞♦ ❛ ✉♠ ♠✉❧t✐✲í♥❞✐❝❡ α t❡♠♦s✿ |α| = α1 +· · ·+αn✱ ✭♦r❞❡♠ ❞❡ α✮❀ α! = α1!· · ·αn!❀

xα _=xα1

1 · · ·xαnn✱ ♦♥❞❡ x= (x1,· · · , xn)∈Rn.

P♦r ✜♠ ❞❡✜♥✐♠♦s Dα ₌ ∂| α|

∂xα1

1 · · ·∂xαnn

.

❉❡✜♥✐çã♦ ✼ ❙❡❥❛♠ T ∈ D′_{(Ω) ❡ Ω ⊂ R}N _{❛❜❡rt♦✳ ❉❡✜♥✐♠♦s ♣❛r❛ t♦❞♦ ♠✉❧t✐✲í♥❞✐❝❡ α ❛}

❞✐str✐❜✉✐çã♦ Dα_{T ♣♦r}

(Dα_{T)(φ) = (−1)}|α|_T(Dα_{φ) ∀φ∈ D(Ω)}

❊①❡♠♣❧♦ ✸ ❈♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ ❞❡ ❍❡❛✈✐s✐❞❡ ❡♠ R

H(x) =

⎧ ⎨ ⎩

✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✸ ❊st❛ ❢✉♥çã♦ é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ❡ ♣♦r ✐ss♦ ❞❡✜♥❡ ✉♠❛ ❞✐str✐❜✉✐çã♦✱ ❛ q✉❛❧ ❞❡♥♦t❛r❡♠♦s ♣♦r TH✳ ❙❡❥❛ φ∈ D(R) ❡♥tã♦

dTH

dx (φ) =−TH

dφ dx =− R Hdφ

dxdx=−

+∞

0

dφ

dxdx=−y→lim+∞

y

0

dφ dxdx

=φ(0) =δ(φ)

∴ dTH

dx =δ ✭ δ ❞❡ ❉✐r❛❝✮✳

✷✳✷ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈

❙❡❥❛Ω ✉♠ ❛❜❡rt♦ ❞♦ RN_{✱1≤p <+∞ ❡m ∈N✳ ❙❡ u∈L}p_{(Ω)✱ é s❛❜✐❞♦ q✉❡ u♣♦ss✉✐}

❞❡r✐✈❛❞❛s ❞❡ t♦❞❛s ❛s ♦r❞❡♥s ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✱ ♠❛s ♥ã♦ é ✈❡r❞❛❞❡✱ ❡♠ ❣❡r❛❧✱ q✉❡Dα_{u s❡❥❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ❞❡ L}p_(Ω)✳

◗✉❛♥❞♦Dα_{ué ❣❡r❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ❞❡ L}p_{(Ω)✱ ❞❡✜♥✐✲s❡ ✉♠ ♥♦✈♦ ❡s♣❛ç♦ ❞❡♥♦♠✐♥❛❞♦}

❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈✱ ♦ q✉❛❧ r❡♣r❡s❡♥t❛♠♦s ♣♦rWm,p_{(Ω)♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s}

u∈Lp_{(Ω)✱ t❛✐s q✉❡ ♣❛r❛ t♦❞♦ |α| ≤m✱ D}α_u∈Lp_{(Ω)✱ ✐st♦ é}

Wm,p_{(Ω) ={u∈L}p_(Ω);Dα_u∈Lp_{(Ω),∀|α| ≤m}}

s❡♥❞♦ Dα_{u ❛ ❞❡r✐✈❛❞❛ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✳}

P❛r❛ ❝❛❞❛ u∈Wm,p_{(Ω) ❞❡✜♥✐♠♦s ❛ ♥♦r♠❛ ❞❡ u ❞❛ ❢♦r♠❛✿}

up m,p =

|α|≤m

Ω

|Dαu|p

dx.

❖ ❡s♣❛ç♦ ♥♦r♠❛❞♦ (Wm,p_(Ω),.

m,p)é ❞❡♥♦♠✐♥❛❞♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈✳

❘❡♣r❡s❡♥t❛✲s❡ Wm,2_{(Ω) = H}m_{(Ω) ❞❡✈✐❞❛ ❛ ❡str✉t✉r❛ ❤✐❧❜❡rt✐❛♥❛ ❞❡ L}2_{(Ω)✱ ❛ q✉❛❧ é}

❤❡r❞❛❞❛ ♣❡❧♦s ❡s♣❛ç♦s Hm_(Ω)✳

❚❡♦r❡♠❛ ✷ ❖s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ Wm,p_{(Ω) sã♦ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳}

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ uν ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ Wm,p(Ω)✳ ▼♦str❡♠♦s q✉❡ uν

❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ u∈Wm,p_{(Ω). ❉❡ ❢❛t♦✱ ❝♦♠♦ u}

ν é ❞❡ ❈❛✉❝❤② t❡♠♦s✿

uν −uµpm,p =

|α|≤m

Ω

|Dα_u

✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✹ ❙❡❣✉❡ q✉❡ (Dα_u

ν)é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❞♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ Lp(Ω)✳ ▲♦❣♦✱ ♣❛r❛

❝❛❞❛ |α| ≤ m✱ ❡①✐st❡ uα ∈ Lp(Ω) t❛❧ q✉❡✿ Dαuν →uα ❡♠ Lp(Ω)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦

α= (0,· · · ,0) t❡♠♦s q✉❡ uν →u ❡♠ Lp(Ω)✳

❇❛st❛ ♠♦str❛r q✉❡ Dα_{u = u}

α✳ ❈♦♠ ❡❢❡✐t♦✱ ❞❛s ❝♦♥✈❡r❣ê♥❝✐❛s ❛♥t❡r✐♦r❡s✱ t❡♠♦s ❛s

s❡❣✉✐♥t❡s ❝♦♥✈❡r❣ê♥❝✐❛s ❡♠ D′ _(Ω)

Dα_u

ν →uα ❡Dαuν →Dαu

♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡ ❝♦♥❝❧✉í♠♦s ♦ ❞❡s❡❥❛❞♦✳

❈♦r♦❧ár✐♦ ✶ ❖s ❡s♣❛ç♦s Hm_{(Ω) sã♦ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦}

(u, v)Hm_(Ω) =

|α|≤m

(Dα_{u, D}α_v) L2_(Ω).

