(LUẬN văn THẠC sĩ) nghiệm yếu của hệ phương trình p laplace phân thứ trên miền bị chặn với số mũ tới hạn

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ◆❣✉②➵♥ ❚❤à ❇➻♥❤ ◆●❍■➏▼ ❨➌❯ ❈Õ❆ ❍➏ P❍×❒◆● ❚❘➐◆❍ P✲▲❆P▲❆❈❊ P❍❹◆ ❚❍Ù ❚❘➊◆ ▼■➋◆ ❇➚ ❈❍➄◆ ❱❰■ ❙➮ ▼Ô ❚❰■ ❍❸◆✳ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵ download by : skknchat@gmail.com ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ◆❣✉②➵♥ ❚❤à ❇➻♥❤ ◆●❍■➏▼ ❨➌❯ ❈Õ❆ ❍➏ P❍×❒◆● ❚❘➐◆❍ P✲▲❆P▲❆❈❊ P❍❹◆ ❚❍Ù ❚❘➊◆ ▼■➋◆ ❇➚ ❈❍➄◆ ❱❰■ ❙➮ ▼Ô ❚❰■ ❍❸◆✳ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ ❚➼❝❤ ▼➣ số ữớ ữợ ❞➝♥ ❦❤♦❛ ❤å❝✿ ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ❚❤➻♥ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✷✵ download by : skknchat@gmail.com ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➟♣ ợ t ỗ t sỷ t ỗ t ❈→❝ t❤æ♥❣ t✐♥✱ t➔✐ ❧✐➺✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ữủ ró ỗ ố t ✷✵✷✵ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ◆❣✉②➵♥ ❚❤à ❇➻♥❤ ❳→❝ ♥❤➟♥ ổ ữớ ữợ ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ❚❤➻♥ download by : skknchat@gmail.com ✐✐ ▲í✐ ỡ ữủ t ữợ sỹ ữợ t t ữợ ỳ t ú ù tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ▼ët ❧➛♥ ♥ú❛ tæ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ♥❤➜t ✤➳♥ t❤➛②✦ ỗ tớ tổ ỷ ỡ ❈❤õ ◆❤✐➺♠ ❦❤♦❛ ❚♦→♥ ✈➔ ❝→❝ t❤➛② ❝æ tr♦♥❣ tê ❇ë ♠æ♥ ●✐↔✐ t➼❝❤ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ✤÷đ❝ ❧➔♠ ❧✉➟♥ ✈➠♥✱ ✤➣ q✉❛♥ t➙♠ ✈➔ ✤ỉ♥ ✤è❝ tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ▲✉➟♥ ✈➠♥ ❧➔ s↔♥ ♣❤➞♠ ❝õ❛ ✤➲ t➔✐ ✏◆❣❤✐➺♠ ②➳✉ ❝õ❛ ♠ët sè ợ ữỡ tr ữỡ tr r ự t♦→♥ tû ♣✲▲❛♣❧❛❝❡ t❤ù ✈➔ t♦→♥ tû ❇❡ss❡❧✑ ✈ỵ✐ ♠➣ sè ❇✷✵✷✵✲❚◆❆✲✵✻✳ ❚ỉ✐ ①✐♥ ❝↔♠ ì♥ sü ❤é trđ ✈➲ ❦✐♥❤ ♣❤➼ tø ✤➲ t➔✐ ❣â♣ ♣❤➛♥ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✷✵ ◆❣✉②➵♥ ❚❤à ❇➻♥❤ download by : skknchat@gmail.com ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ▲í✐ ❝↔♠ ì♥ ▼ð ✤➛✉ ✶ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ữỡ tr ự t tỷ tự ợ ữủ t ợ t t♦→♥ ✈➔ ♠ët sè ❦➳t q✉↔ ♣❤ư trđ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ự t tỷ tự ợ ữủ t✉②➳♥ ✤ê✐ ❞➜✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✐ ✐✐ ✶ ✸ ✸ ✷✵ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù ❝❤ù❛ sè ụ tợ ữủ ỗ ợ t❤✐➺✉ ✈➲ ❜➔✐ t♦→♥ ✈➔ ♠ët sè ❦➳t q✉↔ ♣❤ö trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù ự số ụ tợ ữủ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✷ ✺✶ ✺✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ download by : skknchat@gmail.com ỵ ❧✉➟♥ ✈➠♥ ❍✐➺♥ ♥❛②✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ ❞➔♥❤ sü q✉❛♥ t➙♠ ✈➔♦ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t♦→♥ tû ❦❤æ♥❣ ữỡ t ỗ t tỷ ♣❤➙♥ t❤ù (−∆)s ✮ tr♦♥❣ ❝↔ ♥❣❤✐➯♥ ❝ù✉ t♦→♥ ❤å❝ t❤✉➛♥ tó② ✈➔ t♦→♥ ❤å❝ ù♥❣ ❞ư♥❣✳ ◆❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♣✲❧❛♣❧❛❝❡ ❦❤→ q✉❛♥ trå♥❣ tr♦♥❣ ♥❤✐➲✉ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ♥❤÷ ❝→❝ ♥❣➔♥❤ ✤✐➺♥ tø tr÷í♥❣✱ t❤✐➯♥ ✈➠♥ ❤å❝✱ ❝ì ❝❤➜t ❧ä♥❣✱✳✳✳ ❚r♦♥❣ ❜è✐ ❝↔♥❤ ✤à❛ ♣❤÷ì♥❣ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲❧❛♣❧❛❝❡ ♠➔ ❤➔♠ ♣❤✐ t✉②➳♥ ❝â ✤ë t➠♥❣ tỵ✐ ❤↕♥ t❤æ♥❣ q✉❛ ✤❛ t↕♣ ◆❛❤❛r✐✳ ▼ët ♠ð rë♥❣ ❝õ❛ (−∆)s ❧➔ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù (−∆)sp u (x) = lim →0 Rn \B (0) (−∆)sp ✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐✿ |u (x) − u (y)|p−2 (u (x) − u (y)) dy, x ∈ Rn n+ps |x − y| t♦→♥ tû ♥➔② ✈➔ ♠ð rë♥❣ ❝õ❛ ♥â ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ♠ët sè t→❝ ❣✐↔ tr➯♥ t❤➳ ❣✐ỵ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙②✳ ❈→❝ ❜➔✐ t♦→♥ ❞↕♥❣ r ổ t ởt số tữủ t ỵ r ❬✶✸❪ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤✿ L  ∂ u p0 E ρ − + ∂t h 2L  ∂u ∂ 2u  dx =0 ∂x ∂x2 ♠ët ♠ð rë♥❣ ❝õ❛ ♣❤÷ì♥❣ tr✉②➲♥ sâ♥❣ ❉✬ ❆❧❛♠❜❡rt✱ ♠æ t↔ sü t❤❛② ✤ê✐ ✤ë ❞➔✐ ❝õ❛ ❞➙② tr♦♥❣ q✉→ tr➻♥❤ r✉♥❣ ✤ë♥❣✱ tr♦♥❣ ✤â ρ, p0 , h, E, L tr➻♥❤ tr➯♥ ❝❤ù❛ ✤↕✐ ❧÷đ♥❣ ❦❤ỉ♥❣ ✤à❛ ♣❤÷ì♥❣ L tr✉♥❣ ❜➻♥❤ (1.1) ∂u ∂u tr➯♥ ∂x dx ❝õ❛ ✤ë♥❣ ♥➠♥❣ ∂x p0 h + E 2L L ❧➔ ❝→❝ ❤➡♥❣ sè✳ P❤÷ì♥❣ ∂u ∂x dx✱ ♣❤ư t❤✉ë❝ ✈➔♦ [0, L]✳ ❍ì♥ ♥ú❛ ❝→❝ ❜➔✐ t♦→♥ ❞↕♥❣ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ♥❤✐➲✉ ♠ỉ ❤➻♥❤ t ỵ s tr õ u ữủ ♠ỉ t↔ ♥❤÷ ♠ët q✉→ tr➻♥❤✳ ❈â ♥❤✐➲✉ ❜➔✐ t♦→♥ r ữủ ự ợ t tû ❦❤→❝ ♥❤❛✉✳ ❚❤í✐ ❣✐❛♥ ❣➛♥ ✤➙②✱ ❜➔✐ t♦→♥ ❑✐r❝❤❤♦❢❢ ✤➣ ✤÷đ❝ download by : skknchat@gmail.com ✷ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ t♦→♥ tû ▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù✱ ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù tr♦♥❣ ❬✷❪✱ ❬✸❪✱ ❬✽❪✱ tr➯♥ ♠✐➲♥ ❜à ❝❤➦♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❉✐r✐❝❤❧❡t ❤❛② ◆❡✉♠❛♥♥✳ ❱➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ❝ơ♥❣ ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ ❜ð✐ ♥❤✐➲✉ t→❝ ❣✐↔ ❜➡♥❣ ♥❤✐➲✉ ❝→❝❤ ❦❤→❝ ♥❤❛✉ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ♣❤➙♥✱ ✤❛ t↕♣ ◆❡❤❛r✐ ❬✶❪✱ ❬✶✶❪✱ ❬✶✸❪✱ ❬✶✹❪✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ t✐➳♣ tử ữợ ự tr ú tổ t ✏◆❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù tr➯♥ ♠✐➲♥ ❜à ❝❤➦♥ ✈ỵ✐ sè ♠ơ tỵ✐ ❤↕♥✑ ❧➔♠ ❧✉➟♥ ✈➠♥ ❝❛♦ ❤å❝✳ ✷✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ▲✉➟♥ ✈➠♥ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❝ì ❜↔♥✳ ✸✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲❧❛♣❧❛❝❡ ♣❤➙♥ t❤ù tr➯♥ ♠✐➲♥ ❜à ❝❤➦♥ ✈ỵ✐ sè ụ tợ ữủ t ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ t➻♠ ❤✐➸✉ ✈➲ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲❧❛♣❧❛❝❡ ♣❤➙♥ t❤ù ợ số ụ tợ ữủ ỗ ỗ ữỡ ✲ ❈❤÷ì♥❣ ✶✳ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t tỷ tự ợ ữủ t ✤è✐ ❞➜✉ ✳ ✲ ❈❤÷ì♥❣ ✷✳ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù ❝❤ù❛ sè ụ tợ ữủ ỗ download by : skknchat@gmail.com ✸ ❈❤÷ì♥❣ ✶ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ự t tỷ tự ợ ữủ t✉②➳♥ ✤ê✐ ❞➜✉ ✶✳✶ ●✐ỵ✐ t❤✐➺✉ ✈➲ ❜➔✐ t♦→♥ ✈➔ ♠ët sè ❦➳t q✉↔ ♣❤ư trđ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tổ ự sỹ tỗ t ữỡ tr s (P,à ) tr õ ()sp ()sp u (x)   p  |u(x)−u(y)|  (−∆)sp u (x) M  n+ps dxdy  |x−y|  2n     2α   |u|α−2 u|v|β = λf (x) |u|q−2 u + α+β   tr♦♥❣ , p |v(x)−v(y)| s M (−∆)  n+ps dxdy p v (x)  |x−y|  2n     2β   = µg (x) |v|q−2 v + α+β |u|α |v|β−2 v     u = v = tr♦♥❣ Rn \Ω tr♦♥❣ f, g Ω ❧➔ ❝→❝ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐✿ = lim →0 Rn \B (0) |u (x) − u (y)|p−2 (u (x) − u (y)) dy, x ∈ Rn , n+ps |x − y| M (t) = a + bt, a, b > o, p ≥ 2, < q < p✱ 2p < r ≤ p∗s , ps < n < 2ps s ∈ (0, 1) , λ, µ ❧➔ ❝→❝ sè t❤ü❝✱ Ω ∈ Rn ❧➔ ♠✐➲♥ ❣✐ỵ✐ ❤↕♥ ✈ỵ✐ ❣✐ỵ✐ ❤↕♥ trì♥ ✈➔ tr♦♥❣ ✤â ✈ỵ✐ Ω t❤ä❛ ♠➣♥ ❝→❝ ❣✐↔ ✤à♥❤ s❛✉✿ download by : skknchat@gmail.com ✹ α+β (f1 ) f, g ∈ Lγ (Ω) ✈ỵ✐ γ = α+β−q ; (f2 ) f + (x) = max {f (x) , 0} = tr♦♥❣ Ω ✈➔ g + (x) = max {g (x) , 0} = Ω ✭f ✈➔ ❣ ❝â t❤➸ ✤ê✐ ❞➜✉ tr➯♥ Ω✮✳ tr♦♥❣ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ ♠✐♥❤ sü tỗ t ổ ố ợ ữỡ tr r t ợ t tỷ t❤ù ✈➔ ✤↕✐ ❧÷đ♥❣ ♣❤✐ t✉②➳♥ ✤ê✐ ❞➜✉ ❜➡♥❣ ❝→❝❤ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t❤✉ë❝ t➼♥❤ ❝õ❛ ✤❛ t↕♣ ◆❡❤❛r✐ ✤è✐ ✈ỵ✐ t❤❛♠ sè λ X = {u|u : Rn → R ❧➔ ❚r♦♥❣ ✤â ✈➔ µ✳ < p < ∞, ❈❤♦ ✤♦ ✤÷đ❝, u|Ω Q = R2n \ (CΩ × CΩ) t❛ ①➨t ❦❤æ♥❣ ❣✐❛♥ u (x) − u (y) ∈ Lp (Ω) , |x − y| CΩ = Rn \Ω✳ ✈➔ n p +s ∈ Lp (Q)} ❑❤✐ ✤â✱ ❳ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ✈ỵ✐ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐✿  p1  |u(x) − u(y)|p  dxdy  n+ps |x − y|  ||u||X = ||u||Lp (Ω) + ✭✶✳✶✮ Q ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ỵ X0 = {h X : h = tr♦♥❣ Rn \ Ω} ❑❤æ♥❣ ❣✐❛♥ X0 ởt ổ ợ ữủ ✤à♥❤ ❜ð✐  p1  ||u||X0 |u(x) − u(y)|p  dxdy  n+ps |x − y|  = ✭✶✳✷✮ Q ữ ỵ r tr ữỡ tr t õ t ữủ rở (P,à ) R2n ✱ tø u=0 tr♦♥❣ Rn \Ω✳ ❝❤ó♥❣ t❛ ①➨t ❦❤ỉ♥❣ ❣✐❛♥ t➼❝❤ (u, v) = ❍➔♠ ♥➠♥❣ ❧÷đ♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❇➙② ❣✐í ✤➸ t➻♠ r❛ ❝→❝ ♥❣❤✐➺♠ t E := X0 ì X0 (P,à ) 1 Jλ,µ (u, v) = M u pX0 + M v p p  − λ f (x)|u|q dx + µ q Ω p X0 u + v ✈ỵ✐ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐ p p X0 ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿ p X0  g(x)|v|q dx − Ω α+β |u|α |v|β dx, Ω t tr♦♥❣ ✤â M (t) = M (s) ds ❧➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ▼✳ download by : skknchat@gmail.com ✺ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈➦♣ ❤➔♠ (u, v) ∈ E ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ (Pλ,µ ) ♥➳✉ |u(x) − u(y)|p−2 (u(x) − u(y))(ϕ(x) − ϕ(y)) dxdy |x − y|n+ps M (||u||pX0 ) R2n +M |v(x) − v(y)|p−2 (v(x) − v(y))(ψ(x) − ψ(y)) dxdy |x − y|n+ps p X0 v R2n f (x) |u|q−2 uϕ (x) dx+µ =λ g (x) |v|q−2 vψ (x) dx Ω + Ω 2α α+β |u|α−2 u|v|β ϕ (x) dx + 2β α+β Ω ✈ỵ✐ ♠å✐ |u|α |v|β−2 vψ (x) dx Ω (ϕ, ψ) ∈ E ✣❛ t↕♣ ◆❡❤❛r✐ N,à t ợ t (P,à ) ữủ ♥❣❤➽❛ ❜ð✐✿ Nλ,µ = (u, v) ∈ E\ {0} : Jλ,µ (u, v) , (u, v) = , tr♦♥❣ ✤â , ❧➔ t➼❝❤ ✤è✐ ♥❣➝✉ ❣✐ú❛ E ✈➔ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉✳ ✣➦t (λ, µ) ∈ R \ {(0, 0)} : (|λ| f SΓ = γ) p p−q + (|µ| g γ ) p p−q p−q p − (α + β − q) p p p (α + β − q) − pq α+β−q S |Ω| α+β λ p−q + µ p−q (uk , vk ) < q (α + β − p) ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ Jλ,µ (uk , vk ) E∗ →0 ❦❤✐ p−q p k → ∞✳ q−p p q ✭✷✳✸✸✮ p−q ❈è ✤à♥❤ k ∈ N✳ ⑩♣ zk = (uk , vk ) t❛ t➻♠ ✤÷đ❝ ❤➔♠ sè ξk : B (0, k ) → R+ ✱ ❝❤♦ k > s❛♦ ❝❤♦ ξk (h) (zk − h) ∈ Nλ,µ ✳ ▲➜② < ρ < k ✱ ❝❤♦ w ∈ E ✈ỵ✐ w = ✈➔ ρw ∗ ∗ ∗ ✤➦t h = w ✳ ❚❛ ✤➦t hρ = ξk (h ) (zk − h )✳ ❑❤✐ ✤â hρ ∈ Nλ,µ ✈➔ tø ✭✷✳✸✵✮✱ t❛ ❞ư♥❣ ❇ê ợ õ J,à (h ) J,à (zk ) ≥ − hp − zk k ❚❤❡♦ ỵ tr tr t ữủ J,à (zk ) , hp − zk + o ( hp − zk ) ≥ − hp − zk k ❑❤✐ ✤â✱ t❛ ❝â Jλ,µ (zk ) , −h∗ + (ξk (h∗ ) − 1) Jλ,µ (zk ) , zk − h∗ ≥− hp − zk + o ( hp − zk ) k download by : skknchat@gmail.com ữ ỵ r k (h ) (zk h ) ∈ Nλ,µ ✱ w w −ρ Jλ,µ (zk ) , ữ + (k (h ) 1) J,à (zk ) − Jλ,µ (hρ ) , zk − h∗ ≥− hp − zk + o ( hp − zk ) k ❉♦ ✤â✱ t❛ ♥❤➟♥ ✤÷đ❝ o ( hp − zk ) o ( hp − zk ) hp − zk + + kρ ρ ρ ∗ (ξk (h ) − 1) Jλ,µ (zk ) − Jλ,µ (hρ ) , zk − h∗ + ρ w w Jλ,µ (zk ) , ≤ ✭✷✳✸✹✮ ❚ø hp − zk ≤ ρ |ξk (h∗ )| + |ξk (h∗ ) − 1| zk ∗ ✈➔ lim |ξk (hρ )−1| ≤ ξk (0) ✈ỵ✐ ρ→0 ✭✷✳✸✸✮✱ t❛ ❝â t❤➸ ❝❤å♥ C>0 k ∈N ρ→0 ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ Jλ,µ (zk ) , ✣✐➲✉ ✤â ✤ó♥❣ ♥➳✉ ❦❤✐ ❝❤♦ w w supk∈N ξk (0) ≤ E∗ ρ tr♦♥❣ ✭✷✳✸✹✮✱ ❦❤✐ ✤â t❤❡♦ s❛♦ ❝❤♦ C + ξk (0) k < ∞✳ ❇ð✐ ✭✷✳✷✾✮✱✭✷✳✸✸✮ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍♦❧❞❡r✱ t❛ ❝â ξk (0) , h C1 h ≤ (p − q) (uk , vk ) p − (α + β − q) |uk |α |vk |β dx Ω ✈ỵ✐ ♠é✐ C1 > ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ (p − q) (uk , vk ) p |uk |α |vk |β dx ≥ C2 − (α + β − q) Ω ✈ỵ✐ ♠é✐ C2 > ✈➔ ❦ ✤õ ợ ữủ sỷ tỗ t (p − q)||(uk , vk )||p − 2(α + β − q) {(uk , vk )}k∈N ✈ỵ✐ |uk |α |vk |β dx = ok (1) ✭✷✳✸✺✮ |uk |α |vk |β dx + ok (1), ✭✷✳✸✻✮ Ω ❚❤❡♦ ✭✷✳✸✺✮ ✈➔ {(uk , vk )}k∈N , (uk , vk ) p = t❛ ❝â (α + β − q) p−q Ω download by : skknchat@gmail.com ✹✹ (uk , vk ) p = (α + β − q) α+β−p (λ|uk |q + µ|vk |q ) dx + ok (1) ✭✷✳✸✼✮ Ω ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✱ t❛ ❝â |uk |α |vk |β dx ≤ S − α+β p α+β (uk , vk ) Ω ❉♦ ✤â tø ❇➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈➔ ✭✷✳✸✻✮✱ t❛ ♥❤➟♥ ✤÷đ❝ |(uk , vk )| ≥ ( α+β p−q S p ) α+β−p + ok (1) 2(α + β − q) ✭✷✳✸✽✮ ❍ì♥ ♥ú❛✱ tø ✭✷✳✸✼✮ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍♦❧❞❡r✱ t❛ ❝â (uk , vk ) p q p p α + β − q α+β−q |Ω| α+β S − p λ p−q + µ p−q ≤ α+β−p p−q p (uk , vk ) q + ok (1) ❉♦ ✤â (uk , vk ) ≤ α + β − q − pq α+β−q S |Ω| α+β α+β−p p−q λ p p−q +µ p p−q p + ok (1) ✭✷✳✸✾✮ ❚ø ✭✷✳✸✽✮ ✈➔ ợ t t ữủ p p λ p−q + µ p−q ≥ p−q (α + β − q) p ▼➙✉ t❤✉➝♥ p α+β−p α + β − q α+β−q |Ω| α+β α+β−p p − p−q α+β q S α+β−p + p−q = Λ1 p < λ p−q + µ p−q < Λ1 ✳ ❈❤♦ ♥➯♥ Jλ,µ (uk , vk ) , w −1 w ≤ C k ◆❤÷ ✈➟② ✭✐✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✶✻✱ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ✭✐✐✮✳ ▼➺♥❤ ✤➲ ✷✳✶✳✶✽✳ ❈❤♦ Λ1 ♥❤÷ tr♦♥❣ ✭✷✳✼✮✳ ●✐↔ sû < λ p p−q + pq < p + õ tỗ t (u1, v1) N,à ợ t t s ✭✐✮ Jλ,µ (u1, v1) = cλ,µ = c+λ,µ < 0, ✭✐✐✮ (u1, v1) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮✳ ❈❤ù♥❣ tỗ t ỹ t {(uk , vk )} ⊂ Nλ,µ t❤ä❛ ♠➣♥ lim Jλ,µ (uk , vk ) = cλ,µ ≤ c+ λ,µ < 0, Jλ,µ (uk , vk ) = ok (1) k→∞ tr♦♥❣ download by : skknchat@gmail.com E ∗ ✹✺ (u1 , v1 ) ∈ E s❛♦ ❝❤♦ ❞➣② ❝♦♥ uk uk → u1 , vk → v1 ♠↕♥❤ tr♦♥❣ Lγ (Ω)✱ ✈ỵ✐ ≤ r < p∗ ✳ u1 , vk õ tỗ t (|uk |q + à|vk |q ) dx → Ω tr♦♥❣ X0 , ❑❤✐ ✤â (λ|u1 |q + µ|v1 |q ) dx ❦❤✐ k → ∞ Ω ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✾ ❚ø v1 (u1 , v1 ) (uk , vk ) ∈ Nλ,µ ✱ Jλ,µ (uk , vk ) = ❧➔ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ✭✷✳✶✮✳ t❛ ❝â α+β−p (uk , vk ) p (α + β) p − α+β−q q (α + β) (λ|uk |q + µ|vk |q ) dx Ω ≥− α+β−q q (α + β) (λ|uk |q + µ|vk |q ) dx Ω cλ,µ < ợ t õ (|u1 |q + à|v1 |q ) dx ≥ − q (α + β) cλ,µ > α+β−q Ω ❈❤♦ ♥➯♥ (u1 , v1 ) ∈ N,à ổ t tữớ t t❛ ❝❤ù♥❣ ♠✐♥❤ (uk , vk ) → (u1 , v1 ) ♠↕♥❤ tr♦♥❣ ❊ ✈➔ Jλ,µ (u1 , v1 ) = c+ λ,µ ✳ ❚ø (u1 , v1 ) ∈ Nλ,µ ✱ →♣ ❞ư♥❣ ❜ê ✤➲ ❋❛t♦✉✱ t❛ ♥❤➟♥ ữủ c,à J,à (u1 , v1 ) +p +q = (u1 , v1 ) p − (λ|u1 |q + µ|v1 |q ) dx p (α + β) q (α + β) Ω   α+β−p α+β−q ≤ lim inf  (uk , vk ) p − (λ|uk |q + µ|vk |q ) dx k→∞ p (α + β) q (α + β) Ω = lim inf Jλ,µ (uk , vk ) = cλ,µ k→∞ ❱➟② Jλ,µ (u1 , v1 ) = cλ,µ ✈➔ (uk − u1 , vk − v1 ) (uk , vk ) p p → (u1 , v1 ) = (uk , vk ) p p ✳ ❚❛ ❝ô♥❣ ❝â − (u1 , v1 ) p + ok (1) + (uk , vk ) → (u1 , v1 ) ♠↕♥❤ tr♦♥❣ ❊✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ (u1 , v1 ) ∈ Nλ,µ ✱ ✈➔ − cλ,µ = c+ λ,µ ✳ ●✐↔ sû ❞♦ ♠➙✉ t❤✉➝♥ (u1 , v1 ) ∈ Nλ,µ ✳ ❚❤❡♦ ❇ê ✤➲ tỗ t t t2 > t1 > s❛♦ ❝❤♦✿ ❈❤♦ ♥➯♥ + − (t1 u1 , t1 v1 ) ∈ Nλ,µ , (t2 u1 , t2 v1 ) ∈ Nλ,µ download by : skknchat@gmail.com ✹✻ ✣➦❝ t t õ t1 < t2 = ữ ỵ r➡♥❣ d d2 Jλ,µ (t1 u1 , t1 v1 ) = 0; Jλ,µ (t1 u1 , t1 v1 ) > 0, dt dt tỗ t t (t1 , 1] s❛♦ ❝❤♦ Jλ,µ (t1 u1 , t1 v1 ) < Jλ,µ (t∗ u1 , t∗ v1 )✳ ❑❤✐ ✤â cλ,µ ≤ Jλ,µ (t1 u1 , t1 v1 ) < Jλ,µ (t∗ u1 , t∗ v1 ) , ≤ J,à (u1 , v1 ) = c,à ổ ỵ ✤â + (u1 , v1 ) ∈ Nλ,µ ❚✐➳♣ t t sỹ tỗ t ỹ t ✤è✐ ✈ỵ✐ S := infu∈X0 \{0} R2n |u(x)−u(y)| n+ps |x−y| |u (x)| np n−ps − ✳ Jλ,µ |Nλ,µ ❈❤♦ p dxdy n−ps n dx Ω < p < , s (0, 1) , n > ps tỗ t↕✐ ❤➔♠ ❝ü❝ t✐➸✉ ✤è✐ ✈ỵ✐ n ❙ ✈➔ ❝❤♦ ỹ t tỗ t x0 ∈ R ✈➔ ❤➔♠ u : R → R ❦❤æ♥❣ ➙♠✱ ❣✐↔♠ s❛♦ ❝❤♦ U (x) = u (|x − x0 |) U = U (r) ◆❤➙♥ ❯ ✈ỵ✐ ♠ët sè ❞÷ì♥❣ ❚ø ❬✹❪✱ t❛ ✤➣ ❜✐➳t ♥➳✉ ❝➛♥✱ t❛ ❝â t❤➸ ❣✐↔ sû ∗ (−∆)sp U = U ps −1 ❈❤♦ >0 tr♦♥❣ Rn ✭✷✳✹✵✮ ❜➜t ❦➻✱ t❛ ❝â ❤➔♠ sè U (x) = n−ps p |x| U ❝ơ♥❣ ❧➔ ♠ët ❝ü❝ t✐➸✉ ✤è✐ ✈ỵ✐ ❙ tọ r t õ ữợ t t s ỗ t↕✐ c1, c2 > 0, θ > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ r > 1✱ t❛ ❝â c1 r n−ps p−1 c2 ≤ U (r) ≤ r n−ps p−1 ●✐↔ sû✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t ♠➔ m ,δ = U (θr) ≤ U (r) ∈ Ω✳ ❈❤♦ , δ > 0✱ ❣✐↔ sû U (δ) U (δ) − U (θδ)      g ,δ (t) = mp,δ (t − U (θδ))    t + U (δ) mp−1 − ,δ , ♥➳✉ ≤ t ≤ U (θδ) ♥➳✉ U (θδ) ≤ t ≤ U (δ) ♥➳✉ t ≥ U (δ) download by : skknchat@gmail.com ✹✼ ✈➔ t G ,δ (t) = g ,δ (τ ) p dτ = ❍➔♠ g ,δ ✈➔      m ,δ (t − U (θδ))    t + U (δ) mp−1 − ,δ G ,δ ♥➳✉ ≤ t ≤ U (θδ) ♥➳✉ U (θδ) ≤ t ≤ U (δ) ♥➳✉ t ≥ U (δ) ❦❤ỉ♥❣ t➠♥❣ ✈➔ ❧✐➯♥ tư❝✳ ❳➨t ❤➔♠ ❦❤æ♥❣ t➠♥❣ ✤è✐ ①ù♥❣ ❝➛✉ u ,δ (r) = G ,δ (U (r)) t❤ä❛ ♠➣♥✿  u (r) u , (r) = õ ữợ t s ❝❤♦ u ,δ ✱ ✭✷✳✹✶✮ ♥➳✉ r≤δ ♥➳✉ r ≥ θδ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❬❬✽❪✱ ❇ê ✤➲ ỗ t ởt số C = C (n, p, s) > t❤ä❛ ♠➣♥ ❝❤♦ ❜➜t ❦➻ 0< ≤ δ ✱ t❛ ❝â ❝→❝ ✤→♥❤ ❣✐→ s❛✉ ✤➙②✿ n−ps n |u ,δ (x) − u ,δ (y)|p ps + O(( ) p−1 ); dxdy ≤ S |x − y|n+ps δ R2n n ∗ n |u ,δ (x)|ps dx ≥ S ps − C(( ) p−1 ) δ Rn ❇ê ✤➲ ✷✳✶✳✷✶✳ ●✐↔ sû ✤✐➲✉ ú õ tỗ t > s❛♦ ❝❤♦ λ, µ t❤ä❛ ♠➣♥ < λ s❛♦ ❝❤♦ p p−q + µ p−q < Λ2 ✱ p tỗ t (u, v) E\ {(0, 0)} ợ u ≥ 0, v ≥ sup Jλ,µ < c∞ t≥0 ❚r♦♥❣ ✤â c∞ ❧➔ ❤➡♥❣ sè ✤➣ ❝❤♦ tr♦♥❣ ✭✷✳✷✷✮✳ t c,à < c ợ , tọ ♠➣♥ < λ + µ < Λ2 ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â Jλ,µ (u, v) = J (u, v) − K (u, v)✱ tr♦♥❣ ✤â ❤➔♠ p p−q J :E→R ✈➔ p p−q K:E→R ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ J (u, v) = (u, v) p p − α+β |u|α |v|β dx; Ω download by : skknchat@gmail.com ✹✽ K (u, v) = q (λ|u|q + µ|v|q ) dx Ω p p u0 := α u ,δ , v0 := β u ,δ ✱ tr♦♥❣ ✤â u ,δ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✹✶✮✳ ⑩♥❤ ①↕ h (t) := J (tu0 , tv0 ) t❤ä❛ ♠➣♥ h (0) = 0, h (t) > ❝❤♦ t > ✤õ ❜➨ ✈➔ h (t) < ❝❤♦ t > ✤õ ❧ỵ♥✳ ❍ì♥ ♥ú❛✱ ❤ ✤↕t ❝ü❝ ✤↕✐ t↕✐   α+β−p ✣➦t  t∗ :=  (u0 , v0 ) p α β |u0 | |v0 | dx   Ω ❉♦ ✤â✱ t❛ ❝â tp∗ sup J (tu0 , tv0 ) = h (t∗ ) = (u0 , v0 ) p t≥0 p 2tα+β ∗ − α+β |u0 |α |v0 |β dx Ω = p(α+β) α+β−p (u0 , v0 ) 1 − p α+β p α+β−p |u0 |α |v0 |β dx Ω = 1 − p α+β (α + β) p α+β−p α (α+β) α+β−p α α+β−p β p(α+β) α+β−p u ,δ β α+β−p X0 p α+β−p |u ,δ |α+β dx Ω  psn  s = n n−ps ps α β β α+β β α + n ps α α+β p X0 u ,δ      ∗ |u ,δ |ps dx    p  ∗ ps  Ω ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✷✵ ✈➔ ✭✷✳✶✽✮✱ t❛ ❝â sup J (tu0 , tv0 ) ≤ t≥0 s n n−ps ps β α+β α β 2s Sα,β ≤ n β α + n ps +O n ps α α+β n−ps p−1 δ   psn n−ps p−1 n ps  S +O  n S ps + C δ n p−1 p p∗ s   δ ✭✷✳✹✷✮ ❈❤♦ δ1 > s ợ c , p tọ 2s Sα,β = n n ps p < λ p−q + µ p−q < δ1 ✱ p ✈➔ p − C0 λ p−q + µ p−q > download by : skknchat@gmail.com ✹✾ ❚❛ ❝â Jλ,µ (tu0 , tv0 ) ≤ (u0 , v0 ) p ❉♦ õ tỗ t t0 (0, 1) p Ctp ❝❤♦ t≥0 ✈➔ λ, µ > s❛♦ ❝❤♦ sup J,à (tu0 , tv0 ) < c 0tt0 ợ ♠å✐ λ, µ p t❤ä❛ ♠➣♥ p < λ p−q + µ p−q < δ1 ✳ α, β > 1✱ ❱➻ tø ✭✷✳✹✶✮ ✈➔ ✭✷✳✹✷✮✱ t❛ ❝â sup Jλ,µ (tu0 , tv0 ) = sup [Jλ,µ (tu0 , tv0 ) − Kλ,µ (tu0 , tv0 )] t≥to t≥to 2s Sα,β ≤ n n ps +O n−ps p−1 δ q q tp0 − λα p + µβ p q |u ,δ |q dx B(0,δ) 2s Sα,β ≤ n n ps +O n−ps p−1 δ tp0 − (λ + µ) q |u ,δ |q dx B(0,δ) δ > ✤õ ❜➨ s❛♦ ❝❤♦ Bθδ (0) ⊂ Ω ✭❣✐↔ sû ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t δ ♠➔ ∈ Ω✮✳ ❱➻ sup (u ,δ ) ⊂ Ω✱ t❤❡♦ ❝æ♥❣ t❤ù❝ ✭✷✳✹✶✮✱ ❇ê ✤➲ ✷✳✶✳✶✾✱ ✈ỵ✐ < ≤ ✱ ❈è ✤à♥❤ t❛ ❝â |u ,δ (x)|q dx = B(0,δ) n− n−ps p q |U (x)|q dx = B(0,δ) |U (x)|q dx B (0, δ ) δ/ ≥ n− n−ps p q δ/ U (r)q rn−1 dr ≥ ωn−1 n− n−ps p q C          n−ps ωn−1 cq1 rn− p−1 q−1 dr n− n−ps p q n− n−ps p q (n−ps)q p(p−1) |log | ♥➳✉ q> ♥➳✉ q= ♥➳✉ q< n(p−1) n−ps n(p−1) n−ps n(p−1) n−ps ❚❤❡♦ ✤✐➲✉ t ữủ 2s S, sup J,à (tu0 , tv0 ) ≤ n t≥t0  n−ps  n− p q −C(λ, µ)  n− n−ps p q |log | n ps +C ♥➳✉ q> ♥➳✉ q= n−ps p−1 n(p−1) n−ps n(p−1) n−ps download by : skknchat@gmail.com ✺✵ = λ ❈❤♦ p p−q +µ p p−q p−1 n−ps ∈ 0, 2δ ✱ t❛ ❝â n p p 2s Sα,β ps sup Jλ,µ (tu0 , tv0 ) ≤ + C λ p−q + µ p−q n t≥t0  p−1 n−ps p p n−ps (n− p q )   λ p−q + µ p−q ♥➳✉ q > −C(λ, µ) n(p−1) p p p p  p(n−ps)  λ p−q + µ p−q ♥➳✉ q = log λ p−q + µ p−q ◆➳✉ n(p−1) n−ps ✱ t❛ ❝❤å♥ q> > , s ợ p n(p1) n−ps n(p−1) n−ps t❤ä❛ ♠➣♥ p < λ p−q + µ p−q < δ2 , C λ p p−q +µ < −C0 λ C0 ❚r♦♥❣ ✤â ❧➔ 1+ ❚❤❛② q= p p−q p p−q − C (λ + µ) λ +µ p p−q p p−q +µ (p−1) (n−ps) p p−q (n− n−ps p q) ✭✷✳✹✸✮ ❧➔ ❤➡♥❣ sè ❞÷ì♥❣ ①→❝ ✤à♥❤ tr♦♥❣ ✭✷✳✶✹✮✳ ❚❤ü❝ t➳ ✭✷✳✹✸✮ ❝â ♥❣❤➽❛ p p−1 n − ps n− q p − q n − ps p n(p−1) n−ps ✱ t❛ ❝❤å♥ δ3 > < p n (p − 1) ⇔q> pq n ps s ợ p , t❤ä❛ ♠➣♥ p < λ p−q + µ p−q < δ3 C λ p p−q +µ p p−q − C (λ + µ) λ p p−q p +µ p p−q n(p−1) p(n−ps) p p log λ p−q + µ p−q p < −C0 λ p−q + µ p−q p ❦❤✐ p log λ p−q + µ p−q → +∞ ( + à) p pq ợ +à , → p p−q ✈➔ n(p−1) p(n−ps) ❑❤✐ ✤â✱ ❝❤å♥ δ Λ2 = δ1 , δ2 , δ3 , ợ , p tọ p p p−q + µ p−q n−ps p−1 >0 p < λ p−q + µ p−q < Λ2 ✱ t❛ ❝â sup Jλ,µ (tu, tv) < c∞ t≥0 download by : skknchat@gmail.com ✭✷✳✹✹✮ ✺✶ (u0 , v0 ) = (0, 0) tỗ t t2 > s❛♦ ❝❤♦✿ − − (t2 u0 , t2 v0 ) ∈ Nλ,µ ✈➔ cλ,µ ≤ Jλ,µ (t2 u0 , t2 v0 ) ≤ sup Jλ,µ (tu0 , tv0 ) < c ợ , tọ 0 s❛♦ ❝❤♦ ❝❤♦ λ, µ t❤ä❛ ♠➣♥ < λ p−q + µ p−q < Λ3 ❤➔♠ Jλ,µ ❝â ❝ü❝ t✐➸✉ (u2 , v2 ) ✭✐✮ Jλ,µ (u2, v2) = c−λ,µ, ✭✐✐✮ (u2, v2) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮✳ p p ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ Λ2 − tr♦♥❣ Nλ,µ t❤ä❛ ♠➣♥✿ Λ3 := ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳✷✶ ✈➔ ✤➦t p Λ2 , q p p p−q Λ1 p (λ, µ) s❛♦ ❝❤♦ < λ p−q + pq < tỗ t ố ợ J,à tữỡ tỹ ữ tr ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✳✶✼ ✭✐✐✮ ✈ỵ✐ − (P S)c−λ {(uk , vk )} N,à tỗ t (u2 , v2 ) ∈ E Jλ,µ (u2 , v2 ) = c− λ,µ ✳ s❛♦ ❝❤♦ ❞➣② uk → u2 , vk → v2 ♠↕♥❤ tr♦♥❣ ❊ ✈➔ (u2 , v2 ) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮✳ − − (u2 , v2 ) ∈ Nλ,µ ✳ ❚ø {(uk , vk )} ∈ Nλ,µ , t❛ ❍ì♥ ♥ú❛✱ ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ ϕuk ,vk (1) = (p − q) (uk , vk ) p ❝â |uk |α |vk |β dx < − ((α + β) − q) Ω ❱➻ uk → u2 , vk → v2 ♠↕♥❤ tr♦♥❣ ❊✱ ❝❤♦ ϕu2 ,v2 (1) = (p − q) (u2 , v2 ) p k → ∞✱ t❛ ✤÷đ❝ |u2 |α |v2 |β dx ≤ − ((α + β) − q) Ω ❱➻ − Nλ,µ = φ✱ t❛ ♥❤➟♥ ✤÷đ❝ ϕu2 ,v2 (1) < ❤❛② − (u2 , v2 ) N,à ữỡ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù ❝❤ù❛ sè ♠ô tợ ữủ ỗ s (0, 1) , p > ✈➔ Ω ❧➔ ♠✐➲♥ ❜à ❝❤➦♥ ❝õ❛ Rn ✳ ❑❤✐ ✤â t❛ ❝â ❦➳t q✉↔ s ỵ sỷ r p s ◆➳✉ (u, 0) ❤♦➦❝ (0, u) ✭✷✳✹✻✮ ❧➔ ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮✱ ❦❤✐ ✤â ✭✷✳✶✮ q✉② ✈➲  (−∆)s u = λ|u|q−2 u p u = tr♦♥❣ Ω tr♦♥❣ Rn \Ω ✭✷✳✹✼✮ ❚❛ ❝â |u (x) − u (y)|p λ dxdy − q |x − y|n+ps Jλ,µ (u, 0) = p Q =− ❚ø ✭✷✳✹✻✮ ✈➔ ✭✷✳✹✽✮ t❛ ❝â |u|q dx ✭✷✳✹✽✮ Ω p−q u pq (u2 , v2 ) p X0 < ❧➔ ♥❣❤✐➺♠ ❦❤æ♥❣ t➛♠ t❤÷í♥❣✳ ❇➙② ❣✐í t❛ (u1 , v1 ) ❧➔ ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✳ ❑❤ỉ♥❣ ♠➜t t➼♥❤ tê♥❣ v1 ≡ 0✳ ❑❤✐ ✤â u1 ❧➔ ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ✭✷✳✹✼✮ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❣✐↔ sû (u1 , 0) p = u1 p X0 |u1 |q dx > =λ Ω ❍ì♥ ♥ú❛✱ t❛ ❝❤å♥ w ∈ X0 \ {0} (0, w) p s❛♦ ❝❤♦ = w p X0 |w|q dx > =à tỗ t ❞✉② ♥❤➜t < t1 < tmax (u1 , w) s❛♦ ❝❤♦✿ + (t1 u1 , t2 w) ∈ Nλ,µ , download by : skknchat@gmail.com q✉→t✱ ✺✸ tr♦♥❣ ✤â   tmax (u1 , w) =  = q q (α + β − q) (λ|u1 | + µ|w| ) dx Ω (α + β − p) (u1 , w) α+β−q α+β−p p  p−q   p−q > ❍ì♥ ♥ú❛ Jλ,µ (t1 u1 , t1 w) = inf0≤t≤tmax (tu1 , tw) ✈➔ + (u1 , 0) ∈ Nλ,µ ❦➨♦ t❤❡♦ + c+ λ,µ ≤ Jλ,µ (t1 u1 , t1 w) ≤ Jλ,µ (tu1 , tw) < J,à (u1 , 0) = c,à , ổ ỵ ✤â (u1 , v1 ) ❝ơ♥❣ ❧➔ ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✳ download by : skknchat@gmail.com ✺✹ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù tr➯♥ ♠✐➲♥ ❜à ❝❤➦♥ ✈ỵ✐ sè ụ tợ ữủ t ố ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ t➻♠ ❤✐➸✉ ✈➲ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù ợ số ụ tợ ữủ ỗ t q ỗ õ ữỡ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ ♣❤➙♥ tự ợ ữủ t ữỡ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t♦→♥ tû ♣✲▲❛♣❧❛❝❡ tự ự số ụ tợ ữủ ỗ ã r ữỡ tổ tr sỹ tỗ t ổ ố ợ ữỡ tr➻♥❤ ❦✐➸✉ ❑✐r❝❤❤♦❢❢ ❧♦↕✐ ❡❧❧✐♣t✐❝ ✈ỵ✐ t♦→♥ tû p✲♣❤➙♥ t❤ù ✈➔ ✤↕✐ ❧÷đ♥❣ ♣❤✐ t✉②➳♥ ✤è✐ ❞➜✉✳ ❈→❝ ❦➳t q✉↔ ỵ ỵ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✶ ✤÷đ❝ ❞ü❛ tr♦♥❣ ❬✶✺❪ ❝õ❛ t➔✐ t ã r ữỡ tổ tr sỹ tỗ t ữỡ tr ự t♦→♥ tû p✲▲❛♣❧❛❝❡ ♣❤➙♥ t❤ù ❝❤ù❛ sè ♠ơ tỵ✐ ❤↕♥ ữủ ỗ t q ỵ ✷✳✷✳✶✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✷ ✤÷đ❝ ❞ü❛ tr♦♥❣ ❬✼❪ ❝õ❛ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ download by : skknchat@gmail.com ✺✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆❞r✐♦✉❝❤✱ ❑✳✱ ❊▲ ❍❛♠✐❞✐✱ ❆✳✱ ✭✷✵✵✻✮✱ ❚❤❡ ◆❡❤❛r✐ ♠❛♥✐❢♦❧❞ ❢♦r s②st❡♠s ♦❢ ♥♦♥❧✐♥❡❛r ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✱ ✻✹✭✶✵✮✱ ✷✶✹✾✲✷✶✻✼✳ ❬✷❪ ❆✉t✉♦r✐✱ ●✳✱ ❋✐s❝❡❧❧❛✱ ❆✳✱ P✉❝❝✐✱ P✳✱ ✭✷✵✶✺✮✱ ❙t❛t✐♦♥❛r② ❑✐r❝❤❤♦❢❢ ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ ❛ ❢r❛❝t✐♦♥❛❧ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦r ❛♥❞ ❛ ❝r✐t✐❝❛❧ ♥♦♥❧✐♥❡❛r✐t②✱ ❆♥❛❧✱ ✶✷✺✱ ✻✾✾✲✼✶✹✳ ◆♦♥❧✐♥❡❛r ❬✸❪ ❇♦③❤❦♦✈✱ ❨✳✱ ▼✐t✐❞✐❡r✐✱ ❊✳✱ ✭✷✵✵✸✮✱ ❊①✐st❡♥❝❡ ♦❢ ♠✉❧t✐♣❧❡ s♦❧✉t✐♦♥s ❢♦r q✉❛s✐✲ ❧✐♥❡❛r s②st❡♠s ✈✐❛ ❢✐❜❡r✐♥❣ ♠❡t❤♦❞ ✱ ❏✳ ❉✐❢❢❡r✳ ❊q✉❛t✐♦♥s✱ ✶✾✵✭✶✮✱ ✷✸✾✲✷✻✼✳ ❬✹❪ ❇r❛s❝♦✱ ▲✳✱ ▼♦s❝♦♥✐✱ ❙✳✱ ❙q✉❛ss✐♥❛✱ ▼✳✱ ✭✷✵✵✻✮✱ ❖♣t✐♠❛❧ ❞❡❝❛② ♦❢ ❡①tr❡♠❛❧ ❢✉♥❝t✐♦♥s ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ❙♦❜♦❧❡✈ ✐♥❡q✉❛❧✐t②✱ t✐❛❧ ❊q✉❛t✐♦♥s✱ ✺✺✱ ✶✲✸✷✳ ❈❛❧❝✳ ❱❛r✳ P❛rt✐❛❧ ❉✐❢❢❡r❡♥✲ ❬✺❪ ❇r❡③✐s✱ ❍✳✱ ▲✐❡❜✱ ❊✳ ❍✳✱ ✭✶✾✽✸✮✱ ❆ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥❛❧s✱ ✹✽✻✲✹✾✵✳ Pr♦❝✳ ❆♠✳ ▼❛t❤s✳ ❙♦❝✳ ✽✽✱ ❬✻❪ ❈❤❡♥✱ ❲✳✱ ❉❡♥❣✱ ❙✳✱ ✭✷✵✵✻✮✱ ❚❤❡ ◆❡❤❛r✐ ♠❛♥✐❢♦❧❞ ❢♦r ❛ ❢r❛❝t✐♦♥❛❧ ♣✲ ▲❛♣❧❛❝✐❛♥ s②st❡♠ ✐♥✈♦❧✈✐♥❣ ❝♦♥❝❛✈❡✲❝♦♥✈❡① ♥♦♥❧✐♥❡❛r✐t✐❡s✱ ✷✼✱ ✽✵✲✾✷✳ ◆♦♥❧✐♥❡❛r ❆♥❛❧✱ ❬✼❪ ❈❤❡♥✱ ❲✳✱ ❙q✉❛ss✐♥❛✱ ▼✳✱ ✭✷✵✶✻✮✱ ❈r✐t✐❝❛❧ ♥♦♥❧♦❝❛❧ s②st❡♠s ✇✐t❤ ❝♦♥❝❛✈❡✲ ❝♦♥✈❡① ♣♦✇❡rs✱ ❆❞✈❛♥❝❡❞ ◆♦♥❧✐♥❡❛r ❙t✉❞✐❡s✱ ✶✻✭✹✮✱ ✽✷✶✲✽✹✷✳ ❬✽❪ ❉✐ ◆❡③③❛✱ ❊✳✱ P❛❧❛t✉❝❝✐✱ ●✳✱ ❱❛❧❞✐♥♦❝✐✱ ❊✳✱ ✭✷✵✶✷✮✱ ❍✐t❝❤❤✐❦❡rs ❣✉✐❞❡ t♦ t❤❡ ❢r❛❝t✐♦♥❛❧ ❙♦❜♦❧❡✈ s♣❛❝❡s ✱ ❇✉❧❧✳ ❙❝✐✳ ▼❛t❤✱ ✶✸✻✭✺✮✱ ✺✷✶✲✺✼✸✳ ❬✾❪ ❉r❛s❜❡❦✱ P✳✱ P♦❤♦③❛❡✈✱ ❙✳ ▲✳✱ ✭✶✾✾✼✮✱ P♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢♦r t❤❡ ♣✲▲❛♣❧❛❝✐❛♥✿ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❢✐❜❡r✐♥❣ ♠❡t❤♦❞✱ ✼✵✸✲✼✷✻✳ Pr♦❝✳ ❘♦②✳ ❙♦❝✳ ❊❞✐♥❜✉r❣❤ ❙❡❝t✳ ❆✱ ✶✷✼✱ download by : skknchat@gmail.com ✺✻ ❬✶✵❪ ❋❛r✐❛✱ ❋✳✱ ▼✐②❛❣❛❦✐✳ ❖✳✱ P❡r❡✐r❛✳ ❋✳✱ ❙q✉❛ss✐♥❛✳ ▼✳✱ ❩❤❛♥❣✳ ❈✳✱ ✭✷✵✶✻✮✱ ❚❤❡ ❇r❡③✐s✲ ◆✐r❡♥❜❡r❣ ♣r♦❜❧❡♠ ❢♦r ♥♦♥❧♦❝❛❧ s②st❡♠s✱ ✽✺✲✶✵✸✳ ❆❞✈✳ ◆♦♥❧✐♥❡❛r ❆♥❛❧✱ ✺✱ ❬✶✶❪ ●♦②❛❧✱ ❙✳✱ ❙r❡❡♥❛❞❤✱ ❑✳✱ ✭✷✵✶✺✮✱ ❊①✐st❡♥❝❡ ♦❢ ♠✉❧t✐♣❧❡ s♦❧✉t✐♦♥s ♦❢ ♣✲ ❢r❛❝✲ t✐♦♥❛❧ ▲❛♣❧❛❝❡ ♦♣❡r❛t♦r ✇✐t❤ s✐❣♥✲❝❤❛♥❣✐♥❣ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✱ ❆♥❛❧✳ ✹✭✶✮✱ ✸✼✲✺✽✳ ❆❞✈✳ ◆♦♥❧✐♥❡❛r ❬✶✷❪ ❍s✉✱ ❚✳ ❙✳✱ ✭✷✵✵✾✮✱ ▼✉❧t✐♣❧❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢♦r ❛ ❝r✐t✐❝❛❧ q✉❛s✐❧✐♥❡❛r ❡❧✲ ❧✐♣t✐❝ s②st❡♠ ✇✐t❤ ❝♦♥❝❛✈❡✲❝♦♥✈❡① ♥♦♥❧✐♥❡❛r✐t✐❡s✱ ✷✻✾✽✳ ◆♦♥❧✐♥❡❛r ❆♥❛❧✱ ✼✶✱ ✷✻✽✽✲ ❬✶✸❪ ❑✐r❝❤❤♦❢❢✱ ●✳✱ ▼❡❝❤❛♥✐❦✳ ❚❡✉❜♥❡r✳ ▲❡✐♣③✐❣ ✭✶✽✽✸✮✳ ❬✶✹❪ ▼❛ss❛r❛✱ ▼✳✱ ❚❛❧❜✐✱ ▼✳✱ ✭✷✵✶✺✮✱ ❖♥ ❛ ❝❧❛ss ♦❢ ♥♦♥❧♦❝❛❧ ❡❧❧✐♣t✐❝ s②st❡♠s ♦❢ ✭♣✱ q✮✲ ❑✐r❝❤❤♦❢❢ t②♣❡✱ ❏✳ ❊❣②♣t✳ ▼❛t❤✳ ❙♦❝✱ ✷✸✭✶✮✱ ✸✼✲✹✶✳ ❬✶✺❪ ▼✐s❤r❛✱ P✳ ❑✳✱ ❙r❡❡♥❛❞❤✱ ❑✳✱ ✭✷✵✶✼✮✱ ❋r❛❝t✐♦♥❛❧ ♣✲❑✐r❝❤❤♦❢❢ s②st❡♠ ✇✐t❤ s✐❣♥ ❝❤❛♥❣✐♥❣ ♥♦♥❧✐♥❡❛r✐t✐❡s✱ ❘❆❈❙❆▼✱ ✶✶✶✱ ✷✽✶✲✷✾✻✳ download by : skknchat@gmail.com ... q ps −1 p X0 |Ω|  v u q −1 − p  + v p X0 |Ω| p p∗ s −q p? ?? s p? ?? s −q p? ?? s  q S − p λ  q S − p µ ✭✷✳✶✻✮ p + C λ p? ??q + µ p? ??q −1 p p ( (u, v) p ) + C λ p? ??q + µ p? ??q , tr♦♥❣ ✤â  C= p − q p. .. t❛ ❝â✿ α+β−q p? ??q Ψu,v (tmax (u, v)) ≥ α+β−q α+β? ?p S − pq |Ω| α+β−q α+β λ p p−q +µ (u, v) p? ??q p p p? ??q S |Ω| α+β−q α+β p? ??(α+β) p? ??q λ p? ??(α+β) p? ??q p( q−(α+β)) p? ??q p p−q ✭✷✳✽✮ p? ??(α+β) p? ??q p? ??q α+β−q =... ♣✱q✱s✱♥✱❙ ✈➔ |Ω| s❛♦ ❝❤♦✿ (u, v) p p Jλ,µ (u, v) ≥ −C0 (λ p? ??q + µ p? ??q ), ✭✷✳✶✸✮ tr♦♥❣ ✤â p p (p? ?? q s −q) p − q (p? ??s − q) p? ??q − p? ??q p? ?? (p? ??q) s C0 := |Ω| S p pqp∗s (p? ??s − p) p? ??q ✭✷✳✶✹✮ ✈ỵ✐ ❙ ❧➔ ❤➔♥❣ sè