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(7) FICHA CATALOGRÁFICA ELABORADA PELA BIBLIOTECA DO IMECC DA UNICAMP Bibliotecário: Maria Júlia Milani Rodrigues – CRB8a / 2116. Diniz, Hugo Alex Carneiro D416u. Unicidade e não-degenerescência para problemas envolvendo plaplaciano em anéis / Hugo Alex Carneiro Diniz -- Campinas, [S.P. : s.n.], 2005. Orientador : Djairo Guedes de Figueiredo Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica. 1. Equações diferenciais parciais. 2. Equações diferenciais elípticas. 3. Dirichlet, Problemas de. I. Figueiredo, Djairo Guedes de. II. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. III. Título.. Título em inglês: Uniqueness and nondegeneracy for problems involving p-laplacian in annuli. Palavras-chave em inglês (Keywords): 1. Partial differential equations. 2. Partial differential elliptic. 3. Dirichlet problem. Área de concentração: Análise – Equações Diferenciais Parciais Elípticas Titulação: Doutorado em Matemática Banca examinadora: Prof. Dr. Djairo Guedes de Figueiredo (IMECC-UNICAMP) Prof. Dr. João Marcos Bezerra do Ó (UFPB) Prof. Dr. Olímpio Miyagaki (UFV) Prof. Dr. Pedro Ubilla Lopez (Universidade de Santiago – Chile) Prof. Dr. Orlando Francisco Lopes (IMECC-UNICAMP) Data da defesa: 23/08/2005.
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(9) A você, Euna, meu amor, que me sustentou, apoiou e amou, e ao Vitor, meu tesouro, que sempre subia no meu colo enquanto eu digitava esta tese..
(10) Meu querido mestre e amigo, Jesus, o Cristo de Israel, que sempre foi e será fiel para comigo. Seu cuidado e direção foram imprescindíveis nesta jornada. Bendito é o que vem em nome do Senhor. Euna, você a cada dia me faz querer ser um homem melhor. Vitor, tudo o que tenho é seu. Todos os dias lhe darei o meu melhor. Pedro e Ruth, pai e mãe, eu os amo profundamente e reconheço toda dedicação e investimento em minha vida. Pastores, irmãos e igreja, recebi o sustento enviado por vocês. Vocês sempre estiveram comigo. Professor Djairo, sua atenção, conselho e exemplo, eu os levarei com muito carinho. Meus queridos amigos, obrigado por confiarem em mim e deixarem que eu participasse de suas vidas. Professores e funcionários do IMECC, seu trabalho e dedicação são de excelência..
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(140) &G/E&DTM B*D /DT:M +E ! M > E >0 ;b
(141) M!DTM!P/EVP/B *:- fg;767)5 f /n a +6 6 + ILB*KSB*DFE&H
(142) *E&DFE DFEP/ B ! E P/B9G O! E ! G/BM! K P M!P/B M /G/NO!E f (r, s) = a(r)s q > p − 1 > HKSB*H !?I M!P/E&HTP/B*HJILB
(143) )M 9??I ! E#B*HJILB*G/P/B*D E&H KSB*H !?I M!P/E&H B* D 8M!G $ Z!k 9 M!KLM E
(144) M!HSE H 9+B* K ! G/BM!KWB 9+E& H ?6I 87&E !/BHLO!E KSB*HJIL K ?ILE&H M!E ! )M 9! )M
(145) M!G/E
(146) *E&D f M I
(147) &G/E&DTM B*D /DTM +E ! M > B*G/B*KL)M ! M!DFE&H I M!"D *D E&HTKSB*H !?I M!P/E&HFB*D I )M ! E&G )M
(148) * B !! M = 9+ E %H 9/KSE 7!M!DFE&HM /G
(149) P M!P/B G
(150) !/ H 87&B 9 M!KLM4E
(151) M!HSE G O! E M I
(152) &G/E&DFE B
(153) *E&G/ H P/B*KLM!DFE&H E 9/KS E ! B*DTM B*DM!G H*> E >0 ;b <
(154) M!DTM!P/EUP/B *]:- fg;767)5 f /9 &+ a / "# *1 6 / G/E&HSHSE DFEP/ B ! 'E 9 M!KLM MG O! E ! G/BM! K P M!P/BWHSB*K M /G/NO!E f (r, s) = s − a(r)s − C q > m ≥ p − 1 > !/B*HJI O!E P/B /G
(155) P M!P/BWB G O! E P/ B &B*G/B*KSB* H
(156) *G
(157) M 9 M!KLMTB*HJILB 6I 9+ETP/B G O! E ! G/BM! K P M!P/ B 1G/E
(158) M!HSE 9 M!KJ6I
(159) ! M!K P/E ! )M 9! )M
(160) M!G/E f M I
(161) &G/E&DTMXB f (0) = 0 ! B*HJI /P M!P/E B*D E&G M!G = @ > B*HJILB
(162) )M 9??I ! E fB*HJILB*G/P/B*DFE&H E&H KSB*H !?I M!P/E&H B* D 8M!G $ Z!k 9 M!KLM "E
(163) M!HSE H 9+B* K ! G/BM!K B 9+E& H ?6I 87&E !/B HLO!ETKSB*HJIL K ?ILE&HCM!%E ! )M 9! )M
(164) M!G/:E
(165) *E&D f M I
(166) &G/E&DTMRB*D /DT%M +E ! M > B*G/B*KL)M ! M!DFE&H I M!"D *DE M!KJ6I &E P/B I )M ! E&G )M
(167) * B !! M = 9/KSE 7!M!G/P/E -M /G
(168) P M!P/B 9 M!KLM E
(169) M!HSE G O! E M I
(170) &G/E&DFE 9+B*KS%D ?6I G/P/E %. 0. q. . m. !. q. . !. . . q. m. . . !. . . . . !. q.
(171) =.
(172) . HSB LMRDTM) E&K !/B p − 1 9+B*KSD%?I6 G/P/E E
(173) M!HSE * 6 +6 +)5 . f (r , 0) < 0/ B
(174) *E&G/H P/B*KLM!G/P/ERE%9/KSE ! B*DTM B*DM!G H*> E >0 ;b + 9/KSE 7!M!DFE&H /D KSB*H!?I M!P/E#P/B%/G
(175) P M!P/B:9 M!KLM E ! M)9! M)
(176) M!G/EB*DP/E D%? G E&H 9/K1 = DFE&H 'M /DTM +E ! M .bB*D /D HSB*G8I6 P/EM)9/KSE 9/K M!P/E!/ ! M!G/P/E f WH9+B*K ! G/BM!K*> !. &B*G/B*KLM)! M!DFE&HFE !/B b E 9/KSE 7!M!P/E B*D #E )Z ( 9 M!KLM E
(177) M!HSE 9 M!KJ6I
(178) ! M!KFB*D !/B f /DTM 9+E!6I *G
(179) M > E ),67 & DFE&HJILKLM!KSB*DFE&H !/ B ! M!G/P/' E 9/KSE
(180) /KLM!DFE&HC)M 9+B*G M!H HS E !/ N B*H KLM!P )M H E 9/KS E ! B*DTM (P ) KSB*P HSBWM (P ) > E ),67 & B B*HJI /P M!DFE&HUM B = HJ6I *G
(181) M B /G
(182) P M!P/B9P/E 9/KS E ! B*DTMP/B & M
(183) M!HSHSE
(184) M!P/ERM (P ) !/BWM:9+E!I6*G
(185) M. m. 0. . . !. 0. 0. P/E. 0 − r N −1 |u0 |p−2 u0 = r N −1 f (r, u), r ∈ (r0 , R) u(r0 ) = α u0 (r ) = 0.. E ),67 & ILKLM) )M !/E/>. 0. B*G!/G
(186) M!KSB*DFE&H ILB*E&KSB*DTM!H M = ! M!KSB*H, !/B HLO!E" D%9+E&KJI M!G8ILB*H M!E"! E&G &E.