❖ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s C∞❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ❝♦♥t✐❞♦ ❡♠ Ωé ❞❡♥s♦ ❡♠ Lp_{(Ω)✱ ♠❛s}

♥ã♦ é ✈❡r❞❛❞❡ q✉❡ ❡st❡ ♠❡s♠♦ ❡s♣❛ç♦✱ ♦ q✉❛❧ ♣❛ss❛♠♦s ❛ ❞❡♥♦t❛r ♣♦r C∞

0 (Ω)✱ s❡❥❛ ❞❡♥s♦

❡♠ Wm,p_{(Ω)✳ P♦r ❡st❛ r❛③ã♦ ❞❡✜♥❡✲s❡ ♦ ❡s♣❛ç♦ W}m,p

0 (Ω) ❝♦♠♦ s❡♥❞♦ ♦ ❢❡❝❤♦ ❞❡ C0∞(Ω)

❡♠ Wm,p_{(Ω)✱ ✐st♦ é✱}

W₀m,p(Ω) =C∞

0 (Ω)

.m,p .

❖s ❡s♣❛ç♦s W₀m,p(Ω) t❛♠❜é♠ sã♦ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ❡ ❡♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦ p = 2✱

W₀m,2(Ω) =Hm

0 sã♦ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt✳

✷✳✷✳✶ ❚❡♦r❡♠❛s ❞❡ ❞❡♥s✐❞❛❞❡ ❡ ✐♠❡rsã♦

❆s ❞❡♠♦str❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s ❞❡st❛ s❡çã♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✾❪✳ ❚❡♦r❡♠❛ ✸ ❖ s✉❜❡s♣❛ç♦ C∞_(Ω)∩Wk,p_{(Ω) é ❞❡♥s♦ ❡♠ W}k,p_(Ω)✳

❱❛♠♦s ❛❣♦r❛ ❡♥✉♥❝✐❛r ❛s ❝♦♥❤❡❝✐❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❞❡ ❙♦❜♦❧❡✈ ♣❛r❛ ❢✉♥çõ❡s ❡♠ W₀1,p(Ω)✳ ❚❡♦r❡♠❛ ✹

W₀1,p(Ω) ⊂

⎧ ⎨ ⎩

✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✺ ❆❧é♠ ❞✐ss♦✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C =C(N, p) t❛❧ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r u∈W₀1,p(Ω)✱

u N p

N−p ≤ C Dup ♣❛r❛ p < N,

sup

Ω

|u| ≤ C|Ω|N1−

1

p Du

p ♣❛r❛ p > N.

❉❡✜♥✐çã♦ ✽ ❉✐③❡♠♦s q✉❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ B1 ❡stá ✐♠❡rs♦ ❝♦♥t✐♥✉❛♠❡♥t❡ ♥♦ ❡s♣❛ç♦

❞❡ ❇❛♥❛❝❤ B2 ❡ ❞❡♥♦t❛♠♦s B1 ֒→B2✱ s❡ ❡①✐st✐r ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r ❧✐♠✐t❛❞❛ ❡ ❜✐❥❡t♦r❛

B1 →B2✳

❆ss✐♠✱ ♦ ❚❡♦r❡♠❛ ✹ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❞❛ ❢♦r♠❛

W₀1,p(Ω)֒→LNN p−p(Ω) p < N,

W₀1,p(Ω) ֒→C0(Ω) p > N.

■t❡r❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✹ k ✈❡③❡s ❝❤❡❣❛♠♦s ❛ ✉♠❛ ❡①t❡♥sã♦ ♣❛r❛ ♦s ❡s♣❛ç♦s W₀k,p(Ω)✳

❈♦r♦❧ár✐♦ ✷

W₀k,p(Ω)֒→LNN p−kp(Ω) kp < N,

W₀k,p(Ω) ֒→Cm_{(Ω) kp > N.}

✷✳✸ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛

❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ I : E → R ✉♠ ❢✉♥❝✐♦♥❛❧✳ ❱❡❥❛♠♦s ❛s s❡❣✉✐♥t❡s ❞❡✜♥✐çõ❡s✳

❉❡✜♥✐çã♦ ✾ ❖ ❢✉♥❝✐♦♥❛❧ I é ❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ u ∈ E s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦

❧✐♥❡❛r ❝♦♥tí♥✉❛ L=L(u) :E →R s❛t✐s❢❛③❡♥❞♦✿

∀ǫ >0, ∃δ =δ(ǫ, u)>0 t❛❧ q✉❡ |I(u+v)−I(u)−Lv| ≤ǫv✱ s❡♠♣r❡ q✉❡ v ≤δ✳

❆ ❛♣❧✐❝❛çã♦ L é ✉s✉❛❧♠❡♥t❡ ❞❡♥♦t❛❞❛ ♣♦r I′(u)✳

❉❡✜♥✐çã♦ ✶✵ ❯♠ ♣♦♥t♦ ❝rít✐❝♦ u ❞❡ I é t❛❧ q✉❡ I′(u) = 0✱ ✐st♦ é✱

✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✻ ❖ ✈❛❧♦r ❞❡ I ❡♠ u é ❡♥tã♦ ❝❤❛♠❛❞♦ ❞❡ ✈❛❧♦r ❝rít✐❝♦ ❞❡ I✳

❊♠ ❛♣❧✐❝❛çõ❡s ♣❛r❛ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✱ ♣♦♥t♦s ❝rít✐❝♦s ❞❡ ❢✉♥❝✐♦♥❛✐s ❝♦r✲ r❡s♣♦♥❞❡♠ ❛ s♦❧✉çõ❡s ❢r❛❝❛s ❞❡ ❡q✉❛çõ❡s✳

❊①✐st❡♠ r❡s✉❧t❛❞♦s q✉❡ ❣❛r❛♥t❡♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ ❢✉♥❝✐♦♥❛✐s✳ ❖ ❚❡♦✲ r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ é ✉♠ ❞❡ss❡s r❡s✉❧t❛❞♦s ❡ ❡stá ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ✭P❙✮✳

❉❡✜♥✐çã♦ ✶✶ ❙❡❥❛ I ∈ C1_{(E,R)✱ E ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❉✐③❡♠♦s q✉❡ I s❛t✐s❢❛③ ❛ ❝♦♥✲}

❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ✭P❙✮ s❡ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ {un} ❡♠ E t❛❧ q✉❡ {I(un)} é ✉♠❛

s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❡ I′_(u

n)→0q✉❛♥❞♦ n→ ∞✱ ♣♦ss✉❛ ✉♠❛ s✉❜s❡q✉ê♥✲

❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳

❱❛♠♦s ♣r♦✈❛r ❛ ✈❡rsã♦ ✉s✉❛❧ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳ ❆ ♣r♦✈❛ s❡ t♦r♥❛ s✐♠♣❧❡s ✉t✐❧✐③❛♥❞♦ ❛ s❡❣✉✐♥t❡ ✈❡rsã♦ ❞♦ ▲❡♠❛ ❞❛ ❞❡❢♦r♠❛çã♦✳

▲❡♠❛ ✶ ✭▲❡♠❛ ❞❛ ❞❡❢♦r♠❛çã♦✮ ❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❙✉♣♦♥❤❛ q✉❡ ♦ ❢✉♥❝✐✲

♦♥❛❧ I ∈ C1_{(E,R) é t❛❧ q✉❡ s❛t✐s❢❛ç❛ ❛ ❝♦♥❞✐çã♦ ✭P❙✮✳ ❙❡ c ∈ R ♥ã♦ é ✉♠ ✈❛❧♦r ❝rít✐❝♦}

❞❡ I✱ ❡♥tã♦ ❞❛❞♦ ǫ >0 ❡①✐st❡ ✉♠ ǫ ∈ (0, ǫ) ❡ η ∈ C([0,1]×E, E) t❛✐s q✉❡ ♣❛r❛ q✉❛❧q✉❡r

u∈E ❡ t∈[0,1] t❡♠✲s❡✿

✶♦_{✮ η(t, u) = u s❡✱ u /∈I}−1_{([c−ǫ, c+ǫ])❀}

✷♦_{✮ η(1, I}c+ǫ_)⊂Ic−ǫ_✳

❉❡♠♦♥str❛çã♦✿

❈♦♠♦ c∈R ♥ã♦ é ✉♠ ✈❛❧♦r ❝rít✐❝♦ ❞❡ _I✱ ❞❡✈❡♠ ❡①✐st✐r ❝♦♥st❛♥t❡s _{α, β >0}t❛✐s q✉❡ s❡

u ∈I−1_{([c−α, c+α]) ✐♠♣❧✐❝❛ q✉❡ I}′_{(u) ≥β✱ ❝❛s♦ ❝♦♥trár✐♦✱ ♣❛r❛ q✉❛✐sq✉❡r α, β > 0}

❡①✐st✐rá u∗ _∈I−1_{([c−α, c+α])❝♦♠ I}′_(u∗_{)< β.}

❚♦♠❡ ♣❛r❛ ❝❛❞❛ n∈N _α= 1

n ❡β =

1

n✱ ❛ss✐♠ ❡①✐st✐rá u

∗

n t❛❧ q✉❡

c− 1

n ≤I(u

∗

n)≤c+

1

n ❝♦♠ I

′

(u∗n)<

1

n ∀n∈N.

▲♦❣♦✱ ♣❛ss❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ n→ ∞ ♦❜t❡♠♦s q✉❡ I(u∗n)→c ❡I′(u∗n)→0✳

❈♦♠♦ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭P❙✮ t❡♠♦s q✉❡ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛s u∗

n →u ❡♠ E✳

❯♠❛ ✈❡③ q✉❡ I ∈C1_{(E,R)✱ t❡♠♦s I(u}∗

n)→I(u) ❡I′(u∗n)→I′(u)✳

P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡ s❡❣✉❡ q✉❡ I(u) =c❡ I′_{(u) = 0✱ ✐st♦ é✱ cé ✉♠ ✈❛❧♦r ❝rít✐❝♦ ❞❡}

✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✼ P♦rt❛♥t♦✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡s α, β > 0 t❛✐s q✉❡✱ s❡ u ∈ I−1_{([c−α, c+α]) ❡♥tã♦}

I′_{(u) ≥β.}

❆❣♦r❛ ❝♦♥s✐❞❡r❡ ǫ∈(0, α] ✜①❛❞♦✱ ǫ∈(0, ǫ) ❡ δ= 4ǫ

β✳

❙❡❥❛♠

A=I−1([c−ǫ, c+ǫ])✱B =I−1([c−ǫ, c+ǫ]) ❡ Y ={u∈E;I′(u)= 0}.

❆❧é♠ ❞✐ss♦✱ ❝♦♥s✐❞❡r❡ V :Y →X ✉♠ ❝❛♠♣♦ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡ ♣❛r❛ I ❡♠ Y✱ ✐st♦ é✱ V

é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✐❛♥❛ ❡ t❛❧ q✉❡✱ ♣❛r❛ ❝❛❞❛ u∈Y s❛t✐s❢❛③✿

(i) V(u) ≤2I′(u),

(ii) I′_{(u), V(u) ≥ I}′_(u)2_.

❈♦♥s✐❞❡r❡ t❛♠❜é♠ 0≤ρ≤1✉♠❛ ❢✉♥çã♦ ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✐❛♥❛ ❞❡✜♥✐❞❛ ♣♦r

ρ : E −→ R

u −→ ρ(u) = _d(u,Ed(_\u,E_A)+\A_d()_u,B).

❉❡✜♥❛ f :E →E ♣♦r

f(u) =

⎧ ⎨ ⎩

−ρ(u)_VV(₍u_u)₎, s❡u∈A

0, s❡ u /∈A.

❚❡♠♦s q✉❡ f é ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✐❛♥❛ ❡ f ≤1♣❛r❛ t♦❞❛ u∈E✳

❚❡♠♦s ❛✐♥❞❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②

⎧ ⎨ ⎩

d

dtw(t, u) = f(w(t, u))

w(0, u) = u,

t❡♠ s♦❧✉çã♦ ú♥✐❝❛✱ ❛ q✉❛❧ ❞❡♥♦t❛♠♦s ♣♦r w(t, u)✱ q✉❡ ❡stá ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ t∈R❡ ♣❛r❛ ❝❛❞❛ u∈E✳

