❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ✲✲✲✲✲✲ ✲✲✲✲✲✲ ▼❆■ ❚❍➚ ❚❘❆◆● ❙Ü ❚➬◆ ❚❸■ ◆●❍■➏▼ ❨➌❯ ❈Õ❆ ▼❐❚ ▲❰P ❇⑨■ ❚❖⑩◆ ❇■➊◆ ❊▲■P❚■❈ ❚✃◆● ◗❯⑩❚ ❑■➎❯ P ✲▲❆P▲❆❈■❆◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣❤➺ ❆♥ ✲ ✷✵✶✻ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ✲✲✲✲✲✲ ✲✲✲✲✲✲ ▼❆■ ❚❍➚ ❚❘❆◆● ❙Ü ❚➬◆ ❚❸■ ◆●❍■➏▼ ❨➌❯ ❈Õ❆ ▼❐❚ ▲❰P ❇⑨■ ❚❖⑩◆ ❇■➊◆ ❊▲■P❚■❈ ❚✃◆● ◗❯⑩❚ ❑■➎❯ P ✲▲❆P▲❆❈■❆◆ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳ ✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ữớ ữợ ◆❣❤➺ ❆♥ ✲ ✷✵✶✻ ✶ ▼Ư❈ ▲Ư❈ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❑✐➳♥ t❤ù❝ ❜ê trñ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✺ ✶✳✶✳ ❑❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ♣❤✐➳♠ ❤➔♠ pỗ ổ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸✳ P❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝ tê♥❣ q✉→t ❦✐➸✉ p✲▲❛♣❧❛❝✐❛♥ ✶✼ ✷✳✶✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t ❜➔✐ t♦→♥ ✭✷✳✶✮ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷ ▼Ð ✣❺❯ ◆❤÷ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t✱ ♠ët sè ♠æ ❤➻♥❤ ✈➲ ❝→❝ ❜➔✐ t♦→♥ ✤ë❝ ❧➟♣ t❤í✐ ❣✐❛♥ tr♦♥❣ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❦➽ t❤✉➟t ❦❤→❝ ♥❤❛✉ s➩ ❞➝♥ ✤➳♥ ❝→❝ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ ❱✐➺❝ ♥❣❤✐➯♥ ự sỹ tỗ t ợ t ♥➔② s➩ tr↔ ❧í✐ ♥❤✐➲✉ ❝➙✉ ❤ä✐ ❧✐➯♥ q✉❛♥✱ tø ❝→❝ ✈➜♥ ✤➲ ①✉➜t ❤✐➺♥ tr♦♥❣ ♥ë✐ t↕✐ ♥❣➔♥❤ t♦→♥ ❤å❝ ❝❤♦ ✤➳♥ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❦➽ t❤✉➟t ❦❤→❝ ✭①❡♠ ❬✶✵❪✮✳ ❱➻ t❤➳ tr♦♥❣ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙② ❝â t tr ữợ q t ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝✱ ❝❤➤♥❣ ❤↕♥✱ ❉✳▼✳ ✣ù❝ ❬✾❪✱ ◆✳▼✳ ❈❤÷ì♥❣ ❬✽❪✱ ❍✳◗✳ ❚♦➔♥ ❬✼❪✱ ❏✳ ▼❛✇❤✐♥ ❬✶✸❪✱ ❉✳ ▼♦tr❡❛♥✉ ✈➔ ◆✳❙✳ P❛♣❛❣❡♦r❣✐♦✉ ❬✶✹❪✱ ❆✳ ❆♠❜r♦s❡tt✐ ✈➔ P✳❍✳ ❘❛❜✐♥♦✇✐t③ ❬✷❪✱ P✳ ❞❡ ◆❛♣♦❧✐ ❬✶✺❪✱ ❆✳ ❑r✐st❛❧② ❬✶✷❪✱ ✳✳✳ ❈â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤÷❛ r❛ ❞ò♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝✱ ✤â ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❜➟❝ tỉ♣ỉ✱ ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➺♠ tr➯♥ ✲ ữợ ữỡ t ữỡ ♣❤➙♥✱ ✳✳✳ ▼é✐ ♣❤÷ì♥❣ ♣❤→♣ ❝â ♥❤ú♥❣ ÷✉ ✤✐➸♠ ✈➔ ❤↕♥ ❝❤➳ r✐➯♥❣✱ ❞♦ ✤â ❝❤➾ →♣ ❞ư♥❣ ✤÷đ❝ ❝❤♦ ởt ợ t t r số ỳ ữỡ ♣❤→♣ ❦➸ tr➯♥✱ ❝❤ó♥❣ tỉ✐ ✤➦❝ ❜✐➺t q✉❛♥ t➙♠ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ♣❤➙♥✳ ❱➲ ♥❣✉②➯♥ t➢❝✱ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ♥➔②✱ ✤➸ t➻♠ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♠ët ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝✱ t❛ q✉② ✈➲ t➻♠ ✤✐➸♠ tỵ✐ ❤↕♥ ❝õ❛ ♠ët ♣❤✐➳♠ ❤➔♠ ♥➔♦ ✤â tr♦♥❣ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ t❤➼❝❤ ❤ñ♣✳ P❤✐➳♠ ❤➔♠ ♥➔② s➩ t❤ä❛ ♠➣♥ ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤➸ ❝â t❤➸ ❦❤↔ ✈✐ ✈➔ →♣ ❞ư♥❣ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❜✐➳♥ ♣❤➙♥ ♥❤➡♠ t❤✉ ✤÷đ❝ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ◆❤÷ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t✱ ❜ð✐ ♥❤✐➲✉ ❦❤â ❦❤➠♥ ①✉➜t ❤✐➺♥ tr♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✱ ♥❤ú♥❣ ❦➳t q t ữủ ố ợ t t ổ t✉②➳♥ t➼♥❤ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳✳ ❈ị♥❣ ✈ỵ✐ sü ♣❤→t tr✐➸♥ ♠↕♥❤ ♠➩ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝ ❦❤ỉ♥❣ t✉②➳♥ ✸ t➼♥❤✱ ❝→❝ ❝ỉ♥❣ ❝ư ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ♣❤➙♥ ♥❣➔② ❝➔♥❣ ✤÷đ❝ ❝↔✐ t✐➳♥✳ ✣➦❝ ❜✐➺t✱ ❝❤ó♥❣ tỉ✐ q✉❛♥ t➙♠ ✤➳♥ ✤à♥❤ ❧➼ ✧◗✉❛ ♥ó✐✧ ❝õ❛ ❆✳ ❆♠❜r♦s❡tt✐ ✈➔ P✳❍✳ ❘❛❜✐♥♦✇✐t③ tr♦♥❣ ❝æ♥❣ tr➻♥❤ ❬✷❪✱ ð ✤â ❝→❝ t→❝ ❣✐↔ ✤➣ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ t ởt số sỹ tỗ t tợ ỹ tr trữớ ủ ổ ữợ ú ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ❝â t❤➯♠ ♠ët ❝ỉ♥❣ ❝ư ❤ú✉ ❤✐➺✉ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❧ỵ♣ ❝→❝ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝ ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤ t❤❛② ✈➻ ❝❤➾ ❞ị♥❣ ❝→❝ ♥❣✉②➯♥ ❧➼ ❝ü❝ t✐➸✉✳ ✣➸ ♥➢♠ ✤÷đ❝ ❝→❝ ♥❣✉②➯♥ ❧➼ ❜✐➳♥ ♣❤➙♥ ✈➔ q✉→ tr➻♥❤ ♣❤→t tr✐➸♥ ❝õ❛ ♥â ❝ơ♥❣ ♥❤÷ ù♥❣ ❞ư♥❣ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❝✉è♥ s→❝❤ ❝õ❛ ▼✳ ❙tr✉✇❡ ❬✶✼❪✳ ❚r♦♥❣ ❦❤✉ỉ♥ ú tổ ữợ ợ t ởt ♠ỉ ❤➻♥❤ t✐➯✉ ❜✐➸✉ ✈➲ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ♣❤➙♥✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❞ü❛ tr➯♥ ✈✐➺❝ t❤❛♠ ❦❤↔♦ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❜➔✐ ❜→♦ ❬✶✺❪✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ỗ õ ữỡ ữỡ tự ❈❤÷ì♥❣ ♥➔② ❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ pỗ ổ ởt số ❦❤→✐ ♥✐➺♠ ✈➔ ♥❣✉②➯♥ ❧➼ ❝ì ❜↔♥ tr♦♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ♣❤➙♥ s➩ ✤÷đ❝ ❞ị♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ❈❤÷ì♥❣ ✷✿ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝ tê♥❣ q✉→t ❦✐➸✉ p✲▲❛♣❧❛❝✐❛♥✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ỗ ợ t t ự sỹ tỗ t t ②➳✉ ✤è✐ ✈ỵ✐ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝ ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤ ❦✐➸✉ p✲▲❛♣❧❛❝✐❛♥ ✭①❡♠ ✣à♥❤ ❧➼ ✷✳✷✳✷ ✈➔ ✣à♥❤ ❧➼ ✷✳✷✳✾✮✳ P❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ❞ị♥❣ ð ✤➙② ❧➔ ✤à♥❤ ❧➼ ✧◗✉❛ ♥ó✐✧ ✈➔ ❞↕♥❣ ✤è✐ ①ù♥❣ ❝õ❛ ♥â ✭①❡♠ ▼➺♥❤ ✤➲ ✶✳✸✳✽ ✈➔ ▼➺♥❤ ✤➲ ✶✳✸✳✾✮✳ ❚r♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✱ tỉ✐ ✤➣ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ ❣✐ó♣ ✤ï tø ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❦❤♦❛ ❙÷ ♣❤↕♠ ❚♦→♥ ❤å❝✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ ❣✐❛ ✤➻♥❤ t t ữợ ❚❙✳ ◆❣✉②➵♥ ✹ ❚❤➔♥❤ ❈❤✉♥❣✳ ▼➦❝ ❞ò ✤➣ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ✈➔ ❝ỉ♥❣ sù❝ ✤➸ t➻♠ ❤✐➸✉✱ ♥❣❤✐➯♥ ❝ù✉ ♥❤÷♥❣ ❝❤➢❝ ❝❤➢♥ ❧✉➟♥ ✈➠♥ s➩ ❝á♥ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ♥❤➜t ✤à♥❤✳ ❘➜t ♠♦♥❣ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ õ õ ỵ ữủ t ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❱✐♥❤✱ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✻ ❚→❝ ❣✐↔ ▼❛✐ ❚❤à ❚r❛♥❣ ✺ ❈❍×❒◆● ✶ ❑■➌◆ ❚❍Ù❈ ❇✃ ❚❘Ñ ✶✳✶✳ ❑❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ỡ pỗ ●✐↔ sû X ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ✳ P❤✐➳♠ ❤➔♠ A : X → R ✤÷đ❝ A (x + (1 )y) ỗ tr t ỗ C X A(x) + (1 λ)A(y) ✈ỵ✐ ♠å✐ x, y ∈ C ✈➔ λ ∈ [0, 1]✳ P❤✐➳♠ ❤➔♠ A : X → R ✤÷đ❝ ❣å✐ ❧➔ C ⊂ X ♥➳✉ A x+y tr➯♥ t ỗ 1 A(x) + A(y) 2 ợ ♠å✐ x, y ∈ C ✳ P❤✐➳♠ ❤➔♠ A : X R ữủ ợ ỗ t ỳ ỗ tr t ỗ C X > tỗ t số ( ) > s❛♦ ❝❤♦ A x+y 1 A(x) + A(y) − δ( ) 2 ✈ỵ✐ ♠å✐ x, y ∈ C t❤ä❛ ♠➣♥ x − y > ✳ ●✐↔ sû p ∈ (1, +∞)✳ P❤✐➳♠ ❤➔♠ A : X R ữủ pỗ t ỗ C X tỗ t số k(p, C) > s❛♦ ❝❤♦ A x+y ✈ỵ✐ ♠å✐ x, y ∈ C ✳ 1 A(x) + A(y) − k(p, C) x − y 2 p ✤➲✉ tr➯♥ ✻ ✶✳✶✳✷ ú ỵ A ởt ỗ ỗ pỗ tr t ỗ C X t A ỗ t ỳ tr➯♥ C ✳ ✷✳ ◆➳✉ A ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ pỗ tr C t A ỗ ✤➲✉ tr➯♥ C ✳ ◆❣♦➔✐ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ✈➲ ♣❤✐➳♠ ỗ pỗ ú t õ t ũ ổ ỗ ởt ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ❝→❝ ❜➔✐ ❜→♦ ❬✶✽❪✳ ▼➺♥❤ ✤➲ s❛✉ ✤➙② ❝❤♦ t❛ ♠ët ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ♠ët ỗ t ỳ s ởt ỗ tr ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ◆➳✉ ♣❤✐➳♠ ❤➔♠ A : X → R ❧➔ ❧✐➯♥ tö❝ ✈➔ ỗ t ỳ tr t ỗ C X t A ỗ tr C q pỗ ú tổ ợ t sỡ ữủ ởt sè ❦➳t q✉↔ s❛✉ ✤➙②✱ ①❡♠ ❬✶✺❪✳ ✶✳✶✳✹ ▼➺♥❤ ✤➲✳ ●✐↔ sû f ∈ C 1(R, R)✱ < p < +∞ ✈➔ ✤↕♦ ❤➔♠ f ❧➔ p✲✤ì♥ ✤✐➺✉ ♠↕♥❤✱ tự tỗ t c > s f (y) − f (x) (y − x)p−1 c, ✈ỵ✐ ♠å✐ x, y ∈ R s❛♦ ❝❤♦ y > x✳ ❑❤✐ õ f ởt pỗ ự sû y > x✱ ✤➦t z = x+y ✱ t❛ ❝â y f (y) − f (z) = f (t)dt, z z f (z) − f (x) = f (s)ds x ❙✉② r❛ y f (y) + f (x) − 2f (z) = z f (t) dt − z ✣➦t t = s + y−x x ✭♥➳✉ x > y t❛ ✤➦t s = t + 1 f (x) + f (y) − F (z) = 2 x−y ✮ z f x f (s) ds s+ t❛ ❝â y−x − f (s) ds ✭✶✳✶✮ ✼ ❚❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â f s+ y−x − f (s) y−x c p−1 ✈ỵ✐ ♠å✐ x, y ∈ R s❛♦ ❝❤♦ y > x✳ ❉♦ ✤â tø ✭✶✳✶✮ s✉② r❛ 1 f (x) + f (y) − f (z) 2 z x y−x c ❤❛② p−1 ds c y−x 2 x+y f (x) + f (y) − f 2 ❚ø ✤➙② s✉② r❛ f ởt pỗ p ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ A : X R ởt pỗ > õ [A(x)] qỗ ợ q = αp✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❣✐↔ t❤✐➳t A ❧➔ ♠ët pỗ tỗ t số c > s❛♦ ❝❤♦ A p x+y x−y +c 2 1 A(x) + A(y), 2 ✭✶✳✷✮ ∀x, y ∈ X ❱➻ α > ♥➯♥ t❛ ❝â sα + tα ✈ỵ✐ ♠å✐ s, t (s + t)α 0✳ ❱ỵ✐ s = A x+y x+y A ✈➔ t = c α α +c x−y p ✱ x−y q t❛ ❝â x+y x−y A +c 2 p α ❉♦ ✤â✱ ✈➻ ❤➔♠ sè t t ỗ ỗ ợ t ∈ [0, +∞) ♥➯♥ tø ✭✶✳✷✮ s✉② r❛ x+y A A q ỗ +c x−y q α 1 A(x) + A(y) 2 1 A(x)α + A(y)α 2 ✽ ✶✳✶✳✻ ❱➼ ❞ư✳ ✶✳ ❳➨t ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X = Rd ✈ỵ✐ ❝❤✉➞♥ ❊✉❝❧✐❞❡ d |xj |2 |x| = , xj ∈ R, x = (x1 , x2 , , xd ), j = 1, 2, , d i=1 ❑❤✐ ✤â✱ ♣❤✐➳♠ ❤➔♠ A : X → R ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ A(x) = |x|2 p ❧➔ 2ỗ tr X ứ s r A(x) = |x|p pỗ ✈ỵ✐ ♠å✐ p 2✳ ✷✳ ❳➨t ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X = L2 () ỗ ữỡ t tr➯♥ t➟♣ Ω ⊂ Rd ✈ỵ✐ ❝❤✉➞♥ 2 |f (x)| dx f = , f ∈ L2 (Ω) Ω ❑❤✐ ✤â ♣❤✐➳♠ ❤➔♠ A : X → R ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ |f (x)|2 dx A(f ) = p 2ỗ tr X ữỡ tü ♥❤÷ tr➯♥ A(x) = x p ❧➔ ♣❤✐➳♠ pỗ ợ p > qt ♥➳✉ H ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈ỵ✐ ❝❤✉➞♥ t❤➻ tø ✤➥♥❣ t❤ù❝ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤ ✈➔ ▼➺♥❤ ✤➲ ✶✳✶✳✺ s✉② r❛ ♣❤✐➳♠ ❤➔♠ A : H → R A(x) = x p pỗ ✈ỵ✐ ♠å✐ p 2✳ ✶✳✷✳ ❑❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ●✐↔ sû d ∈ N∗ ✳ ❑➼ ❤✐➺✉ Rd := {x = (x1 , , xd ) : xj ∈ R, j = 1, 2, , d}✱ Ω ❧➔ ♠ët ♠✐➲♥ ✭♠ð ✈➔ ❧✐➯♥ t❤æ♥❣✮ tr♦♥❣ Rd ✱ α = (α1 , α2 , , αd ) ∈ Zd+ ❧➔ ♠ët ✤❛ ❝❤➾ sè ✈ỵ✐ |α| = ♠å✐ α t❤ä❛ ♠➣♥ |α| d j=1 αj ✳ ◆➳✉ u ❧➔ ❤➔♠ ❦❤↔ ✈✐ n ❧➛♥ t❤➻ ✈ỵ✐ n✱ ✤↕♦ ❤➔♠ ❝➜♣ α ❝õ❛ u ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ ∂ |α| u(x) D u(x) = α1 α2 αd ∂x1 ∂x2 ∂xd α ✸✷ ❱➻ ❞➣② {un } ❤ë✐ tö ②➳✉ ♥➯♥ ♥â ❜à tự tỗ t số r > s❛♦ ❝❤♦ un < r✳ ❱➻ J ❜à ❝❤➦♥ tr➯♥ ❝→❝ t➟♣ ❜à ❝❤➦♥ ♥➯♥ s✉② r❛ ❞➣② sè t❤ü❝ {J(un )} õ tỗ t ởt ❝♦♥ ❝õ❛ {un } ✭✈➝♥ ❞ò♥❣ ❦➼ ❤✐➺✉ ❧➔ {un }✮ s❛♦ ❝❤♦ {J(un )} ❤ë✐ tö✱ tù❝ ❧➔ J(un ) → c ❦❤✐ n → ∞✳ ❚❤❡♦ 1,p ❇ê ✤➲ ✷✳✷✳✹✱ J ❧➔ ♣❤✐➳♠ ❤➔♠ ♥û❛ ❧✐➯♥ tö❝ ②➳✉ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ W0 (Ω)✱ s✉② r❛ J(u) lim inf J(un ) = c n→∞ ▼➦t ❦❤→❝✱ t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✷✳✷✳✹ t❤➻ J ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ 1,p ỗ tr W0 () J(u) ứ ✭✷✳✶✺✮ t❛ ❝â J(u) un u ♥➯♥ un +u ✭✷✳✶✺✮ J(un ) + J (un )(u − un ) c✳ ◆❤÷ ✈➟②✱ J(u) = c✳ ▲↕✐ ❝â ✈➻ u n t ỷ tử ữợ ②➳✉ ❝õ❛ ♣❤✐➳♠ ❤➔♠ J ✱ t❛ ❝â c = J(u) lim inf J n→∞ un + u ✭✷✳✶✻✮ ●✐↔ t❤✐➳t ♣❤↔♥ ❝❤ù♥❣ r➡♥❣ {un } ❦❤æ♥❣ ❤ë✐ tử u õ tỗ t > ✈➔ ♠ët ❞➣② ❝♦♥ ❝õ❛ ❞➣② {un }✱ ✈➝♥ ❞ò♥❣ ❦➼ ❤✐➺✉ {un } s❛♦ ❝❤♦ u − un W01,p (Ω) ✳ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✷✳✷✳✹ t❛ ❝â u + un 1 J(u) + J(un ) − J 2 c2 u − un p W01,p (Ω) c2 p ✭✷✳✶✼✮ ❚r♦♥❣ ✭✷✳✶✼✮ ❝❤♦ n → ∞✱ t❛ ❝â lim sup J n→∞ u + un c − c2 p ✭✷✳✶✽✮ ❚ø ✭✷✳✶✻✮ ✈➔ ✭✷✳✶✽✮ t❛ t❤➜② ♠➙✉ t❤✉➝♥✳ ❙✉② r❛ ❞➣② {un } ❤ë✐ tư ♠↕♥❤ 1,p ✤➳♥ u tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ W0 (Ω) ❦❤✐ n → ∞✳ ◆❤÷ ✈➟② t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ J t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (S+ )✳ ✷✳✷✳✼ ❇ê ✤➲✳ P❤✐➳♠ ❤➔♠ I ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝ ✭✷✳✻✮ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ P❛❧❛✐s✲❙♠❛❧❡✳ ✸✸ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ (un) ⊂ W01,p(Ω) ❧➔ ♠ët ❞➣② P❛❧❛✐s✲❙♠❛❧❡✱ tù❝ ❧➔ {I(un)} 1,p ❜à ❝❤➦♥ ✈➔ I (un ) → tr♦♥❣ ổ ố (W0 ()) rữợ t 1,p t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ {un } ❧➔ ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ W0 (Ω)✳ ❚❤➟t ✈➟②✱ t❛ ❝â I(un ) − I (un )(un ) θ A(x, ∇un ) dx − = Ω F (x, un ) dx − Ω + θ θ a(x, ∇un )∇un dx Ω f (x, un )un dx ✭✷✳✶✾✮ Ω A(x, ∇un ) − a(x, ∇un )∇un dx θ Ω f (x, un )un − F (x, un ) dx + Ω θ = ❚ø ❣✐↔ t❤✐➳t ✭❆✹✮ t❛ ❝â a(x, ∇un )∇un pA(x, ∇un ), ∀n ∈ N∗ ❉♦ ✤â✱ tø ✭✷✳✶✾✮ s✉② r❛ I(un ) − I (un )(un ) θ 1− p θ + Ω A(x, ∇un ) dx Ω f (x, un )un − F (x, un ) dx θ ❱➻ f : Ω × R → R ❧✐➯♥ tư❝ ♥➯♥ M = sup f (x, t)t − F (x, t) : x ∈ Ω, θ |t| t0 < +∞ ❉♦ ✤â 1− p θ A(x, ∇un ) dx Ω I(un ) − I (un )(un ) θ − f (x, un )un − F (x, un ) dx + M µ(Ω) θ {x∈Ω: |un |>t0 } ✸✹ ❚ø ❣✐↔ t❤✐➳t ✭❢✷✮ t❛ ❝â f (x, un )un − F (x, un ) θ 0, ∀x ∈ Ω : |un (x)| > t0 ❙✉② r❛ 1− p θ A(x, ∇un ) dx Ω I(un ) − I (un )(un ) + M µ(Ω) θ ✭✷✳✷✵✮ ❚ø ❣✐↔ t❤✐➳t ✭❆✺✮ t❛ ❝â A(x, ∇un ) c3 |∇un |p , ∀n ∈ N∗ , ❞♦ ✤â tø ✭✷✳✷✵✮✱ c3 − p θ tr♦♥❣ ✤â ∗ |∇un |p Ω I(un ) − I (un )(un ) + M µ(Ω) θ I(un ) + I (un ) ∗ un W01,p (Ω) + M µ(Ω), θ 1,p ✤÷đ❝ ❤✐➸✉ ❧➔ ❝❤✉➞♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ (W0 (Ω))∗ ✳ ❚â♠ ❧↕✐ t❛ ❝â c3 − p θ un p W01,p (Ω) I(un ) + I (un ) θ ∗ un W01,p (Ω) + M µ(Ω) ✭✷✳✷✶✮ ●✐↔ t❤✐➳t ♣❤↔♥ ❝❤ù♥❣ r➡♥❣ ❞➣② {un } ❦❤æ♥❣ ❜à ❝❤➦♥✳ ❑❤✐ ✤â t❛ ❝â t❤➸ tr➼❝❤ ♠ët ❞➣② ❝♦♥ ❝õ❛ {un }✱ ✤➸ ✤ì♥ ❣✐↔♥ t❛ ✈➝♥ ❦➼ ❤✐➺✉ ❧➔ {un } s❛♦ ❝❤♦ un un W01,p (Ω) W01,p (Ω) → +∞✳ ❚ø ✭✷✳✷✶✮ ✈➔ < p < θ✱ ❝❤✐❛ ❝↔ ❤❛✐ ✈➳ ❝❤♦ ✈➔ ❝❤♦ n → ∞✱ t❛ t❤➜② ♠➙✉ t❤✉➝♥✳ ◆❤÷ ✈➟② ❞➣② {un } ❜à 1,p ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ W0 (Ω)✳ 1,p ❱➻ W0 (Ω) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ ♥➯♥ t❛ ❝â t❤➸ tr➼❝❤ ♠ët ❞➣② ❝♦♥ ❝õ❛ {un } ✭✈➝♥ ❞ị♥❣ ❦➼ ❤✐➺✉ ❧➔ {un }✮ ❤ë✐ tư ②➳✉ ✤➳♥ ❤➔♠ u ∈ W01,p (Ω) ♥➔♦ ✤â✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❞➣② {un } ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tư 1,p ♠↕♥❤ ✤➳♥ u tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ W0 (Ω)✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✻✱ J t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (S+ )✳ ❉♦ ✤â t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ lim sup J (un )(un − u) n→∞ ✸✺ ❤❛② a(x, ∇un )(∇un − ∇u) dx lim sup n→∞ ✭✷✳✷✷✮ Ω ❚❛ ❝â a(x, ∇un )(∇un − ∇u) dx = I (un )(un − u) + f (x, un )(un − u) dx Ω Ω ❚❛ ❝â I (un ) → ❦❤✐ n → ∞ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ u ❦❤✐ n → ∞ ♥➯♥ un − u un ✭✷✳✷✸✮ (W01,p (Ω))∗ ✳ ❍ì♥ ♥ú❛✱ ❦❤✐ n → ∞ ✈➔ ❞♦ ✤â ❞➣② {un − u} 1,p ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ W0 (Ω)✳ ❚ø ✤â s✉② r❛ ✭✷✳✷✹✮ lim I (un )(un − u) = n→∞ ▼➦t ❦❤→❝✱ t❤❡♦ ❣✐↔ t❤✐➳t ✭❢✶✮ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎ ♦❧❞❡r t❛ ❝â |f (x, un )||un − u| dx f (x, un )(un − u) dx Ω Ω (1 + |un |q−1 )|un − u| dx c4 Ω c4 µ(Ω) q−1 q + un q−1 Lq (Ω) un − u Lq (Ω) 1,p ❱➻ ♣❤➨♣ ♥❤ó♥❣ W0 (Ω) → → Lq (Ω) ❧➔ ❝♦♠♣❛❝t ✈ỵ✐ p < q < p∗ ✭①❡♠ ▼➺♥❤ ✤➲ ✶✳✷✳✾✮ ♥➯♥ ❞➣② {un } ❤ë✐ tư ♠↕♥❤ ✤➳♥ u tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Lq (Ω)✳ ❚ø ✤â s✉② r❛ lim n→∞ Ω ✭✷✳✷✺✮ f (x, un )(un − u) dx = ❑➳t ❤ñ♣ ✭✷✳✷✸✮✲✭✷✳✷✺✮ t❛ ❝â lim n→∞ Ω a(x, ∇un )(∇un − ∇u) dx = ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✻✱ t♦→♥ tû J t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (S+ ) ♥➯♥ s✉② r❛ ❞➣② {un } ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ♠↕♥❤ ✤➳♥ u tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ W01,p (Ω)✳ ❱➻ 1,p ✈➟② ♣❤✐➳♠ ❤➔♠ I t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ P❛❧❛✐s✲❙♠❛❧❡ tr➯♥ W0 () ỗ t sè α, ρ > s❛♦ ❝❤♦ I(u) = α > ✈ỵ✐ ♠å✐ u ∈ W01,p(Ω) t❤ä❛ ♠➣♥ u ỗ t e W01,p() ợ = W01,p (Ω) e W01,p (Ω) >ρ s❛♦ ❝❤♦ I(e) < 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ✶✳ ❚❤❡♦ ❣✐↔ t❤✐➳t ✭❢✸✮✱ t❛ ❝â < λ < λ1pc3✳ ❈❤å♥ ♥❤ä s❛♦ ❝❤♦ c3 > > + p1 ứ tỗ t ❤➡♥❣ sè δ = δ( ) > s❛♦ ❝❤♦ f (x, t) |t|p−2 t ✭✷✳✷✻✮ λ+ ✈ỵ✐ ♠å✐ < |t| < δ ✈➔ ✈ỵ✐ ♠å✐ x ∈ Ω✳ ❙✉② r❛ (λ + )tp−1 ♥➳✉ < t < δ, p−1 −(λ + )t ♥➳✉ −δ < t < f (x, t) ❇➡♥❣ ❝→❝❤ ❧➜② t➼❝❤ ♣❤➙♥ t❤❡♦ tø♥❣ ✈➳ t❛ ❝â F (x, t) (λ + )|t|p p ✭✷✳✷✼✮ ✈ỵ✐ ♠å✐ |t| < δ ✈➔ x ∈ Ω✳ ❚ø ✭✷✳✽✮✱ ✈ỵ✐ δ > ♥â✐ ð tr➯♥ t❛ ❧✉ỉ♥ t➻♠ ✤÷đ❝ ❤➡♥❣ sè c(δ) > ♣❤ư t❤✉ë❝ ✈➔♦ δ s❛♦ ❝❤♦ F (x, t) ✈ỵ✐ ♠å✐ |t| c(δ)|t|q ✭✷✳✷✽✮ δ ✈➔ x ∈ Ω✳ 1,p ❚❤❡♦ ❣✐↔ t❤✐➳t ✭❆✺✮ ✈➔ ✤à♥❤ ❧➼ ♥❤ó♥❣ tø ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ W0 (Ω) ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ Lq (Ω)✱ p < q < p∗ ✱ t❛ ❝â A(x, ∇u) dx − I(u) = Ω F (x, u) dx Ω A(x, ∇u) dx − = Ω {x∈Ω:|u| ✤õ ♥❤ä s❛♦ ❝❤♦ α > ✈ỵ✐ ♠å✐ u ∈ W01,p (Ω) ❝â ❝❤✉➞♥ u W01,p () = rữợ t ú t s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ A(x, ξ)tp A(x, tξ) ✈ỵ✐ ♠å✐ t ✭✷✳✷✾✮ 1✱ x ∈ Ω✱ ξ ∈ Rd ✳ ❱ỵ✐ ♠é✐ x ∈ Ω ✈➔ ξ ∈ Rd ❝è ✤à♥❤✱ ①➨t ❤➔♠ sè ϕ : [1, +∞) → R ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ ϕ(t) = A(x, tξ) ❚❛ ❝â ϕ (t) = a(x, tξ)ξ = a(x, tξ)(tξ) t ❚ø ❣✐↔ t❤✐➳t ✭❆✹✮ t❛ ❝â p A(x, tξ) t p ϕ(t) t ϕ (t) ❙✉② r❛ ϕ (t) ϕ(t) p , t ∀t ▲➜② t➼❝❤ ♣❤➙♥ ❝↔ ❤❛✐ ✈➳ tø ✤➳♥ t t❛ ✤÷đ❝ ln ϕ(t) − ln ϕ(1) p ln t, ∀t ❚ø ✤â t❛ ❝â ϕ(t) ϕ(1)tp , ∀t 1 ✸✽ ❤❛② ✭✷✳✷✾✮ t❤ä❛ ♠➣♥✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ s➩ ❝❤➾ r❛ F (x, t)sθ F (x, ts) ✈ỵ✐ ♠å✐ |t| t > 0✱ s ✭✷✳✸✵✮ ✈➔ x ∈ Ω✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ |t| t0 > ❝è ✤à♥❤✱ t❛ ①➨t ❤➔♠ sè s → ψ(s) = F (x, st) tr➯♥ ♠✐➲♥ [1, +∞)✳ ❚❛ ❝â |st| = s|t| t0 > 0✳ ❚❤❡♦ ❣✐↔ t❤✐➳t ✭❢✷✮✱ t❛ ❝â ψ (s) = f (x, st)t = f (x, st)(st) s θ θF (x, st) = ψ(s) > s s ❉♦ ✤â ψ (s) ψ(s) θ , s ∀s ▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ tø ✤➳♥ s t❛ ✤÷đ❝ ψ(s) ψ(1)sθ , ❚ø ✤â s✉② r❛ ✭✷✳✸✵✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ |t| ∀s t > 0✱ s ✈➔ x ∈ Ω✳ ●✐↔ sû u0 ∈ C0∞ (Ω)\{0} ❝è ✤à♥❤ s❛♦ ❝❤♦ µ{x ∈ Ω : |u0 (x)| t0 } > 0✳ ❑❤✐ ✤â ✈ỵ✐ s > 1✱ t❤❡♦ ✭✷✳✷✾✮ ✈➔ ✭✷✳✸✵✮ t❛ ❝â A(x, s∇u0 ) dx − I(su0 ) = Ω F (x, su0 ) dx Ω sp A(x, ∇u0 ) dx − sθ Ω F (x, u0 ) dx + M µ(Ω) {x∈Ω: |u0 | t0 } ✈ỵ✐ M = sup{|F (x, t)| : x ∈ Ω, |t| t0 } < +∞ ✈➻ F (x, t) ❧✐➯♥ tư❝ tr➯♥ Ω × [−t0 , t0 ]✳ ❱➻ θ > p ♥➯♥ t❛ ❦❤✐ s → +∞ t❤➻ I(su0 ) → −∞✳ ❉♦ ✤â tỗ t s > s s u0 W01,p (Ω) > ρ ✈➔ I(s∗ u0 ) < 0✳ ✣➦t e = s∗ u0 t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ❧➼ s❛✉ ✤➙② ❝❤♦ ❝❤ó♥❣ t❛ ♠ët ❦➳t q✉↔ ✈➲ t➼♥❤ ✤❛ ♥❣❤✐➺♠ ✤è✐ ✈ỵ✐ ❜➔✐ t♦→♥ ✭✷✳✶✮✳ ✸✾ ✷✳✷✳✾ ✣à♥❤ ❧➼✳ ●✐↔ sû r➡♥❣ ❝→❝ ❤➔♠ a : Ω × Rd → Rd ✈➔ f : Ω × R → R t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✣à♥❤ ❧➼ ✷✳✷✳✷✳ ❍ì♥ ♥ú❛✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥ ✭❆✻✮ a(x, −ξ) = −a(x, ξ), ∀x ∈ Ω, ξ ∈ Rd ✈➔ ✭❢✹✮ f (x, −t) = −f (x, t), ∀x ∈ Ω, t ∈ R✳ ❑❤✐ ✤â ❜➔✐ t♦→♥ ✭✷✳✶✮ ❝â ♠ët ❞➣② ✈æ ❤↕♥ ♥❣❤✐➺♠ ②➳✉ {un} s❛♦ ❝❤♦ limn→∞ I(un ) → +∞✱ tr♦♥❣ ✤â I ❧➔ ♣❤✐➳♠ ❤➔♠ ❧✐➯♥ ❦➳t ✈ỵ✐ ❜➔✐ t♦→♥ ✭✷✳✶✮ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ ✭✷✳✻✮✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ❦✐➸♠ tr❛ ❝→❝ ❣✐↔ tt rữợ t t õ ✤➲ s❛✉✳ ✷✳✷✳✶✵ ❇ê ✤➲✳ ●✐↔ sû X1 ⊂ W01,p(Ω) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ❑❤✐ ✤â t➟♣ ❤ñ♣ S ❣✐❛♥ W01,p(Ω)✳ = {u ∈ X1 : I(u) 0} tr ổ ự rữợ t ú t s ự r tỗ t C(Ω)✱ γ(x) > ✈ỵ✐ ♠å✐ x ∈ Ω s❛♦ ❝❤♦ F (x, t) ✈ỵ✐ ♠å✐ x ∈ Ω ✈➔ |t| γ(x)|t|θ t0 ✳ ❚❤➟t ✈➟②✱ tø ✤✐➲✉ ❦✐➺♥ ✭❢✷✮✱ f (x, t) F (x, t) ✈ỵ✐ ♠å✐ x ∈ Ω ✈➔ t ✭✷✳✸✶✮ θ t t0 > 0✳ ▲➜② t➼❝❤ ♣❤➙♥ ❝↔ ❤❛✐ ✈➳ tø t0 ✤➳♥ t✱ t❛ ✤÷đ❝ ln F (x, t) − ln F (x, t0 ) ln tθ − ln tθ0 ❤❛② F (x, t) ✈ỵ✐ ♠å✐ x ∈ Ω ✈➔ t γ1 (x)tθ t0 > 0✱ tr♦♥❣ ✤â γ1 (x) = F (x,t0 ) ✳ tθ0 ✹✵ ❚ø ✤✐➲✉ ❦✐➺♥ ✭❢✷✮✱ γ1 ∈ C(Ω) ✈➔ γ1 (x) > ✈ỵ✐ ♠å✐ x ∈ C(Ω)✳ ❈❤ù♥❣ ♠✐♥❤ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü✱ t❛ ❝â γ2 (x)(−t)θ F (x, t) ✈ỵ✐ ♠å✐ x ∈ Ω ✈➔ t −t0 ✱ tr♦♥❣ ✤â γ2 = F (x,−t0 ) tθ0 ∈ C(Ω) ✈➔ γ2 (x) > ✈ỵ✐ ♠å✐ x ∈ Ω✳ ✣➦t γ(x) = min{γ1 (x), γ2 (x)} ✈ỵ✐ x ∈ Ω t❛ ❝â γ ∈ C(Ω) ✈➔ γ(x) > ✈ỵ✐ ♠å✐ x ∈ C(Ω)✱ ❞♦ ✤â ✭✷✳✸✶✮ ✤ó♥❣✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ I(u) c1 u p W01,p (Ω) + c5 u W01,p (Ω) γ(x)|u|θ + c6 − ✭✷✳✸✷✮ Ω 1,p ✈ỵ✐ ♠å✐ u W0 () ợ ci số ữỡ i = 5, 6✳ 1,p ●✐↔ sû u ∈ W0 () tũ ỵ < = {x Ω : |u(x)| < t0 }, Ω = Ω\Ω< ứ p < q tỗ t↕✐ ❤➡♥❣ sè c7 > s❛♦ ❝❤♦ c7 (|t|q + 1) |F (x, t)| ✭✷✳✸✸✮ ✈ỵ✐ ♠å✐ x ∈ Ω ✈➔ t ∈ R✳ ❉♦ ✤â (|u|q + 1) dx −c7 F (x, u) dx Ω< Ω< −c7 Ω ✭✷✳✸✹✮ (tq0 + 1) dx = −c7 (tq0 + 1)µ(Ω) ❚ø ✭✷✳✸✶✮✱ t❛ ❝â γ(x)|u|θ dx F (x, u) dx Ω Ω ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✷✳✷✳✸ ✈➔ ✭✷✳✸✹✮✱ t❛ ❝â A(x, ∇u) dx − I(u) = Ω F (x, u) dx + Ω< F (x, u) dx Ω ✹✶ c1 γ(x)|u|θ dx + c7 (tq0 + 1)µ(Ω) ∇u W01,p (Ω) − p Ω p−1 c1 c1 u pW 1,p (Ω) + µ(Ω) p u W01,p (Ω) − γ(x)|u|θ dx p Ω c1 u p W01,p (Ω) + Ω< c1 u + γ(x)|u|θ dx + c7 (tq0 + 1)µ(Ω) p W01,p (Ω) + c5 u W01,p (Ω) γ(x)|u|θ + c6 , − Ω tr♦♥❣ ✤â c6 ❧➔ ❤➡♥❣ sè✳ ❉♦ ✤â ✭✷✳✸✷✮ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ 1,p ❚❤❡♦ ✭✷✳✸✶✮ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✷✳✷✳✸✱ ♥➳✉ u ∈ W0 (Ω) t❤➻ γ(x)|u|θ dx F (x, u) dx Ω c4 u Lq (Ω) µ(Ω) q−1 q + c4 u q q Lq (Ω) < +∞ 1,p ✈➻ ♣❤➨♣ ♥❤ó♥❣ W0 (Ω) → → Lq (Ω) ❧✐➯♥ tư❝ ✈➔ ❝♦♠♣❛❝t✳ ❍➔♠ γ : W01,p (Ω) → R ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ u γ γ(x)|u|θ dx = θ , Ω u ∈ W01,p (Ω) ✭✷✳✸✺✮ 1,p ❧➔ ♠ët ❝❤✉➞♥ tr♦♥❣ W0 (Ω)✱ tr♦♥❣ ✤â γ(.) ❧➔ ❤➔♠ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✸✶✮✳ 1,p ◆❤÷ ✈➟②✱ ♥➳✉ X1 ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❝õ❛ W0 (Ω) t❤➻ ❝→❝ ❝❤✉➞♥ W01,p (Ω) ✈➔ γ ❧➔ tữỡ ữỡ õ tỗ t số c8 = c8 (X1 ) > ♣❤ö t❤✉ë❝ ✈➔♦ X1 s❛♦ ❝❤♦ u ✭✷✳✸✻✮ ∀u ∈ X1 c8 u γ , W01,p (Ω) ❑➳t ❤ñ♣ ✭✷✳✸✷✮ ✈➔ ✭✷✳✸✻✮✱ t❛ ❝â I(u) c1 cp8 u p γ + c5 c8 u γ − u θ γ + c6 ❉♦ ✤â c1 cp8 u p γ + c5 c8 u ✈ỵ✐ ♠å✐ u ∈ S := {u ∈ X1 : I(u) 1,p ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ W0 (Ω)✳ γ − u θ γ + c6 0}✳ ❱➻ θ > p ♥➯♥ s✉② r❛ t➟♣ ❤ñ♣ S ✹✷ ❈❤ù♥❣ ♠✐♥❤ ✣à♥❤ ❧➼ ✷✳✷✳✾✳ ❚ø ❣✐↔ t❤✐➳t ✭❆✻✮ ✈➔ ✭❢✹✮ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤✐➳♠ ❤➔♠ I s✉② r❛ I ❧➔ ♣❤✐➳♠ ❤➔♠ ❝❤➤♥✱ I(0) = 0✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✼✱ I t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ P❛❧❛✐s✲❙♠❛❧❡✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✽✱ tỗ t số , > s ❝❤♦ J(u) α✱ u W01,p (Ω) = ρ✳ ❈✉è✐ ❝ò♥❣✱ t❤❡♦ ❇ê ✤➲ ✷✳✷✳✶✵✱ ✈ỵ✐ ♠é✐ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ X1 ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ W01,p (Ω)✱ t➟♣ ❤đ♣ S := {v ∈ X1 : I(u) 0} ❜à ❝❤➦♥✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✸✳✾✱ ♣❤✐➳♠ ❤➔♠ I ❝â ♠ët ❞➣② ✈ỉ ❤↕♥ ❝→❝ ✤✐➸♠ tỵ✐ ❤↕♥ {un } s❛♦ ❝❤♦ I (un ) = ✈➔ limn→∞ I(un ) = +∞✱ tù❝ ❧➔ ❜➔✐ t♦→♥ ✭✷✳✶✮ ❝â ♠ët ❞➣② ✈æ ❤↕♥ ♥❣❤✐➺♠ ②➳✉ {un } t❤ä❛ ♠➣♥ limn→∞ I(un ) = +∞✳ ✹✸ ❑➌❚ ▲❯❾◆ ❙❛✉ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ t t ữợ sỹ ữợ t t t ❈❤✉♥❣✱ ❝❤ó♥❣ tỉ✐ ✤➣ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ♥❤÷ s❛✉✿ ✶✳ ❚➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ pỗ ổ ởt sè ♥❣✉②➯♥ ❧➼ ❝ì ❜↔♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ♣❤➙♥✳ ✷✳ ❚r➻♥❤ ❜➔② ❧↕✐ ❝❤✐ t✐➳t ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ tr♦♥❣ ❜➔✐ ❜→♦ ❝õ❛ P✳ ❉❡ ◆❛♣♦❧✐ ✈➔ ▼✳❈✳ ▼❛r✐❛♥✐ ❬✶✺❪✱ ✤÷❛ r❛ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ ✈➼ ❞ö ✤â✳ ❚ø ♥❤ú♥❣ ✈➜♥ ✤➲ ✤↕t ✤÷đ❝ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ ❞ü ✤à♥❤ s➩ t✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❜➔✐ t♦→♥ ❜✐➯♥ ❡❧❧✐♣t✐❝ ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤ ✭✷✳✶✮ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ♣❤➙♥✱ ✤➦❝ ❜✐➺t tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤✐➲✉ ❦✐➺♥ ✭❢✷✮ ❦❤ỉ♥❣ t❤ä❛ ♠➣♥ ✭①❡♠ ❬✼✱ ✶✷❪✮✱ ❜➔✐ t♦→♥ ❡❧❧✐♣t✐❝ ❦❤æ♥❣ ✤➲✉ ✭①❡♠ ❬✾❪✮ ❤♦➦❝ ①❡♠ ①➨t ❜➔✐ t♦→♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♠✐➲♥ Ω ❦❤æ♥❣ ❜à ❝❤➦♥✳ ✹✹ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❘✳ ❆❞❛♠s ✭✶✾✼✺✮✱ ❙♦❜♦❧❡✈ ❙♣❛❝❡s✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦✳ ❬✷❪ ❆✳ ❆♠❜r♦s❡tt✐ ❛♥❞ P✳❍✳ ❘❛❜✐♥♦✇✐t③ ✭✶✾✼✸✮✱ ❉✉❛❧ ✈❛r✐❛t✐♦♥❛❧ ♠❡t❤✲ ♦❞s ✐♥ ❝r✐t✐❝❛❧ ♣♦✐♥ts t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳✱ ✵✹✱ ✸✹✾✲✸✽✶✳ ❬✸❪ ❍✳ ❇r❡③✐s ✭✶✾✽✼✮✱ ❆♥❛❧②s❡ ❢♦♥❝t✐♦♥♥❡❧❧❡✱ ❚❤➨♦r✐❡ ❡t ❛♣♣❧✐❝❛t✐♦♥s✱ ▼❛ss♦♥✱ ◆❡✇ ❨♦r❦✳ ❬✹❪ ❆✳ ❈❛♥❛❞❛✱ P✳ ❉r❛❜❡❦ ❛♥❞ ❏✳ ▲✳ ●❛♠❡③ ✭✶✾✾✼✮✱ ❊①✐st❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢♦r s♦♠❡ ♣r♦❜❧❡♠s ✇✐t❤ ♥♦♥❧✐♥❡❛r ❞✐❢❢✉s✐♦♥✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✸✹✾✭✶✵✮✱ ✹✷✸✶✲✹✷✹✾✳ ❊❧❧✐♣t✐❝ ♣r♦❜❧❡♠s ✇✐t❤ ❧❛❝❦ ♦❢ ❝♦♠✲ ♣❛❝t♥❡ss ✈✐❛ ❛ ♥❡✇ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠✱ ❏✳ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❬✺❪ ❙✳ ❈❛r❧ ❛♥❞ ❙✳ ❍❡✐❦❦✐❛ ✭✷✵✵✷✮✱ ✶✽✻✭✶✮✱ ✶✷✷✲✶✹✵✳ ▼♦rs❡ t❤❡♦r② ❢♦r ✐♥❞❡❢✐♥✐t❡ ❬✻❪ ❑✳❈✳ ❈❤❛♥❣ ❛♥❞ ▼✳❨✳ ❏✐❛♥❣ ✭✷✵✵✾✮✱ ♥♦♥❧✐♥❡❛r ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s✱ ❆♥♥✳ ■✳ ❍✳ P♦✐♥❝❛r➨ ✲ ❆◆✱ ✷✻✱ ✶✸✾✲✶✺✽✳ ❖♥ ❛ ♥♦♥❧✐♥❡❛r ❛♥❞ ♥♦♥✲ ❤♦♠♦❣❡♥❡♦✉s ♣r♦❜❧❡♠ ✇✐t❤♦✉t ✭❆✲❘✮ t②♣❡ ❝♦♥❞✐t✐♦♥ ✐♥ ❖r❧✐❝③✲ ❙♦❜♦❧❡✈ s♣❛❝❡s✱ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳✱ ✷✶✾✭✶✹✮✱ ✼✽✷✵✲✼✽✷✾✳ ❬✼❪ ◆✳❚✳ ❈❤✉♥❣ ❛♥❞ ❍✳◗✳ ❚♦❛♥ ✭✷✵✶✸✮✱ ❬✽❪ ◆✳▼✳ ❈❤✉♦♥❣ ❛♥❞ ❚✳❉✳ ❑❡ ✭✷✵✵✹✮✱ ❊①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ❢♦r ❛ ♥♦♥❧✐♥❡❛r ❞❡❣❡♥❡r❛t❡ ❡❧❧✐♣t✐❝ s②st❡♠✱ ❊❧❡❝tr♦♥✐❝ ❏✳ ❉✐❢❢✳ ❊q✉❛✳✱ ❱♦❧✳ ✷✵✵✹✭✾✸✮✱ ✶✲✶✺✳ ✹✺ ❬✾❪ ❉✳▼✳ ❉✉❝ ❛♥❞ ◆✳❚✳ ❱✉ ✭✷✵✵✺✮✱ p✲▲❛♣❧❛❝✐❛♥ ◆♦♥✉♥✐❢♦r♠❧② ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥s ♦❢ t②♣❡✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✭❚▼❆✮✱ ✻✶✱ ✶✹✽✸ ✲ ✶✹✾✺✳ ❬✶✵❪ ❉✳ ●✐❧❜❛r❣ ❛♥❞ ◆✳❙✳ ❚r✉❞✐♥❣❡r ✭✶✾✽✸✮✱ ❊❧❧✐♣t✐❝ ♣❛rt✐❛❧ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ s❡❝♦♥❞ ♦r❞❡r✱ ✷♥❞ ❡❞✳✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥✳ ❬✶✶❪ ▼✳❆✳ ❑r❛s♥♦❡s❧s❦✐✐ ❛♥❞ ❨❛✳ ❇✳ ❘✉t✐❝❦✐✐ ✭✶✾✻✶✮✱ ❛♥❞ ❖r❧✐❝③ s♣❛❝❡s✱ ❈♦♥✈❡① ❢✉♥❝t✐♦♥s P✳ ◆♦♦r❞❤♦❢❢ ▲t❞✳✱ ●r♦♥✐♥❣❡♥✱ ◆❡t❤❡r❧❛♥❞s ✭❚r❛♥s❧❛t❡❞ ❢r♦♠ t❤❡ ❢✐rst ❘✉ss✐❛♥ ❡❞✐t✐♦♥ ❜② ▲✳❋✳ ❇♦r♦♥✮✳ ❬✶✷❪ ❆✳ ❑r✐st❛❧②✱ ❍✳ ▲✐s❡✐ ❛♥❞ ❈✳ ❱❛r❣❛ ✭✷✵✵✽✮✱ p−▲❛♣❧❛❝✐❛♥ t②♣❡ ❡q✉❛t✐♦♥s✱ ▼✉❧t✐♣❧❡ s♦❧✉t✐♦♥s ❢♦r ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✭❚▼❆✮✱ ✻✽✱ ✶✸✼✺✲ ✶✸✽✶✳ ❬✶✸❪ ❏✳ ▼❛✇❤✐♥✱ ●✳ ❉✐♥❝❛ ❛♥❞ P✳ ❏❡❜❡❧❡❛♥ ✭✶✾✾✺✮✱ ❆ r❡s✉❧t ♦❢ ❆♠❜r♦s❡tt✐✲❘❛❜✐♥♦✇✐t③ t②♣❡ ❢♦r p✲▲❛♣❧❛❝✐❛♥✱ ✐♥ ❈✳ ❈♦r❞✉♥❡❛♥✉ ✭❊❞✳✮✿ ◗✉❛❧✐t❛t✐✈❡ ♣r♦❜❧❡♠s ❢♦r ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛♥❞ ❝♦♥tr♦❧ t❤❡♦r② ✱ ❲♦r❧❞ ❙❝✐❡♥t✐❝✱ ❙✐♥❣❛♣♦r❡✱ ✷✸✶✲✷✹✷✳ ❬✶✹❪ ❉✳ ▼♦tr❡❛♥✉✱ ❱✳❱✳ ▼♦tr❡❛♥✉ ❛♥❞ ◆✳ ❙✳ P❛♣❛❣❡♦r❣✐♦✉ ✭✷✵✶✹✮✱ ❚♦♣♦❧♦❣✐❝❛❧ ❛♥❞ ✈❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ♥♦♥❧✐♥❡❛r ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s✱ ❙♣r✐♥❣❡r ◆❡✇ ❨♦r❦ ❍❡✐❞❡❧❜❡r❣ ❉♦r❞r❡❝❤t ▲♦♥❞♦♥✳ ❬✶✺❪ P✳ ❞❡ ◆❛♣♦❧✐ ❛♥❞ ▼✳❈✳ ▼❛r✐❛♥✐ ✭✷✵✵✸✮✱ ▼♦✉♥t❛✐♥ ♣❛ss s♦❧✉t✐♦♥s t♦ ❡q✉❛t✐♦♥s ♦❢ p−▲❛♣❧❛❝✐❛♥ t②♣❡✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✭❚▼❆✮✱ ✺✹✱ ✶✷✵✺✲ ✶✷✶✾✳ ❬✶✻❪ P✳ ❘❛❜✐♥♦✇✐t③ ✭✶✾✽✹✮✱ ▼✐♥✐♠❛① ♠❡t❤♦❞s ✐♥ ❝r✐t✐❝❛❧ ♣♦✐♥t t❤❡♦r② ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❊①♣♦s✐t♦r② ▲❡❝t✉r❡s ❢r♦♠ t❤❡ ❈❇▼❙ ❘❡❣✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ❤❡❧❞ ❛t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ▼✐❛♠✐✱ ❆♠❡r✲ ✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✳ ✹✻ ❬✶✼❪ ▼✳ ❙tr✉✇❡ ✭✷✵✵✽✮✱ ❱❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞s✱ ❱♦❧✳ ✸✹✱ ✹t❤ ❡❞✳✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✳ ❬✶✽❪ ❈✳ ❩❛❧✐♥❡s❝✉ ✭✶✾✽✸✮✱ ❖♥ ✉♥✐❢♦r♠❧② ❝♦♥✈❡① ❢✉♥❝t✐♦♥s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✾✺✱ ✸✹✹✲✸✼✹✳ ... (tq0 + 1)µ(Ω) ∇u W01 ,p (Ω) − p Ω p? ??1 c1 c1 u pW 1 ,p (Ω) + µ(Ω) p u W01 ,p (Ω) − γ(x)|u|θ dx p Ω c1 u p W01 ,p (Ω) + Ω< c1 u + γ(x)|u|θ dx + c7 (tq0 + 1)µ(Ω) p W01 ,p (Ω) + c5 u W01 ,p (Ω) γ(x)|u|θ +... |ξ |p? ??1 tp−1 )|ξ| dt c1 c1 |ξ| + c1 p |ξ| p ✈ỵ✐ ♠å✐ x ∈ Ω ✈➔ ξ ∈ Rd ✳ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎ ♦❧❞❡r✱ s✉② r❛ A(x, ∇u) dx Ω c1 |∇u| + Ω c1 u W01 ,p (Ω) c1 |∇u |p p µ(Ω) p? ??1 p + dx c1 u p p W01 ,p (Ω)... ▲✐♣s❝❤✐t③✱ p < ∞✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ♣❤➨♣ ♥❤ó♥❣ s❛✉✿ dp ✳ ✶✮ ◆➳✉ kp < d t❤➻ W k ,p( Ω) → Lq (Ω)✱ ✈ỵ✐ ♠å✐ q p? ?? = d−kp dp ❍ì♥ ♥ú❛✱ W k ,p( Ω) → → Lq (Ω) ✈ỵ✐ ♠å✐ q < p? ?? = d−kp ✳ ✷✮ ◆➳✉ k = dp t❤➻ W k ,p( Ω)