(187) . . . . $B LM!D. N ≥2. B. - . . . > M!KLM.
(188) *E&G/H P/B*KSB*DFE&H ERP/E&D%? G E. p>1 0 ≤ r0 < R < ∞ {x ∈ RN ; r < |x| < R} 0 Br0 ,R = {x ∈ RN ; |x| < R}. HSB r > 0 HSB r = 0. $ 9+E&G M!DFE&H !/B u ∈ C (B ) /DTM HSE !/NO!ERKLM!P M)! DFB*G8ILBCH D' ILK
(189) M /P E%9/KSE ! B*DTM B*D B −4 u = f (|x|, u) 1. 0. r0 ,R. p. E&G/P/BWM /G/NO!E. 0. r0 ,R. u>0 ∂u (x) = 0 ∂ν u(x) = 0. M!KLM 9 M!KLM 9. |x| = r0 ,. HSB. . =&>?= /. r0 > 0. HLMgI6 H7M !H9 1!ILB*HSB*H 9+E&HSH -P/B* K 87!M!P M B*D r
(190) *E&G8I6? G! MRB f (r, s) ≤ 0 ∀s ≥ 0
(191) (M ) f ∈ C ([0, ∞) × [0, ∞)) $B 9 M!KLMFM)! #/D (r, s) ∈ [r , R] × [0, sup u] ILB*DFE&H f (r, s) = 0 /B*G8I O!E F (r, s) ≤ 0 /E&G/P/B (M ) R > F (r, s) := f (r, t) dt 1. |x| = R,. f (r, s). 0. r. 0. 2. s 0.
(192) = j.
(193)
(194) . @ *B*DFE&H !/B. u ∈ C 1 (B r0 ,R ). HLMgI6 H7M RB ! M!NO!E −4p u = f (|x|, u). HSB. Z. |∇u|. p−2. ∇u · ∇ϕ dx =. Z. B*D. Br0 ,R. f (|x|, u)ϕ dx,. ∀ϕ ∈ C01 (Br0 ,R ).. KSE 7!M!KSB*DFE&H E&H HSB # G8ILB*H KSB*H!?I M!P/E&H 0 #" f $&%')(*$+%-, (M ) "/. '10324'52&67%8$-29;:<0324=>%76()%9?6 " . =&>?= /A@ f1 0 7! . Br0 ,R. 1. E = " $ E /" . ' "&F. "/.. f1 0 ` '1032. Br0 ,R. u. " @ : BA%G$-29;:<032G=>%76()%9H6 /" . " " " "/. &" F @ =>%76()%9;B ' 6 E = $ E ' #". u ∈ C 1 (B r0 ,R ). M!KLM&/DTM DFB !/E&K 7 H M)! M!NO!E P M&9 1!ILB*HSB B = B*D%9! E&H P/"B /G/ N B*H f !/BWM HLMgI6 H7M *B*D& . f. (M2 ). . =&>?= /. f. $&%')(*$+%-,. "/. C " . ' 032D. (M1 ) − (M2 ). E&H HSB # G8ILB*H & K +
(195) *E&HWDFE&HJILKLM!D. !KSBM. f. s. ". =>%76()%9;B. f. s. I1 ≥. !KSBM. I2. LMJ LMJ LMJ LMJ MJ MJ MJ MLML LMLJMJ LMJ L L LMJ LMJ LL MLML MJ MJ LMJ LMJ MLJ I2MJ IKJ I I I I KJ J K J K J K K L L L L ML J M MJ IKJ IKJ IKJ KJ KIJ KJ KIKI IKJ I I I J K IKIJKJ IKJ II KJ II KIKI KJ IKJ I 1 KJ IKJ IKJ IKJ KIJ KJ KJ KIKI IKIJKJ IKJ IKJ IKJ KJ I I I I KJ I KJ I KIKI KIJKJ. s. M!G/P/ECKSB*HJILK ?ILE M HSE !/N B*HKLM!P M) H E 9/KSE ! B*DTM. =&>?= /-KSB*P HSB M!ECHSB # G8ILB /9 KSE ! B*DTM !/BXB*G 7&E !87&B / DTM B ! M!NO!EFPbB*KSB*G
(196) M)! E&KSP G !K M .7&B*K 9A*G/P
(197) *B "/ 0 − r N −1 |u0 (r)|p−2 u0 (r) = r N −1 f (r, u(r)), u>0 u0 (r ) = u(R) = 0, 0. r ∈ (r0 , R). . =&>[Z#/.
(198)
(199)
(200) . E&G/P/B. . .
(201) . . r = |x|. . . = Y. 84
(202) . . > 4B ! M!NO!ETB*D . =&>[Z#/eI M!D" *D +9 EP/B HSB*K B*H
(203) *K ?I M:
(204) *E&DFE 0 N − 1 0 p−2 0 |u0 |p−2 u0 + |u | u + f (r, u) = 0, r. . =&> k /. r ∈ (r0 , R).. !M KLM E:
(205) M!HSE 9 M!KJI6
(206) ! M!K B*D !/BXE P/E&D%? G E "/DTM"+E ! M . r = 0/ /B = HJILB*D KSB*H!?I M!P/E&H !/B P O!E
(207) *E&G/P N B*H H
(208) B*G8ILB*H'9 M!KLM !/B -,5 HSE !/NO!E P/B . =&>?= / HSB LMdKLM!P M)! BUKLM!P M)! DFB*G8ILB P/B
(209) *KSB*H
(210) *B*G8ILB!> EP/B*DFE&H
(211) ?I M!K5E&H5ILKLM) M)!/E&H5P/B P M!H KSB*G+B*K = . 9 M!KLM p = 2/ KSE
(212) . 9 M!KLM p > 1/ B*HLM 7!M!G )M
(213) * B !! M = ( . 9 M!KLM p = N / B @ M!DTM!H
(214) *B !! M)
(215) *B !! M Y ( . 9 M!KLM 1 < p < /
(216) SE&H5B*!G /G
(217) M!P/E&HB*G
(218) *E&G8ILKLM!:D HSBeG/E 9A*G/P
(219) *B &C> %D 9+E&KJI M!G8ILB G/E!I M!K !/BeE&H5QB*E&KSB*DTM!He=&>?= 2 B =&>[Z G O!E 9/KSE 7!M!D !/0B /DTM P/BILB*KS%D G M!P M HS E !/NO!E eKLM!P )M ! DTM!H5 H D 9/KSE 7!M!D4M DFE&G/E!ILE&G
(220) P M!P/B P/B /DTMTHS E !/NO!ETKLM!P )M !5P/E 9/KS E ! B*DTM . =&>?= / 9 M!KLM /DT%M
(221) ! M!HSHSB P/B /G/ N B*H f +G O!E G
(222) !? P MRG/E&H ILKL)M )M !/E&H
(223) ?I M!P/E& H /B I M!"D *D 9 M!KLM E
(224) M!HSEFB*D !/BXEFP/E&%D ? G E "/DM!G/ B !0. r > 0/ > M!HSBM!DFE&H M!H P M!H P/B*HJIL"B
(225) )M 9??I ! ETG/E&H M!KJ6I &E&H P/B KLM!G
(226) 5M!G
(227) *E&G/ B !! $B*KS K G = k B $B*KS K G Q M!G Z fE&G/P/B B*G
(228) *E&G8ILKLM!DFE& H 9 M!KLM E 9/KS E ! B*DTM M I
(229) &G/E&DF'E . f G O!EXP/ B 9+B*G/P/B P/B r / /D KSB*H !?I M!P/E P/B DFE&G/E!ILE&G
(230) P M!P/B 9 M!KLM HS E !/ N B*H KLM!P )M HP/E 6I 9+E &KS#E /G/P HJI MgILB +) HJILE u(r) → 0 ! M!G/P/E r → ∞ > . 0. . . . $ . . " !. . . . 0. . . !. . .