❙❡❥❛η: [0,1]×E →E ❞❡✜♥✐❞❛ ♣♦rη(t, u) =w(δt, u)✳ ❱❛♠♦s ♠♦str❛r q✉❡ η(t, u) = u

s❡ u /∈I−1_{([c−ǫ, c+ǫ]) =A✳}

❉❡ ❢❛t♦✱ s❡❥❛ w1(t, u) = u ♣❛r❛ t♦❞♦ t ∈ R✳ ◆♦t❡ q✉❡

d

dtw1(t, u) = 0 =f(w1(t, u))✱

♣♦✐su /∈A✱ ♦ q✉❡ ✐♠♣❧✐❝❛

⎧ ⎨ ⎩

d

dtw1(t, u) = f(w1(t, u))

✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✽ P♦r ✉♥✐❝✐❞❛❞❡ w1(t, u) = u = w(t, u) ♣❛r❛ t♦❞♦ t ∈ R✳ ▲♦❣♦ s❡ u /∈ A t❡♠♦s q✉❡

η(t, u) = w(δt, u) = u ♣❛r❛ t∈[0,1]✱ ♦ q✉❡ ♠♦str❛ ♦ ♣r✐♠❡✐r♦ ✐t❡♠ ❞♦ ▲❡♠❛✳

P❛r❛ ♦ s❡❣✉♥❞♦ ✐t❡♠ ❞♦ ▲❡♠❛ ♥♦t❡ q✉❡ ♣❛r❛ ❝❛❞❛ u∈E ✜①❛❞♦ ❛ ❢✉♥çã♦ I(w(t, u))é ❞❡❝r❡s❝❡♥t❡ ❡♠ t✳ ❙❡❥❛ u∈Ic+ǫ _{✐r❡♠♦s ❝♦♥s✐❞❡r❛r ❞♦✐s ❝❛s♦s✿}

✭✶✮ P❛r❛ ❛❧❣✉♠ t ∈[0, δ]✱ t❡♠♦s I(w(t, u))≤c−ǫ✳

❆ss✐♠ ❝♦♠♦ I(w(t, u)) é ❞❡❝r❡s❝❡♥t❡ I(w(δ, u)) ≤ I(w(t, u)) ≤ c− ǫ✱ ❞♦♥❞❡ s❡❣✉❡ η(1, u) =w(δ, u)∈Ic−ǫ_✳

✭✷✮ P❛r❛ t♦❞♦ t∈[0, δ]✱ t❡♠♦s I(w(t, u))≥c−ǫ✳

❱✐st♦ q✉❡ I(w(t, u)) é ❞❡❝r❡s❝❡♥t❡ I(w(t, u))≤I(w(0, u)) =I(u)≤c+ǫ✳

▲♦❣♦ w(t, u)∈I−1_{([c−ǫ, c+ǫ]) =B ♣❛r❛ t♦❞♦ t∈[0, δ].}

❯s❛♥❞♦ q✉❡ I(w(t, u)) é ❞❡❝r❡s❝❡♥t❡ ❡ q✉❡ ρ= 1 ❡♠ B✱ ♦❜t❡♠♦s

I(w(δ, u)) = I(u) +

δ

0

d

dtI(w(t, u))dt

≤ I(u) + 1 2

δ

o

I′(w(t, u))dt

≤ c+ǫ− 1

2 4ǫ

δ δ =c−ǫ.

▼♦str❛♥❞♦ q✉❡

I(w(δ, u))≤c−ǫ.

P♦rt❛♥t♦✱ ❡♠ q✉❛❧q✉❡r ❝❛s♦✱ t❡♠♦s q✉❡ η(1, u) = w(δ, u) ∈ Ic−ǫ _{s❡ u ∈ I}c+ǫ_{✱ ✐st♦ é✱}

η(1, Ic+ǫ_)⊂Ic−ǫ_.

❚❡♦r❡♠❛ ✺ ✭❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✮ ❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ s✉✲

♣♦♥❤❛ q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ I ∈C1_{(E,R) ❡ s❛t✐s❢❛ç❛ ❛ ❝♦♥❞✐çã♦ ✭P❙✮✳}

❙✉♣♦♥❤❛ ❛✐♥❞❛ q✉❡ I(0) = 0 ❡

I1) ❡①✐st❡♠ ❝♦♥st❛♥t❡s ρ, α >0 t❛✐s q✉❡ I|∂Bρ ≥α✱ ❡ I2) ❡①✐st❡ e∈E\Bρ t❛❧ q✉❡ I(e)≤0✳

❊♥tã♦ I ♣♦ss✉✐ ✉♠ ✈❛❧♦r ❝rít✐❝♦ c≥α✳ ❆❧é♠ ❞✐ss♦ c ♣♦❞❡ s❡r ❝❛r❛❝t❡r✐③❛❞♦ ❝♦♠♦

c= inf

✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✾ ♦♥❞❡ Γ ={g ∈C([0,1], E);g(0) = 0 ❡ g(1) =e}✳

❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ c= inf

g∈Γtmax∈[0,1]I(g(t))<∞✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ g ∈ C([0,1], E)✱

g([0,1]) ⊂ E é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦✱ ❧♦❣♦ ♦ ❝♦♥❥✉♥t♦ {I(g(t));t ∈ [0,1]} ❛t✐♥❣❡ s❡✉

♠á①✐♠♦✳

❆✜r♠❛çã♦✿ max

t∈[0,1]I(g(t))≥α✱ ♣❛r❛ t♦❞❛ g ∈Γ.❈♦♠ ❡❢❡✐t♦✱ s❡❥❛ g ∈Γ ❡ ❞❡✜♥❛

h : [0,1]→R ♣♦r _{h(t) = g(t)}✳

➱ ❝❧❛r♦ q✉❡ h é ❝♦♥tí♥✉❛ ❡ ❛✐♥❞❛✱ h(0) =g(0)=0= 0 < ρ✱ ❡ ❝♦♠♦ e∈E\Bρ s❡❣✉❡

q✉❡h(1) =g(1)=e> ρ✳

❉❡st❛ ❢♦r♠❛ h(0) < ρ < h(1)✱ ♣❡❧♦ t❡♦r❡♠❛ ❞♦ ✈❛❧♦r ✐♥t❡r♠❡❞✐ár✐♦ ❡①✐st❡ t0 ∈ (0,1)

t❛❧ q✉❡ h(t0) =ρ✱ ✐st♦ é✱ g(t0)=ρ✳

▲♦❣♦ g(t0) ∈ ∂Bρ ❡ ♣❡❧❛ ❝♦♥❞✐çã♦ I1) s❡❣✉❡ q✉❡ I(g(t0)) ≥ α✳ ❈♦♠♦ g ∈ Γ é q✉❛❧q✉❡r

♦❜t❡♠♦s q✉❡ max

t∈[0,1]I(g(t))≥α✱ ♣❛r❛ t♦❞❛ g ∈Γ.

❊♥tã♦ ♦ ❝♦♥❥✉♥t♦ H =

max

t∈[0,1]I(g(t));g ∈Γ

é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r α✳

P♦rt❛♥t♦ c= inf

g∈Γtmax∈[0,1]I(g(t)) ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ ❛✐♥❞❛ α s❡♥❞♦ ✉♠❛ ❝♦t❛ ✐♥❢❡r✐♦r ❞♦

❝♦♥❥✉♥t♦ H s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ c q✉❡c≥α✳

❇❛st❛ ❛❣♦r❛ ♠♦str❛r q✉❡ c é ✈❛❧♦r ❝rít✐❝♦ ❞❡ I✳

❙✉♣♦♥❤❛ q✉❡ ♥ã♦ s❡❥❛✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ❞❛ ❞❡❢♦r♠❛çã♦✱ t♦♠❛♥❞♦ ǫ = α₂ > 0 ❡①✐st❡ ✉♠ ǫ ∈ (0, ǫ) ❡ η ∈ C([0,1]× E, E) t❛✐s q✉❡ η(1, u) = u s❡ I(u) ∈/ [c− ǫ, c+ǫ] ❡

η(1, Ac+ǫ)⊂Ac−ǫ✳

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥✜♠♦✱ ❡s❝♦❧❤❛ g ∈Γ ❞❡ ♠♦❞♦ q✉❡

max

t∈[0,1]I(g(t))≤c+ǫ, ✭✷✳✶✮

❡ ❝♦♥s✐❞❡r❡h∗_{(t) =η(1, g(t))✳ ➱ ❝❧❛r♦ q✉❡ h}∗ _{∈C([0,1], E)✳}

◆♦t❡ q✉❡✱ s❡♥❞♦ g(0) = 0❡♥tã♦ I(g(0)) =I(0) = 0✱ ♣♦r ❤✐♣ót❡s❡✳

▲♦❣♦ I(g(0)) = 0< α

2 ≤c−ǫ ❡ ❛ss✐♠ I(g(0)) ∈/ [c−ǫ, c+ǫ]✳ P❡❧♦ ✶♦✮ ✐t❡♠ ❞♦ ▲❡♠❛ ❞❛

❞❡❢♦r♠❛çã♦ s❡❣✉❡ q✉❡ h∗(0) =η(1, g(0)) =η(1,0) = 0.

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♥♦t❡♠♦s q✉❡ g(1) = e ❡ ❛ss✐♠ I(g(1)) = I(e) ≤ 0✱ ♣♦r ❤✐♣ót❡s❡✳ ▲♦❣♦ I(g(1)) ≤0 < α

2 ≤ c−ǫ ♦ q✉❡ ✐♠♣❧✐❝❛ I(g(1)) ∈/ [c−ǫ, c+ǫ]✳ ◆♦✈❛♠❡♥t❡ ♣❡❧♦ ✶♦✮

✐t❡♠ ❞♦ ▲❡♠❛ ❞❛ ❞❡❢♦r♠❛çã♦ s❡❣✉❡ q✉❡ h∗_{(1) =η(1, g(1)) =η(1, e) =e✳}

✷✳ Pr❡❧✐♠✐♥❛r❡s ✸✵ ▲♦❣♦

max

t∈[0,1]I(h

∗_{(t))≥c. ✭✷✳✷✮}

P♦r (2.1) t❡♠♦s q✉❡ g([0,1]) ⊂ Ac+ǫ✱ ♣♦✐s ♣❛r❛ t♦❞♦ t ∈ [0,1] t❡♠✲s❡ I(g(t)) ≤ c+ǫ✳

❉❡st❛ ❢♦r♠❛ h∗_{([0,1]) = η(1, g([0,1])) ⊂ η(1, A}

c+ǫ) ⊂ Ac−ǫ✱ ♣❡❧♦ ✷♦✮ ✐t❡♠ ❞♦ ▲❡♠❛ ❞❛

❞❡❢♦r♠❛çã♦✳

❈♦♠♦ h∗_([0,1])⊂A

c−ǫ t❡♠♦s q✉❡ I(h∗(t))≤c−ǫ✱ ♣❛r❛ t♦❞♦ t ∈[0,1]✱ ❡♠ ♣❛rt✐❝✉❧❛r

max

t∈[0,1]I(h

∗_{(t))≤c−ǫ,}

♦ q✉❡ ❝♦♥tr❛❞✐③ (2.2)✳

P♦rt❛♥t♦ I ♣♦ss✉✐ ✉♠ ✈❛❧♦r ❝rít✐❝♦ c≥α.

❈❛♣ít✉❧♦

✸

❊①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❡st✉❞❛r q✉❡stõ❡s s♦❜r❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s q✉❡ ❛♣r❡s❡♥t❛♠ ❢❡♥ô♠❡♥♦ ❞❡ ❝♦♥❝❡♥tr❛çã♦✱ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ (1.2)✱ ♦♥❞❡ ǫ > 0✱ N ≥ 5 ❡ ♦ ♣♦t❡♥❝✐❛❧ V

s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s (V1) ❡(V2)✳ ❱❛♠♦s ❛❞♠✐t✐r q✉❡ f s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s (f1)−(f5)✳

❍á❞✐✜❝✉❧❞❛❞❡s ♥❛ ❛❜♦r❞❛❣❡♠ ❞✐r❡t❛ ❞♦ ♣r♦❜❧❡♠❛ (1.2)✱ ✉♠❛ ❞❡st❛s ❞✐✜❝✉❧❞❛❞❡s é ❞❡

s❡ ✈❡r✐✜❝❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡✳ P♦r ❡ss❛ r❛③ã♦✱ ♣❛r❛ ❣❛r❛♥t✐r♠♦s ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦✱ ✈❛♠♦s ❢❛③❡r ✉♠❛ ♠♦❞✐✜❝❛çã♦ ♥❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❞❡ ❢♦r♠❛ ❛ r❡❝✉♣❡r❛r ❛ ❝♦♥❞✐çã♦ ✭P❙✮✳ P♦r ✜♠ ♠♦str❛r❡♠♦s q✉❡✱ ♣❛r❛ ǫ

s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❛ s♦❧✉çã♦ ❞♦ ♣❛ss♦ ❞❛ ♠♦♥t❛♥❤❛ ❞♦ ♣r♦❜❧❡♠❛ ♠♦❞✐✜❝❛❞♦✱ é ❞❡ ❢❛t♦ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ♦r✐❣✐♥❛❧✳

✸✳✶ ❖ ♣r♦❜❧❡♠❛ ♠♦❞✐✜❝❛❞♦

❈♦♥s✐❞❡r❡V0❞❛❞♦ ♣❡❧❛ ❝♦♥❞✐çã♦(V2)✳ ❙❡❥❛♠k > 2V0❡a >0t❛✐s q✉❡max

f(a)

a ,

f(−a)

−a

≤ V0

k .