(231) . .
(232) !#"$#%&'!() *
(233) . $B LM. /DTMdHSE !/NO!E P/E49/KSE ! B*DTMdB*D . =&>[Z#/ > /B*KSB*DFE&HTP/B*DFE&G/HJILKLM!K !/B u (r) ≤ 0 B*D [r , R] > HFHSB # G8ILB*H%! B*DTM!H ILKLM *B*D GbE&KSDTM!N B*H" D%9+E&KJI M!G8ILB*HRHSE /KSB#E
(234) *E&D%9+E&KJI M!DFB*G8ILE P/B*HSHLMRHSE !/NO!E M!H ! M) H HSB*KLO!E ILB H G M P/B*DFE&G/HJILKLM!NO!EFP/E ILB*E&KSB*DTM > u ∈ C 1 [r0 , R]. 0. c 0 i. 0. #",+ %. ψ(r) := |u0 (r)|p−2 u0 (r). F'- .. '1032. ψ ∈ C 1 [r0 , R]. −f (r0 , u(r0 )) 0 ψ (r0 ) = − 1 f (0, u(0)) N. " $. $ ". ". r0 > 0, r0 = 0..
(235) =(.
(236)
(237) . E&K. =&>[Z#//ILB*DFE&H !/B.
(238) . −(r N −1 ψ(r))0 = r N −1 f (r, u(r)),. CHSH D&9+B ! M
(239) *E&G8I6 G! P. . =&> !/. r ∈ (r0 , R).. M!P/BXP/B f ψ B = HJILBWB
(240) *E&G8I6? G! MFB*D r 1HSB r > 0 B r ∈ [r , R] > M!KLMFE
(241) M!HSE B*D !/B r > 0 7M *B*G/P/E r → r B*D . =&> k / fE ILB*DFE&HeMWB = 9/KSB*HSHLO!E P/B ψ (r ) B ILB*KSD% G M!DFE&H MTP/B*DFE&G/HJILKLM!NO!E/> M! H HSB r = 0 B*G8I O!E 9/KSB
(242) HLM!DFE&H 9/KSE 7!M!K MTB = HJI6*G
(243) MP/B ψ (0) > HJILE E !/B"7M!KSB*DFE&H M HS B # K*> E&K . =&> !/ )M 9!
(244) M!G/P/ERM B &KLMRP/B !
(245) 9?I )M !f E ILB*DFE&H 0. 0. 0. 0. 0. %. 0. 0. &. E&DFE. ψ(0) = 0. 9+E H. −. 0. (r N −1 ψ(r))0 ψ(r) 1 f (0, u(0)) = lim = lim . N 0 r→0 r→0 r N (r ). u0 (0) = 0.
(246) *E&G
(247) !? DFE&H. ψ 0 (0) = −. 1 f (0, u(0)). N . c 0 . . "C"/." + : < 03286 = ()%. E(r) :=. p−1 0 |u (r)|p + F (r, u(r)) p. E 0 (r) = Fr (r, u(r)) −. @ .. " /" . " 032D E = $ E ' E 2B. N −1 0 |u (r)|p ≤ 0. r. H *KSB &B*G/P/EUM /G/NO!E E G M bE&KSDTM.
(248) 5
(249) 7. 7&B*DFE&H 9+B ! E -B*DTMU=&> k !/B. E(r) =. p p − 1
(250)
(251) 0 p−2 0
(252)
(253) p−1 |u | u + F (r, u), p. E ∈ C 1 [r0 , R]. > QB*DFE&H !/B. 0 E 0 (r) = u0 |u0 |p−2 u0 + Fr (r, u) + f (r, u)u0 h i 0 = Fr (r, u) + u0 |u0 |p−2 u0 + f (r, u). & E&G
(254) !? DFE&H, !/B. N −1 0 p |u | r Z s Fr (r, s) = fr (r, t) dt. E 0 (r) = Fr (r, u) − E 0 (r) ≤ 0. 9+E H. 0. B. 9+E&K. . =&> k /. fr (r, u) ≤ 0. . 9 1!ILB*HSB. (M1 ). />. .
(255)
(256)
(257) . . . c 0 <.
(258) . . #".
(259) . I M) H !/B. . . u0 (r1 ) = 0. =@. 84
(260) . . . %=>%8%9 : B. r1 ∈ [r0 , R). "/.. '1032. $ 9+E&G M!DFE&H !/B E"! B*DTM 7M)! HSE/> KSDTM!DFE&H0 !/B B = HJILB*D u0 (r1 ) = 0, u(r1 ) = u(r2 ), u(r) ≥ u(r1 ). B. u(r1 ) ≥ u(r). 9. M!KLM r B*D. %=>% '52&632. r ≥ r1. F. r1 < r2 ∈ [r0 , R). . =&> #/. [r1 , r2 ]. G O!E%
(261) *E&G/HJI M!G8ILBWB*D [r , r ]. . =&> j / @CB7MgILE
(262) *E&DFEE ! B*DTM 7M)! HSE M /G/NO!E u(r) 9+E&HSH /D 9+E&G8ILE
(263) *K ??I6
(264) *E r¯ ∈ [r , R) B B = HJILB I M) H- !/B u(¯r ) < u(¯r ) > $B LM r ≥ r¯ E 9+E&G8ILE
(265) *K ??I6
(266) *E !/B) EUD%? G DFEFP/B u(r) r¯ ∈ (r , R) B*D [¯r , r¯ ] > & E&DFE u(r ) > 0 B u(R) = 0 B = HJILB r ∈ (¯r , R) HLMg6I H 7M *B*G/P/E . =&> #/ B . =&> j / > u. 1. 2. 1. 2. 1. 1. 1. 2. 2. 1. 1. 0. 1. 2. 2. u(r). B ! E%-B*DTMU=&>. Z. ILB*DFE&H !/B. E(r2 ) − E(r1 ) =. G8I O!E 5. r1. r2. Fr (t, u(t)) − r1. r2. Z. r. R. r2. Fr (t, u(t)) − r1. N −1 0 p |u (t)| dt. t. N −1 0 p p−1 0 |u (t)| dt = |u (r2 )|p + F (r2 , u(r2 )) − F (r1 , u(r1 )) t p Z r2 p−1 0 p = |u (r2 )| + Fr (t, u(r1 )) dt. p r1. -E &E/HLM!G/P/ETERQB*E&KSB*DTM. p−1 0 |u (r2 )|p + p. P/E eM)! E&K *P E #. Z. r2 r1. %. N −1 0 p |u (t)| dt = t =. Z. r2. Fr (t, u(t)) − Fr (t, u(r1 )) dt. Zr1r2 r1. fr (t, ξ(t))[u(t) − u(r1 )] dt,.