❉❡✜♥❛

˜

f(ξ) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−f(−a)

a ξ, s❡ξ < −a

f(ξ), s❡|ξ| ≤a

f(a)

a ξ, s❡ξ > a

❡ g(x, ξ) = χΩ(x)f(ξ) + (1−χΩ(x)) ˜f(ξ)✳

P♦❞❡✲s❡ ✈❡r✐✜❝❛r q✉❡ g s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

(g1) g(x, ξ) =o(|ξ|)✱ q✉❛♥❞♦ξ →0✱

✸✳ ❊①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ✸✷ (g2) ❡①✐st❡♠ ❝♦♥st❛♥t❡s c1, c2 >0 ❡ 1≤p < N_N+4₋₄ s❡N ≥5✱ t❛✐s q✉❡

|g(x, ξ)| ≤c1|ξ|+c2|ξ|p,

♣❛r❛ t♦❞♦ ξ ∈R ❡_x∈RN✱

(g3) ❡①✐st❡ 2< θ < p+ 1 t❛❧ q✉❡

✐✮ 0< θG(x, ξ)≤g(x, ξ)ξ✱∀x∈Ω ❡ξ ∈R✱ ✐✐✮ 0≤2G(x, ξ)≤g(x, ξ)ξ ≤ 1

kV(x)ξ

2_{✱∀ξ∈R❡x /∈Ω✱ ♦♥❞❡G(x, ξ) =}

ξ

0

g(z, t)dt,

(g4) ❛ ❢✉♥çã♦ ξ →

g(x, ξ)

ξ é ♥ã♦✲❞❡❝r❡s❝❡♥t❡ ♣❛r❛ ξ >0❡ ♥ã♦✲❝r❡s❝❡♥t❡ ♣❛r❛ ξ <0✱ ♣❛r❛

t♦❞♦x∈RN_✳

❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛

⎧ ⎨ ⎩

ǫ4_∆2_{u+V(x)u=g(x, u) ❡♠ R}N

u∈H2_(RN_). ✭✸✳✶✮

❖❜s❡r✈❡ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❛❝✐♠❛ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

⎧ ⎨ ⎩

∆2_{v +V(ǫx)v =g(ǫx, v)❡♠ R}N

v ∈H2_(RN_), ✭✸✳✷✮

♦♥❞❡ s✉❛s s♦❧✉çõ❡s uǫ ❡vǫ ❞❡ (3.1)❡ (3.2)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ♣♦r

vǫ(x) = uǫ(ǫx)✳

❖ ❡s♣❛ç♦ ❛❞❡q✉❛❞♦ ♣❛r❛ tr❛t❛r ❞♦ ♣r♦❜❧❡♠❛ (3.2) é ♦ s❡❣✉✐♥t❡ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt

Eǫ =

H2_(RN_{), <·,·>} ǫ

✱ ❝✉❥♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ é ❞❛❞♦ ♣♦r

< u, v >ǫ=

RN

(∆u∆v+V(ǫx)uv)dx,

♦ q✉❛❧ ❞á ♦r✐❣❡♠ ❛ ♥♦r♠❛ u2ǫ =

RN

|∆u|2 +V(ǫx)u2

dx✳

❖ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞♦ ❛ (3.2)❞❛❞♦ ♣♦r

Jǫ(v) =

1 2

RN

(|∆v|2_+V(ǫx)v2_)dx−

RN

G(ǫx, v)dx,

❡stá ❜❡♠ ❞✐✜♥✐❞♦ ❡♠ Eǫ ❡ é ❞❡ ❝❧❛ss❡ C1✳ P❛r❛ ❣❛r❛♥t✐r♠♦s ❛s ❝♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s ❞♦

✸✳ ❊①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ✸✸ ▲❡♠❛ ✷ ❆ss✉♠❛ q✉❡ ❛ ❝♦♥❞✐çã♦ (V1) s❡ ✈❡r✐✜❝❛ ❡ ❝♦♥s✐❞❡r❡ ❛s ♣r♦♣r✐❡❞❛❞❡s (g1)−(g3)✳

❊♥tã♦ ♣❛r❛ ❝❛❞❛ ǫ >0 ❡①✐st❡♠ ❝♦♥st❛♥t❡s ρ✱ β >0 ❡ φ ∈Eǫ ❝♦♠ φǫ > ρ✱ t❛✐s q✉❡

i) Jǫ(v)≥ β✱ s❡♠♣r❡ q✉❡ vǫ=ρ✳

ii) Jǫ(φ)<0✳

❉❡♠♦♥str❛çã♦✿ ❯s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s (g1) ❡ (g2)✱ ♣♦❞❡✲s❡ ✈❡r✐✜❝❛r q✉❡ ♣❛r❛ t♦❞♦

η >0 ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C(η)>0 ❞❡ ♠♦❞♦ q✉❡

|G(x, ξ)| ≤ η

2|ξ|

2_+C(η)|ξ|p+1_{, ∀ξ ∈R.}

❆ss✐♠ ♣❡❧❛s ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈✱ s❡❣✉❡ q✉❡

RN

|G(x, u)|dx ≤ η

2

RN

|u|2_dx+C(η)

RN

|u|p+1

= η 2u

2

L2_(RN₎+C(η)u

p+1

Lp+1_(R)

≤ C1

η

2u

2

ǫ +C2C(η)upǫ+1

≤ Cu2_ǫη

2+C(η)u

p−1

ǫ

.

❊s❝♦❧❤❡♥❞♦ uǫ <

η

2C(η)

_p−11

=γ✱ s❡❣✉❡ q✉❡

RN

|G(x, u)|dx≤Cu2

ǫη.