(267) !Z _
(268) *E&D.
(269)
(270) . E&K. u(r1 ) ≤ ξ(t) ≤ u(t). p−1 0 |u (r2 )|p + p. Z. r2. (M1 ). B . =&> #/. N −1 0 p |u (t)| dt = t. Z. r2. fr (t, ξ(t))[u(t) − u(r1 )] dt ≤ 0.. @ )M ? E ILB*DFE&H !/B u ≡ 0 B*D [r , r ] /E' !/B
(271) *E&G8ILKLM!P . =&> j / > .U +) $&./ ,5 f1 0 70 $B u (r) > 0 9 M!KLM#M)! #/D r ∈ (r , R) 5B*G8I O!E B = HJILB /D 9+E&G8ILE
(272) * K ??6I
(273) *E r ∈ [r , r)
(274) *E&D u(r ) < u(r)
(275) *E&G8ILKLM!K M!G/P/E E'-B*DTMU=&> > r1. 0. 1. r1. 2. 0. 1.
(276) . 0. 1.
(277) !#"$)%&'!() . H /9 K1 = DFE&H P/E H ! B*DTM!H. c 0 +.
(278) . B*HJI M)+B ! B
(279) *B*DE *
(280) E&D%9+E&KJI M!DFB*G8ILE P/B u /9 K1 = DFE P/B r > 0. F (r0 , u(r0 )) > 0. QB*DFE&H !/B.
(281) . &. 0. E(R) =. p−1 0 p−1 0 |u (R)|p + F (R, u(R)) = |u (R)|p ≥ 0. p p. E(r0 ) =. p−1 0 |u (r0 )|p + F (r0 , u(r0 )) = F (r0 , u(r0 )) ≥ 0. p. E&DFE E XG O!E
(282) *KSB*H
(283) *B*G8ILB 9+B ! E%-B*DTM=&> B*G8I O!E. KSDTM!DFE&H !/B F (r , u(r )) > 0 > @CB 7MgILE8HSB E(r ) = F (r , u(r )) = 0 B*G8I O!E E
(284) *E&G/HJI M!G8ILB B*D [r , R] B E ≡ 0 > HJILE /DTM
(285) *E&G8ILKLM!P NO!E 9+E H D%9!
(286) M 9+B ! E -B*DTM=&> 9B (M ) !/B > u ≡0 . 0. 0. 0. 0. 0. 0. 1. 0. c 0 . 0. f (r0 , u(r0 )) > 0.
(287)
(288) . .
(289) . . . $B.
(290) . CHSH D. . . Z=. 84
(291) . . f (r0 , u(r0 )) < 0. B*G8I O!E 9+B ! E%-B*DTM=&> k.
(292)
(293) d 0 p−2 0 (|u (r)| u (r))
(294)
(295) > 0. dr r=r0. %/D +9 E&G8ILE#P/BRD%? G DFE ! E
(296) M)! B*HJILK ?ILE P/B u(r) $
(297) *E&G8ILKLM!K M!G/P/E E -B*DTM =&> > &E&KLM SH B f (r , u(r )) = 0 B*G8I O!E 9+B ! M% 9 1!ILB*HSB (M ) F (r , u(r )) ≤ 0
(298) *E&G8ILKLM!K M!G/P/EFE -B*DTM =&> j > r0. > f (r , u(r )) > 0 .U +) $&./ ,5 f1 0 ` B ! E QB*E&KSB*DTM =&>?=# u (r) ≤ 0 B*D [r , R] > B ! E&H -B*DTM!HW=&> kTBT=&>[Y u (r) < 0 9/K 1 = DFETM r > HSB # K DFE&HJILKLM!DFE&H !/B P/B7MgILE u (r) < 0 9 M!KLM ILEP/E r ∈ (r , R) > $ 9+E&G M!DFE&H 9+E&K
(299) *E&G8ILKLM!P NO!E' !/B r WE'9/K DFB KSE:9+E&G8ILE'
(300) *K ??I6
(301) *ERP/B B*D > QB*DFE&H IL K *H 9+E&HS H ! P M!P/B* H (r , R) $B f (r , u(r )) > 0 B*G8I O!E 9+E&K . =&>[#Z / r D = DFE ! E
(302) )M ! B*HJIL K ?ILE .
(303) *E&G8ILKLM!P NO!4E
(304) *E&D • -B*DTM=&> #/
(305) $B f (r , u(r )) < 0 B*G8I O!E 9+E&K . =&>[#Z / r 9%D ? G DFE ! E
(306) )M ! B*HJIL K ?ILE .
(307) *E&G8ILKLM!P NO!1E
(308) *E&D • -B*DTM=&> #/
(309) B*G8I O!E 9+ B ! )M 9 1!ILB*HSB (M ) • Sef (r , u(r )) = 0 -E &E. 0. 0. 0. 2. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 1. 1. 1. 1. 1. 1. 1. 1. 2. E(r1 ) =. p−1 0 |u (r1 )|p + F (R, u(r1 )) = F (r1 , u(r1 )) ≤ 0, p.
(310) *E&G8ILKLM!P *B*G/P/ERE'-B*DTM=&> 9+E H. E(R) ≥ 0. B. E 0 6≡ 0. B*D. [r1 , R]. >. .
(311) . . # $B LM!D. HSE !/NO!ERP/B. . . . N ≥2 p>1. B. . 0 ≤ r0 < R < ∞. > $ 9+E&G M!DFE&H !/B. 0 N −1 0 p−2 0 − r |u | u (r) = r N −1 f (r, u), u>0 u0 (r ) = u(R) = 0,. u ∈ C 1 [r0 , R]. r ∈ (r0 , R). /DTM . Z>?= /. &E G/P/B M /G/NO!E f HLMgI6 H7M !H 9 1!ILB*HSB*H (M ) − (M ) .7&B*K& M)9??I! ET= / > ! *D P HSHSE 7M!KSB*DFE&H P/E H PbB*KSB*G8ILB*H
(312) *E&G/G8ILE&H P/B9 1!ILB*HSB*H 9 M!KLM f > K DFB KLM!DFB*G8ILB )
(313) *E&G/H P/B*KLM!DFE&H E
(314) M!HSEXB*D !/B /H 9+E&G/P/'E !/B f (R, 0) ≥ 0 0. 1. 2. BWB = HJILB*D C, s > 0 I M) H !/B f (r, s) > −Cs 9 M!KLM 0 < s < s /GbE&KSDFB*DFB*G8ILB 9 M!KLM r 9/K 1 = DFE P/B R
(315) 5>= HJILB 0 ≤ γ < 1 I M)! !/B s f (r, s) ! % D ?I M!P M 9 M!KLM s 9/K1 = DFE P/B 0 B /GbE&KSDFB*DFB*G8ILB (H ) 9 M!KLM r 9/K1 = DFE P/B R E&G/P/B f (r, s) ≡ f (r, s) > . (H1 ) f ∈ C 1 ([0, ∞) × (0, ∞)) f (R, 0) ≥ 0. 0. 0. 2. γ. 0. 0. s. p−1.