▲♦❣♦ s❡ρ∈(0, γ)❡ u∈Eǫ é t❛❧ q✉❡uǫ =ρ ♦❜t❡♠♦s q✉❡

Jǫ(u) =

1 2u

2

ǫ −

RN

G(x, u)dx≥ 1

2ρ

2_−Cηρ2_.

❚♦♠❛♥❞♦ η s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❞❡ ♠♦❞♦ q✉❡ ρ2

1 2−Cη

:=β >0✱ ✐st♦ ♣r♦✈❛i)✳ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ (g3) ❡①✐st❡♠ ❝♦♥st❛♥t❡s a1, a2 >0❡ 2< θ < p+ 1 t❛✐s q✉❡

|G(x, ξ)| ≥a1|ξ|θ−a2 ∀x∈RN ❡ξ ∈R.

✸✳ ❊①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ✸✹

Jǫ(tu) =

1 2tu

2

ǫ −

RN

G(x, tu)dx

≤ t

2

2u

2

ǫ −a1|t|θ

RN

|u|θ

dx+

RN a2dx

= t

2

2u

2

ǫ −a1|t|θ

supp(u)

|u|θ_dx+

supp(u)

a2dx

≤ t

2

2u

2

ǫ −a1|t|θ

supp(u)

|u|θ

dx+a2|supp(u)|.

▲♦❣♦ Jǫ(tu)→ −∞✱ q✉❛♥❞♦t →+∞✱ ❥á q✉❡ θ > 2✱ ❡ ❛ss✐♠ t♦♠❛♥❞♦ t0 >0❞❡ ♠♦❞♦

q✉❡ t0uǫ > ρ✱Jǫ(t0u)<0✱ ♦ q✉❡ ♣r♦✈❛ ii)✳

❖ ♣ró①✐♠♦ ▲❡♠❛ ♥♦s ❣❛r❛♥t❡ q✉❡ Jǫ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭P❙✮✳

▲❡♠❛ ✸ ❆ss✉♠❛ s❛t✐s❢❡✐t❛ ❛ ❝♦♥❞✐çã♦ (V1) ❡ ❝♦♥s✐❞❡r❡ ❛s ♣r♦♣r✐❡❞❛❞❡s (g1)−(g3)✳ ❙❡❥❛

{vn} ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ Eǫ t❛❧ q✉❡ {Jǫ(vn)} é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❡

J′

ǫ(vn)→0. ❊♥tã♦ {vn} ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡ ❡♠ Eǫ✳

❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡ {vn} é ❧✐♠✐t❛❞❛ ❡♠ Eǫ✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ {Jǫ(vn)} é ❧✐♠✐t❛❞❛✱

❡①✐st❡ M > 0 t❛❧ q✉❡ |Jǫ(vn)| ≤ M✱ ♣❛r❛ t♦❞♦ n ∈ N✱ ❝♦♠ ✐ss♦ ❡ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ (g3)

t❡♠♦s

M +1

θvnǫon(1) ≥ Jǫ(vn)−

1

θJ

′

ǫ(vn)vn

= 1 2− 1 θ

un2ǫ +

1

θ

RN

(g(x, un)un−θG(x, un))dx

≥ 1 2− 1 θ

vn 2ǫ +

1

θ

Ωc

(g(ǫx, vn)vn−θG(ǫx, vn))dx

≥ 1 2− 1 θ

un2ǫ +

(2−θ)

θ

Ωc

G(x, un)dx

≥ 1 2− 1 θ

vn 2ǫ +

(2−θ) 2kθ

Ωc

V(ǫx)v2ndx

≥

θ−2 2θ RN

|∆vn|2+

1− 1

k

V(ǫx)vn2

dx

≥

θ−2

2θ 1−

1

k

vn2ǫ,

♦ q✉❡ ✐♠♣❧✐❝❛{vn}❧✐♠✐t❛❞❛ ❡♠Eǫ✳ ❈♦♠♦ Eǫ é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡ ♣♦rt❛♥t♦ r❡✢❡①✐✈♦✱

✸✳ ❊①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ✸✺ é ♥❛ ✈❡r❞❛❞❡ ❢♦rt❡✳ ❈♦♠ ❡❢❡✐t♦✱ ♠♦str❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ❞❛❞♦δ >0✱ ❡①✐st❡ ✉♠R >0 ❞❡ ♠♦❞♦ q✉❡

lim sup n→∞ Bc R(0)

|∆vn|2+V(ǫx)vn2

dx < δ. ✭✸✳✸✮

❆ss✉♠❛ q✉❡ R é ❡s❝♦❧❤✐❞♦ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ Ω ⊂BR

2(0)✳ ❙❡❥❛ ηR ∈ C

∞_(RN_{) t❛❧ q✉❡}

ηR = 0 ❡♠ BR

2(0)✱ ηR = 1 ❡♠

RN _\BR(0)✱ _{0 ≤ηR ≤} 1✱ _{|∇ηR| ≤} C

R ❡ |∆ηR| ≤ C

R2✳ ▲♦❣♦✱

❝♦♠♦{vn} ❧✐♠✐t❛❞❛ ❡ J′ǫ(vn)→0 t❡♠♦s

< J′ǫ(vn), ηRvn>=on(1),

❡ ❛ss✐♠

RN

∆vn∆(ηRvn) +V(ǫx)ηRv2n

dx−

RN

g(ǫx, vn)ηRvndx=on(1),

⇒

RN

|∆vn|2+V(ǫx)vn2

ηRdx+

RN

(∆vn∆ηRvn+ 2∇ηR∇vn∆vn)dx =

RN

g(ǫx, vn)ηRvndx

+ on(1).

❯s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ (g3) t❡♠♦s q✉❡

RN

|∆vn|2+V(ǫx)vn2

ηRdx ≤ −

RN

(∆vn∆ηRvn+ 2∇ηR∇vn∆vn)dx

+1

k

Ωc

V(ǫx)vnηRdx+on(1),

❡ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱

1− 1

k Bc R(0)

|∆vn|2+V(ǫx)vn2

dx ≤ C

R2vnL2(RN)∆vnL2(RN)+on(1)

+2C

R ∇vnL2(RN)∆vnL2(RN).