(316) Z. (H1 ). 8M!KLM!G8ILB. !/B. /!>- . A
(317) *E&DFE &7 B*KSB*DFE&HCDTM) H M!P M!G8ILBWG M $B*NO!E Z>?=&> B*HJILB
(318) M!HSE9+B*KSD%?I6 DFE&H !/BFM /G/NO!E f (r, u(r)) HSB LM ! D%?I M!P M:9 M!KLM r 9/K1 = DFEUP/B R ! D%?I M!G/P/E M)9+B*G M!H E HSB
(319) *KSB*H
(320) DFB*G8ILE:9+B ! M 9 1!ILB*HSB (H ) > H HSB # G8ILB*H B = B*D%9! E&H P/B- / G/N B*H
(321) /D%9/KSB*D M!H9 1!ILB*HSB*H (M ) − (M ) B (H ) − (H )
(322) *E&D a ∈ C a ≥ 0 a ≤ 0 q > 0 B C ≥ 0
(323) • f (r, s) = a(r)s + C
(324) *E&D a ∈ C a ≥ 0 B p − 1 ≤ m < q > • f (r, s) = s − a(r)s !?ILB*KSG MgI687!M!DFB*G8ILB 7 HLM!G/P/ETB*HJI/ P M!K 9/KSE ! B*DTM!H P/ERI69+E * 6 +6 +)5 M!P/D%?I6 KSB*DFE&H !/ B f HLMgI6 H7 M 9. 1!ILB*HSB. 8. . u0 (R) < 0 0. 2. 1. 2. 1. 1. q. q. 2. 0. 1. m. (H01 ) f ∈ C 1 ([0, ∞) × (0, ∞)). B. 0. f (R, 0) < 0.
(325). ! %D ?I M!P M 9 !M KLM s /9 K1 = DFE P/B 0 B /GbE&KSDFB*DFB*G8ILB 9 M!KLM r 9/K1 = DFERP/B R > B*HJILB
(326) M!HSE 9+EP/B*DFE&H ILB*K u (R) = 0 > EdB*G8I M!G8ILE M / G/NO!E f (r, u(r)) P/B 7&B#HSB*K ! D%?I M!P M 9 M!KLM r 9/K1 = DFERP/B R > & ?I M!DFE&H
(327) *E&DFERB = B*D%9! E&H P/B" / G/N B*H !/ BXHLMgI6 H7 M * B*D (M ) − (M ) B (H ) − (H )
(328) *E&D a ∈ C a ≥ 0 a ≤ 0 q > 1 B C > 0
(329) • f (r, s) = a(r)s − C
(330) *E&D a ∈ C a ≥ 0 1 ≤ m < q B C > 0 > • f (r, s) = s − a(r)s − C B*HJILB
(331) M)9??I ! E B*HJI/ P M!KSB*DFE&H ERHSB # G8ILB 9/KSE ! B*DTM P/B 7!M)! E&K G
(332) M)! (H02 ) f 0 (r, s). 0. 0. 1. 0 1. 2. 0 2. 1. q. q. m. 0. 1. 0. 0 −(p − 1) r N −1 |u0 |p−2 ω 0 (r) = r N −1 f 0 (r, u)ω(r), r ∈ (r0 , R) . . Z>[Z#/. −(p − 1) r N −1 |u0 |p−2 ω 0 (r)0 = r N −1 f 0 (r, u)ω(r), r ∈ (r , R) 0 ω 0 (r ) = ω(R) = 0. . Z> k /. ω(r0 ) = 1. ω 0 (r ) = 0, 0. !/BXB*HJI FM!HSHSE
(333) M!P/EFM!E ;b ;767)5 f /g6bh8 ,5 P/B . Z>?= / 0.
(334) Z @ KSB*DFE&H, !/BCMWHSE !/NO!E u )5./2 ,5 *1 )5 f ,5 HSBCE 9/KSE ! B*DTM'. Z> k / 9+E&HSH )5 MXHSE !/NO!E ILK 87 M)! . ω ≡ 0/ > 9/K DFB KSE ILB*E&KSB*DTM) /D KSB*H!?I M!P/ERP/BXB = HJI6*G
(335) MRB"/G
(336) P M!P/B!> 0 : 2 . %B 2-$: " f $&%')(*$+%-,$ ( ' " $ " $ (M ) − (M ) " (H ) − (H ) 2: f1 ` 7! F #" " . Z>?= / F . " " . (H ) − (H ) % u ∈ C [r , R] : BA%A$-29;:<03286 '1032 (*$/' : BA%
(337) ( E %A$-29;:<032 " . Z>[#Z / F ω ∈ C [r , R) ∩ C [r , R] 632 = 29 BA% 9/K 1 = DFETILB*E&KSB*DTMTB*HJI )M +B ! B
(338) *B MTP/ B 9+B*G/P *G
(339) M
(340) *E&G86I ? !G MTP/B u B*D KS B ! M!NO!EM!E 9 M!K DFBILKS:E G
(341) )M ! α = u(r ) > . " " . Z>?= / . " . f1 ` ` @ : 2 %B 2-$: u ∈ C [r , R] : BA% $-29;:<032 6 2 6 %4+ : <032 1. 0 1. ,+. 0 2. 1. 1. 0. 0. 2. 1. 2. -. 0. 0. 0. 1. " " f $&%')(*$+%-,$ ( ' $ $ F- . α = u(r0 ) '1032 %=>% . 0. " F #",+ (M1 ) − (M2 ) (H1 ) − (H2 ) 2: (H01 ) − (H02 ) %B / " . " / " . " " / " . " " . h $/: E ( ' B ' : 2 2 = 29 BA%86 %9*2= ( ( E )( %9 0 − r N −1 |u0h |p−2 u0h (r) = r N −1 f (r, uh ) . . Z> !/. uh (r0 ) = α + h. u0 (r ) = 0. h 0. . " . . . " 2-$$/: ( : BA%
(342) ( E %8$-29;:<032 uh 6 ()67% 4 2 ( ' = % 9*2 " " F- . [r0 , Rh ) $ Rh < R uh(Rh ) = 0 %( 76 % " 0 |uh (r) − u(r)| < . %=>%A'52&632. r ∈ [r0 , Rh ]. [r0 , Rh ]. . " 2 6. ". > 0. . Rh ≤ R uh (r) > 0. " B. |uh (r) − u0 (r)| < ,. F. !?I6 DFERILB*E&KSB*DTM "
(343) M &7 B%9. M!KLMME ILB*G/NO!E P/E&H KSB*H!?I M!P/E&H /P B)/G
(344) P M!P/B G/E&H 9/K1 = DFE&H
(345) M)9??I! &E H*> KSE 7!M!KSB*DFE&H- !/B M' /G/NO!E ω MTP/B*K 87!M!P MTP/B u B*D SK B ! M!NO!EUM!E 9 M!K!DFBILKSE G
(346) M)! α = u(r ) > " " " " F . " "/. " f1 ` i 2A%$ ( ' $ $ 632 2= BA% $/: 2 %B 2-$: %=>% h > 0 $/: E ( ' D B. 0. "/. " " /" . ' : 2. Rh = R. F- .. . '10328% + : < 032. Φh (r) :=. uh (r) − u(r) h.