❚♦♠❛♥❞♦ ♦ lim sup ♥❛ ú❧t✐♠❛ ❡①♣r❡ssã♦ s❡❣✉❡ (3.3)✳

❆❣♦r❛✱ ♥♦t❡ q✉❡ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s (g1) ❡ (g2)✱ ♣♦❞❡✲s❡ ♠♦str❛r q✉❡ ❡①✐st❡ C′ >0 ❞❡

♠♦❞♦ q✉❡

Bc R(0)

|g(ǫx, vn)vn|dx≤C′

Bc R(0)

✸✳ ❊①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ✸✻ ◆♦t❡ ❛✐♥❞❛ q✉❡✱ ♣❡❧❛s ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈ ❡ ♣♦r (3.3) t❡♠♦s q✉❡ ♣❛r❛ n s✉✜❝✐❡♥t❡♠❡♥t❡

❣r❛♥❞❡

Bc R(0)

|g(ǫx, vn)vn|dx≤C˜

vn2ǫ +vnpǫ+1

<C˜δ+δp+12

.

P❡❧❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞❡ x→g(ǫx, v)v✱ t❡♠♦s q✉❡ ❞❛❞♦ δ >0❡①✐st❡ ✉♠ Rδ >0t❛❧ q✉❡

Bc R(0)

g(ǫx, v)vdx < δ

4.

❚♦♠❡ δ1 ❡♠ (3.3) t❛❧ q✉❡ C˜

δ1+δ

p+1 2

1

< δ₄.

❊♥tã♦ ❡①✐st❡ R >0 ❡ n0 ∈N✱ ✭s✳♣✳❣✳ R > Rδ✮ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ n≥n0

Bc R(0)

g(ǫx, vn)vndx <C˜

δ1+δ

p+1 2 1 < δ 4.

▲♦❣♦✱ s❡ n≥n0 t❡♠♦s q✉❡

Bc R(0)

g(ǫx, vn)vndx−

Bc R(0)

g(ǫx, v)vdx <

δ

2.

❖❜s❡r✈❡ q✉❡ vn ⇀ v ❡♠ Eǫ ✐♠♣❧✐❝❛ q✉❡ vn → v ❡♠ Lploc(RN)✱ 1 ≤ p < 2 N N−4 ♣❡❧❛

✐♠❡rsõ❡s ❝♦♠♣❛❝t❛s✳❚❡♠♦s ❛✐♥❞❛ q✉❡ vn → v q✳t✳♣✳ ❡♠ RN✳❈♦♠ ✐ss♦✱ ✉s❛♥❞♦ ♦

❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ●❡♥❡r❛❧✐③❛❞♦ ✭✈❡r ❬✽❪✮✱ ♣♦❞❡♠♦s ♠♦str❛r q✉❡

BR(0)

g(ǫx, vn)vndx →

BR(0)

g(ǫx, v)vdx,

❡ ❛ss✐♠ ♣❛r❛ n≥n0 ♦❜t❡♠♦s q✉❡

BR(0)

g(ǫx, vn)vndx−

BR(0)

g(ǫx, v)vdx < δ 2. P♦rt❛♥t♦ RN

g(ǫx, vn)vndx−

RN

g(ǫx, v)vdx < δ.

P♦r ✜♠✱ ♥♦t❡ q✉❡ v é s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (3.2)✳ ❈♦♠ ❡❢❡✐t♦✱ ♣❛r❛ t♦❞❛ φ∈C∞

0 (RN)✱ ♣❡❧❛

❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛ t❡♠♦s q✉❡

RN

(∆vn∆φ+V(ǫx)vnφ)dx →

RN

✸✳ ❊①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ✸✼ P❡❧❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ❡♠ Lp_loc(RN₎ ❡ ✉t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐✲ ♥❛❞❛ ●❡♥❡r❛❧✐③❛❞♦ s❡❣✉❡ q✉❡

supp(φ)

g(ǫx, vn)vndx→

supp(φ)

g(ǫx, v)vdx.

❆ss✐♠✱

0 = lim

n→∞J ′

ǫ(vn)v =

RN

|∆v|2_+V(ǫx)v2

dx−

RN

g(ǫx, v)vdx =J′ǫ(v)v,

⇒ v2

ǫ =

RN

g(ǫx, v)vdx.

❖❜s❡r✈❡ ❛✐♥❞❛ q✉❡

J′ǫ(vn)vn =vn2ǫ −

RN

g(ǫx, vn)vndx=on(1),

⇒ lim

n→∞vn

2 _{= lim}

n→∞

RN

g(ǫx, vn)vndx=

RN

g(ǫx, v)vdx=v2_ǫ.

P♦rt❛♥t♦ vnǫ → vǫ ❡♠ Eǫ ❡ ❝♦♠♦ ❥á s❛❜✐❛♠♦s q✉❡ ❝♦♥✈❡r❣✐❛ ❢r❛❝♦ ❝♦♥❝❧✉í♠♦s q✉❡

vn→v ❡♠ Eǫ✳

P❡❧♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ♣❛r❛ ❝❛❞❛ ǫ > 0 ❡①✐st❡ vǫ ∈ Eǫ s♦❧✉çã♦ ❢r❛❝❛

♥ã♦✲tr✐✈✐❛❧ ❞❡(3.2)t❛❧ q✉❡ Jǫ(vǫ) =cǫ ♦♥❞❡

cǫ = inf g∈Γǫ

max

t∈[0,1]Jǫ(g(t)),

❡Γǫ ={g ∈C([0,1], Eǫ);g(0) = 0 ❡Jǫ(g(1))<0}✳

P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ (g4) ♣♦❞❡♠♦s ❛✐♥❞❛ ❝❛r❛❝t❡r✐③❛r ♦ ♥í✈❡❧ ♠✐♥✐♠❛① cǫ ❝♦♠♦

cǫ = inf u∈Eǫ\{0}

max

t≥0 Jǫ(tu) = infNǫ Jǫ,

♦♥❞❡Nǫ é ❞❡✜♥✐❞❛ ♣♦r

Nǫ ={u∈Eǫ\ {0};Jǫ′(u)u= 0}.

❆❣♦r❛✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ s❡q✉ê♥❝✐❛ {ǫn} t❛❧ q✉❡ǫn→0q✉❛♥❞♦ n→ ∞✳ ❆✜r♠❛✲

♠♦s q✉❡ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ q✉❡ ❝♦♥t✐♥✉❛r❡♠♦s ❛ ❞❡♥♦t❛r ♣♦r {ǫn}✱ t❛❧ q✉❡vn:=vǫn