(347) Z!j. . E 2 . " = " : . ( +&2= B " " @ 67%767% 9*2 2= . . " /" . " B ' % =>% " F7F BA % . ω(r). " B. [r0 , R] :%. .. 632. h. . .
(348) !#"$)%&'!() . '. 8. . /!>-. "/. " " . " . 6 %G, = 2 2 6 % + : < 032. . ω(r). E!ILB*DFE&H !/BUE /9 KSE ! B*DTM . Z>[Z#/ B !87!M)! B*G8ILB M9B*G
(349) *E&G8ILKLM!K)/D +9 E&G8ILE = E 9 !M KLMdM /G/NO!E Ψ : C [r , R] → C [r , R] P M!P M 9+E&K Z Z 1 t . Z> #/ Ψ(ω)(r) = 1 − f (t, u(t))ω(t) dt ds. (p − 1)|u (s)| s @ 87 P KSB*DFE&H MRP/B*DFE&G/HJILKLM!NO!ERG/E&H HS B # G8ILB*H 9 M!HSHSE& H a * 0 /G/NO!E Ψ B*HJI +B*D P/B G P M
(350) a * ` Ψ /DT"M
(351) *E&G8ILKLM!NO!E 9 M!KLM n H
(352) B*G8ILB*DFB*G8ILB &KLM!G/P/B
(353) a * i 5>= HJILB /D /G
(354) *E 9+E&G8IL)E = %E 9 M!KLM Ψ > K DF B KLM!DFB*G8IL B !7M!KSB*DFE&H M!H HS B # G8ILB*H E /HSB*K 7!M! N B* H CH 9 1!ILB*HSB*H (M ) − (M ) 8M!KLM!G8ILB*& D 9+ B ! E -B*DTMU=&>[Y BXERQB*E&KSB*DTM =&>[Z !/B • B B*D (r , R). . Z> j / f (r , u(r )) > 0 u (r) < 0 $B f HLMg6I H 7 M -9 1!ILB*HSB (H ) &B*G8I O!E 9+ B ! E KSB*H !?I M!P/E P/B e M !/ B Z&Y .7&B*K 9A*G/P
(355) *B &/ • ILB*DFE&H u (R) < 0 > & E&D B b B ?ILE 9 M!KLM r 9/K 1 = DFE P/B R /ILB*DFE&H . 0. 0. 0. 0. r. N −1. s. 0. 0. r0. p−2. r0. n. 1. 2. 0. 0. 0. 0. #. $. . 1. 0. E&G/P/B E&K. β(s) = Csp−1. B *
(356) E&D SH HSE Z. 4p u = −f (r, u) < β(u),. 1. (sβ(s)). E# ILKSE ! !M P/E/HSB f HLMgI6 H *B*K. − p1. ds = C. 0. (H01 ). Z. 1. s−1 ds = +∞.. B*G8I O!E%9+EP/B*DFE&H ILB*K u (R) = 0 > 0. 0.
(357)
(358) . . . a * 0.
(359) . . . . Z&Y. 8 . . /G/NO!E Ψ B*HJI +B*D P/B G P M > M!KLM !/B'. Z> #/ B*HJILB LM:+B*D P/B G P/E 9/KSB
(360) HLM!DFE&H P/B B*HJI6 DTMgI687!M!H 9 M!KLM u (r) 9/K1 = DFE P/B r B R ! M!G/P/E p > 2 > KSB
(361) HLM!DFE&HeI M!D" * D P/B / DTM B*HJI6 DTMgI687!M 9 M!KLM f (r, u(r)) 9/K1 = DFE P/B R 9+E H f (R, 0) G O!E9B*HJI 9P/B G P/E
(362) *E&Gb E&KSDFB (H ) E# (H ) > HWILK *H"9/K1 = DFE&H ! B*DTM!H. 8M!KLM!G8I6 KLO!E M!H B*HJI6 DTMgI687!M!H !/ B 9/KSB
(363) HLM!DFE&H*> . 0. 0. 0. 0. 0 1. 1. c ` . -. (*$/'. ". C>0. ". '1%9 :. %=>%. . =. r. ". ( B 2 6. r0. " '. B 2-$. |u0 (r)|p−1 > C(r − r0 ).. E&K. Z>?= / B'. Z> j /eILB*DFE&H !/B.
(364) . 0. E&K . Z> j 7!M)! B r. //B = HJILB. |u (r)|. p−1. C>0. 0. = −|u (r)|. I M)! !/B. Z r N −1 s f (s, u(s)) ds. u (r) = r r0. p−2 0. f (r0 , u(r0 )) > C. > -E &E 9 M!KLM r H
(365) B*G8ILB*DFB*G8ILB /9 K1 = DFE P/B. 0. |u0 (r)|p−1 >. C (r − r0 ). N . c ` < -. (*$/'. ". C>0. '1%9 :. ". %=>%. r. =. . ( B 2 6. ". R. " '. B 2-$. |u0 (r)|p−1 > C(R − r).. HJILB*D P M!H 9+E&HSH ! P M!P/B*H 6 $B f HLMgI6 H7M (H ) B*G8I O!E u (R) < 0
(366) *E&G
(367) ! G/P/EFM!HSH DM P/B*DFE&G/HJILKLM!NO!ERP/E'-B*DTM > 76 6 $B f HLMgI6 H7M (H ) B u (R) = 0 B*G8I O!E%9+E&K . Z>?= /eILB*DFE&H.
(368) 5>= . 0. 1. . 0 1. 0. |u (r)|. 0. p−1. 0. = −|u (r)|. p−2 0. u (r) = −. Z. R r. s N −1 r. f (s, u(s)) ds..
(369) Z)(. . E&K. 1B = HJILB. P/B R ILB*DFE&H (H02 ). C>0. I M)! !/B. f (R, 0) < −C < 0. 8. . /!>- . > & &E G
(370) !? DFE&H !/B 9 !M KLM r /9 K1 = DFE. |u0 (r)|p−1 > C(R − r). . c ` +. -. ". (*$/'. q>1. '1%9 :. ". . " % ( ' =>%9. Z. R. |f 0 (r, u(r))|q dr. @. 9;( B ( '1%767%. F. C@ B 7&B*DFE&HCM!G M)! HLM!K P M!H 9+E&HSH ! P M!P/B*H 6 $B f HLMgI6 H7M (H ) B*G8I O!E u (R) < 0 B"
(371) *E&D HJILE u(r) > C(R − r) 9 M!KLM r 9/K1 = DFEP/B > E&K (H ) B = HJILB q > 1 I )M ! !/B q · γ < 1 > E 7!M!DFB*G8ILB 9+E&K (H ) ILB*KSB*DFE&H R.
(372) . r0. 0. 1. 2. 2. |f 0 (r, u(r))|q <. C C < , q·γ u(r) (R − r)q·γ. !M KLM r 9/K1 = DFERP/B R > & E&DFE f (r, u(r))
(373) *E&G8I6? G! MFB*D [0, R)
(374) *E&G
(375) !? DFE&H MRP/B*DFE&G/H ILKLM!NO!E/> 76 6 $B f HLMg6I H7M (H ) B*G8I O!ETM /G/NO!E f (r, u(r)) ! D%?I M!P M > 9. 0. . 0 2. B ! E&H ! B*DTM!H. 0. M!G8ILB*K E&KSB*H *
(376) E&G
(377) !? DFE&H !/B Z. R. B. 1 dr < ∞ |u0 (r)|p−2. Z. |f 0 (r, u(r))| dr < ∞,. DFE&HJILKLM!G/P/ERM!HSH D !/BWM) /G/NO!E Ψ B*HJI +B*D P/B G P M > a * ` Ψ /DTM"
(378) *E&G8ILKLM!NO!E 9 M!KLM n H
(379) B*G8ILB*DFB*G8ILB &KLM!G/P/B!> $B LM q > 1 P M!P/E 9+B ! E -B*DTM Z> j > KSDTM!DFE&H !/B 9 M!KLMTILEP/E ω ω ILEP/E n ≥ 0 r0. . Z>[Y#/. R r0. n. 1. n. n. |Ψ (ω1 )(r) − Ψ (ω2 )(r)| ≤ K. n. . |r − r0 |n n!. 10. 2. ∈ C 0 [r0 , R]. q. kω1 − ω2 k∞ , r ∈ [r0 , R],. B. . Z> ( /.
(380)
(381) . . .
(382) . . . . Z)@. 8 . . E&G/P/B #. K=. Z. R. 1 dr (p − 1)|u0 (r)|p−2. Z. R. |f 0 (r, u(r))|q dr. *B K
(383) M!KSB*DFE&H MFP/B*H # M)! P M!P/B Z> (%9+E&K G/P/NO!EFB*D n > !M KLM DFB*G8ILB!>0$ 9+E&G M!DFE&H !/B 7 )! P M 9 M!KLM k > 5 G8I O!E r0. r0. q1. n=0. B. 1 1 + 0 = 1. q q. 1B ! MRHSB &7 B*K
(384) M' DFB*P MgI M .
(385) k+1
(386)
(387)
(388)
(389) Ψ (ω1 )(r) − Ψk+1 (ω2 )(r)
(390) =
(391) Ψ(Ψk (ω1 ))(r) − Ψ(Ψk (ω2 ))(r)
(392) ≤
(393)
(394) Z Z s N −1
(395)
(396) r t 1
(397)
(398) 0 k k f (t, u(t)) Ψ (ω )(t) − Ψ (ω )(t) dt ds ≤
(399)
(400) 1 2
(401)
(402) r0 (p − 1)|u0 (s)|p−2 r0 s Z s N −1 Z r
(403) k
(404) 1 t 0
(405) Ψ (ω1 )(t) − Ψk (ω2 )(t)
(406) dt ds |f (t, u(t))| ≤ (p − 1)|u0 (s)|p−2 r0 s Zr0r Z r
(407)
(408) 1 ≤ ds · |f 0 (s, u(s))|
(409) Ψk (ω1 )(s) − Ψk (ω2 )(s)
(410) ds 0 p−2 r0 (p − 1)|u (s)| r0 1 Z r Z r 1 |s − r0 |k q0 k 0 ≤ K ds · kω1 − ω2 k∞ . ds · |f (s, u(s))| 0 p−2 k! r0 (p − 1)|u (s)| r0. 6I ! M!G/P/E. MR@CB*H # M)! P M!P/B P/B ! /P B*K . q10 |s − r0 |k ds kω1 − ω2 k∞ k! r0 1 |r − r0 |k+1 q0 k+1 ≤ K kω1 − ω2 k∞ . (k + 1)!.
(411) k+1
(412)
(413) Ψ (ω1 )(r) − Ψk+1 (ω2 )(r)
(414) ≤ K k+1. E&KJI M!G8ILE. B 9 M!KLM n & KLM!G/P/B -E &E. n. n. kΨ (ω1 ) − Ψ (ω2 )k∞ ≤ K. K. n. . n. . Z. r. |R − r0 |n n!. |R − r0 |n n!. 10. 10. q. < 1.. /DTM"
(415) *E&G8ILKLM!NO!E%9 M!KLM n H
(416) B*G8ILB*DFB*G8ILB & KLM!G/P/B!> a * i 5>= HJILB / D/G
(417) *E 9+E&G8ILE) = E%9 M!KLM Ψ > Ψn. q. kω1 − ω2 k∞.
(418) k&_. . 8. . /!>- . I6! M!G/P/E EXQB*E&KSB*DTM P/B E&G8ILE> = E P/B M!G M)
(419) 9+EP/B*DFE&H 9/KSE 7!M!K, !/B HSB Ψn /DTM
(420) *E&G8ILKLM!NO!E>9 M!KLM9M)! #/D n > 1 B*G8I O!E Ψ 9+E&HSH /D /G
(421) *E 9+E&G8ILE& = E .7&B*K)$E!ILE&DTM &E&K Z!j & E&KSE !!K E > />?= / >1- E &E B = HJILB /DTM /G
(422) M HSE !/NO!E ω ∈ C 0[r , R] ∩ C 1[r , R) 9 M!KLM9E /9 KSE 0 0 ! B*DTM . Z>[Z#/ > .
(423) !#"$)%&'!() . $B LM!D. . M N >0. . I M) H !/B. |f (r, s)| < M, ∀(r, s) ∈ [r0 , R] × [0, α + ]. B. |u0 (r)| < N, ∀r ∈ [r0 , R].. QE&DFB*DFE&H δ > 0 I M)! !/B. 1 1 2 p−1 δ[N + (M δ) p−1 ] < . 4. $B LM h H
(424) B*G8ILB*DFB*G8ILB +9 B !/B*G/EFI M)! !/B HJILE +9 E&HSH ?87&B ! 9+E H. f (r0 , α + h) > 0.. /DTM HSE !/NO!E. uh. g 6/ 0. 9+B ! E'-B*DTM#=&>[Y> B ! M' KSE 9+E&H NO!E C>?=#A9 M!KLM P/E 9/KSE ! B*DTM B*D . Z> !/ P/B G P M B*D [r , r + δ] B f (r0 , α) > 0. . 0. /G/NO!E. 0. |uh (r) − u(r)| < , ∀r ∈ [r0 , r0 + δ]. u(r). 4. +B = HJILB . Z> @ /. WM /G
(425) M HSE !/NO!ERP/E /9 KSE ! B*DTM P/B !7 M)! E&K G
(426) M)!. 0 − r N −1 |u0 |p−2 u0 (r) = r N −1 f (r, u), r ∈ (r0 , R) u(r0 ) = α u0 (r ) = 0. 0. h<. . Z>?= _ /.
(427) . .
(428) . . . . . k=. 8 . . C@ B-7MgILE9+B ! M!H KSE 9+E&H N B*H C>[ZWB C> kG O!EWILB*DFE&H 8M!KLM!G8I6 MXP/B-/G
(429) P M!P/B9 M!KLM p > 2 B G/E&H'9+E&G8ILE&H r E&G/P/B u (r) = 0 B f (r, u(r)) = 0 > & E&DFE f (r , u(r )) > 0 B u (r) < 0 /B*G8I O!E u(r) B*HJI %/G
(430) M!DFB*G8ILBXP/B G P M > ∀r ∈ (r , R) g 6 / ` u → u /G bE&KSDFB*DFB*G8ILB B*D [r , r + δ] ! M!G/P/E h → 0 > $B LM (h ) I )M ! !/B h → 0 > E&K . Z> @ / /M HS B *G
(431) M (u ) "/G bE&KSDFB*DFB*G8ILB ! %D ?I M!P M > E& K . Z> !//ILB*DFE&H Z 0. 0. 0. 0. 0. 0. h. n. n. hn. r. |u0h (r)|p−2 u0h (r) = −. @CBXE&G/P/BXHSB #/B -E &E. uh n. r0. s N −1 r. f (s, uh (s)) ds.. |u0h (r)|p−1 ≤ M δ.. B
(432) *E&G8I6? G! M > B ! EWQB*E&KSB*DTMXP/B CK *B ! CH
(433) *E !8MXDFB*G/E&H P/B/DTM H/HSB *G
(434) M /Gb E&KSDFB*DFB*G8ILB B*D [r , r + δ] > E&K". Z> !//M /G/NO!E u HLMgI6 H7M B ! M!NO!E" G8ILB &KLM)! → u¯ (uhn ). 0. 1M *B*G/P/E . 0. 0. uhn (r) = α + hn − n→∞. u ¯(r) = α −. Z. hn. Z. r. A−1 r0. r. A−1. ! Z s N −1 t f (t, uhn (t)) dt ds. s r0. ! Z s N −1 t f (t, u ¯(t)) dt ds. s r0. . Z>?=&= /. !M H. Z>?=&= / B !87!M)! B*G8ILB %. Z>?= _ / > -E &E 9+B ! M KSDTMgI687!M =# u¯ ≡ u > B 9+BI6 G/P/E EWM!K #/DFB*G8ILE M!G8ILB*K E&K 9 M!KLM ILEP M H/HSB *G
(435) MTP/B (u ) /E ILB*DFE&H !/B u → u /GbE&KSDFB*DFB*G8ILB!> g 6 / i u → u /GbE&KSDFB*DFB*G8ILB B*D [r , r + δ] ! M!G/P/E h → 0 > E& K . Z> !//ILB*DFE&H r0. %. hn. 0 h. hn. 0. 0. 0.
(436) Z r
(437)
(438) 0
(439)
(440)
(441) s N −1
(442) |u (r)|p−2 u0 (r) − |u0 (r)|p−2 u0 (r)
(443) =
(444)
(445) [f (s, u (s)) − f (s, u(s))] ds h h h
(446)
(447) r Z rr0 ≤ |f (s, uh (s)) − f (s, u(s))| ds.. CHSH D&9+B ! M . KSDTMgI687!MRZ 9/KSE 7!M!DFE&H M KSDTMgI687!M k > r0.
(448) k8Z. . 8. /!>-. . . g 6/ u (r) XM /G
(449) M HSE !/NO!ERP/B . Z> !/ B*D [r , r + δ] > B ! M KSDTMgI687!MRZ B -B*DTMU=&>[Y /B = HJILB*D δ > 0 B h I M) H !/B B*D [r , r + δ ] f (r, u (r)) > 0 9 M!KLM |h| < h > CHS H D 0. h. 1. 0. 0. 0. h. 0. 1. 0. Z r N −1 s =− f (s, uh(s)) ds < 0 r r0. > CHSH D&9+B ! M KSDTMgI687!M k B . Z> j / 9 M!KLM h H
(450) B*G8ILB*DFB*G8ILB +9 B !/B*G/E B*D (r , r + δ]. u (r) < 0 B ! M!H KSE 9+E&H N B*H C>[Z B C> k P/B*DFE&G/HJILKLM!DFE&H M KSDTMgI687!M /> $B LM h H
(451) B*G8ILB*DFB*G8ILB 9+ B !/B*G/EFI )M ! !/B 9. M!KLM. |u0h (r)|p−2 u0h (r). r ∈ (r0 , r0 + δ1 ]. 0 h. 0. 0. |uh (r0 + δ) − u(r0 + δ)| <. B. 4. |u0 (r0 + δ)| < N.. B +BI G/P/E RE M!K #/DFB*G8ILERM!G8ILB*K E&K u B*HJI )/G
(452) M!DFB*G8ILB P/B G P M *B D [r + δ, r + 2δ] u → u B u → u /GbE&KSDFB*DFB*G8ILBXB*D [r + δ, r + 2δ] > @CB 9+E HCP/B/D G /DFB*KSE' G?ILE n P/B BI M)9 M!H ILB*KSB*DFE&H 9 6 0 h. $B. 0. h. 0. 0. 0. 0. h. 0. r0 + (n0 − 1)δ < R ≤ r0 + n0 δ.. 9 !M KLM h +9 B !/B*G/E 9+EP/B*DFE&H B*HJILB*G/P/B*K /G
(453) M!DFB*G8ILB u (r) MgI r = R E#VMgI I M)! !/B u (R ) = 0 >$ B"9+E&K E# ILKSE ! M!P/E u (R) = 0 B*G8I O!E 9+EP/B*DFE&H ILB*K u (r) = 0 r=R 9 M!KLMTM)! #/D r 9/K1 = DFETP/B R > M!HCG/B*HJILB"
(454) M!HSE 9+B ! M'9 1!ILB*HSB (H ) f (r, u (r)) < 0 A9 M!KLM r 9/K1 = DFERP/B R B h 9+B !/ B*G/EA 8M!KLM!G8I6 G/P/E% !/ B 9+EP/B*DFE&H B*HJILB*G/P/B*K-/G
(455) M!DFB*G8ILBWM) /G/NO!E u (r) MgI r = R ≤ R
(456) *E&D u (R ) = 0 HSB R < R > u0 (R) < 0 h. h. h. 0. h. %. 0 h. 0 1. h. h. h. h. h. h.
(457) . . . . .
(458) . . . . k&k. 8 . . . .
(459) !#"$#%&'!() . . B ! ERQB*E&KSB*DTMTZ>[Z. ILB*DFE&H !/B B u (r) → u (r) ! M!G/P/E h → 0 . Z>?= Z#/ u (r) → u(r) /GbE&KSDFB*DFB*G8ILBCB*D [r , R] > H 9/K 1 = DFE&H P/E H ! B*DTM!H
(460) LM!H P/B*DFE&G/HJILKLM!N B*H B*G
(461) *E&G8ILKLM!D: HSB G M $B*NO!EZ> bE&KSG/B
(462) *B*D B*HJ6I DTMg6I 87!M!H
(463) *K
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