Sự không tồn tại nghiệm dương của phương trình tích phân

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Tài liệu tham khảo Sự không tồn tại nghiệm dương của phương trình tích phân

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Nội dung Text: Sự không tồn tại nghiệm dương của phương trình tích phân

LU{J.nvan tot nghi~p Trang 24 CHUONG 4 st; KHONG TON T~I NGHIEM DUONG ? "'".., CUA PHUONG TRINH TICH PHAN Val (J = N -1, N > 2 Trang phgn nay chung ta xet sv kh6ng t6n t~i nghit%mdu'dng cua phu'dng trlnh tich phan phi tuye'n sau day (4.1) U( x ) = bN g(y,u(y)) dy "dx E IRN , f N-l' IRN Iy - xl trang do bN = 2((N -l)lUN+ltl voi lUN+1 dit%n la tich cua m~t c~u ddn vi trong IRN+I, N > 2 va g: IRN x IR+ ~ IR la ham lien t\!Ccho tru'oc thoa di~u kit%n: T6n t~i cae hftng s6 a,fJ ~ 0, M > 0 sao cho (4.2) g(x,u) ~ MlxlP ua, "dx E IRN, "du ~ 0, va mQt sf) di~u kit%nph\! sau do. Phudng trlnh tich phan (4.1) duQc thanh l~p tu bai loan Neumann phi tuye'n sau dayvoiN=n-l>2: TIm mQt ham v Ia nghit%mcua bai loan Neumann (4.3) ~v=O, xEIR: ={(xl,xn):xl EIRn-l,xn >O}, (4.4) - vxn(Xl ,0) = g(XI, V(XI ,0)), Xl E IRn-l, thoa cae tinh cha't: (8]) VEC2(IR:)nC(IR:), vxn EC(IR:), (82 ) lim SUP I vex) + R. sup ov (x) I = 0, k--HOO ( Ixl=R,xn>O Ixl=R,xn>O fun J d day g: IRn-1x [0,+00)~ [0,+00)cho tru'oc thoa cac di~u kit%nsau: (G]) g la ham lien t\!e, (G2) 3a~0,3M>0: g(xl,v)~Mva, "dv~O, "dxl EIRn-l. va mQt sf) di~u kit%nph\! se d~t sau.
Lu(jn van tot nghi~p Trang 25 Khi do, n€u g 1a ham lien t\lC va nghi~m v bai loan (4.3), (4.4) co cac tinh cha'"t(SI)' (S2)' thi v 1anghi~mcua phudngtrinhtich phan sau day I - 2 g(l , vel ,0))dl I n (4.5) vex ,xn) - (n-2)OJn f I I 2 2 (n-2)/2 ' VeX ,xn) E IRp Rn-I (1 Y -x' I +Xn ) trang do OJn1a di~n tich cua m~t c~u ddn vi trong IRn. Day 1a k€t qua trong ph~n thi€t l~p phudng trinh tich phan (chudng 2, dinh 1y 2.1), trang do co stf thay d6i cac ky hi~u trang cach vi€t bang cach thay (a/,an) va (xl,xn) 1~n1u'
Lu(in win tot nghifp Trang 26 Ch6 thich 4.1, K€t qua nay m~nh hdn k€t qua tfong [2], [8]. Th~t v~y, vOi CY =N -1, d cling phu'dng trlnh rich phan (4.7), cae gia thi€t sau day dii sa dt,mg trong cae bai baa [2], [8] ma trong ehu'dngnay khong e~n d€n: (G3) g(x,u) la ham khong giam d6i vdi bi€n u, i.e., (g(x,u)-g(x,v))(u-v)~O VxEIRN, Vu~O, Vv~O. (G4) Tich phan J ( g (1,0; ~-I 1/1' 1+ x ) t6n t~i va du'dng. Tru'de h€t ta e~n mQt sO'ba't d&ng thue sau day: B6 d~ 4.1. Vai mQi q ~ 0, X E IRN, fa dijt: (4.8) A[q](x):=A[(1+lylrq](x)= J(1+lylr:_,dy. lRN Iy - x I Khi an (4.9) A[q](x) = +00, ne'u q:::; 1, (4.10) A[q](x) hQifl;l va A[q](x)~ (q-I)2 OJNN-I (1+ 111 -I' x)q ne'u q>1. Chung minh b6 d~ 4.1. a) Gia sa q :::; . Chti 1 Y d€n ba't d&ng thue tam giae (4.11 ) Iy - xl :::;Iyl + Ix! vdi mQi x, y E IRN , ta suy fa tu eong thue (4.8) ding (4.12) A[q](x) = J (1+lyl )-:-~y [RN Iy - x I - (1 + Iyl rq d = +oo (1+ rrqNlr d J d:S > J J 1/' 1III ( Y + x ) - N1Y II ) - lyl=r 0 ( r + x r' trong d6 J dSr la rich phan m~t tren m~t e~u, tam 0, ban kinh r trong IRN. Iyl=r Tich phan n~y ehinh la dit%nrich eua m~t tren m~t e~u Iyl= r, tue la: N-l (4.13) J dSr = r OJN' Iyl=r
LucJn van tot nghifp Trang 27 Do do, ta suy tu (4.12), (4.13) ding (4.14) +00 N-} dr J A[q](x)~wN I( r:'xl)N 1 (1+r)q =wN q' +00 N-I d Tich philo Jq = f (r+ rll 0 x) N-I (1+r)q r philo ky khi q ~ 1 va hQi t1;1 q > 1. khi Do do, rich philo (4.15) A[q](x) philo ky khi q ~ 1. a) Gia sa q > 1. i) Xet t~i x = 0, ta co - (1 + Iylrq dy - (4.16) A[q](O)- f m~ I y N-I I -wN + f oo(1+ rrq 0 r, N-I rl-Ndr = w f + oo~ N 0 (I+r)q . / / A +00 dr A' , Do do, hch Phan f hOI tu VI q > 1. 0 (1+r)q . . V~y, rich philo (4.17) A [q](0) hQi t1;1khi q > 1. ii) Xet t~i x =F chQn R > 31xJ O. Ta vie't l~i A[q](x) thanh t6ng hai tich philo 0, > (4.18) A[q](x)= f (1+IYI)~q_~y+ f (1+IYI)~q_~y =J~I>CX)+J~2)(X). IY-Xl$/?Iy - xl Jy-xl"/? Iy - xl (1+lylrqdy U)Banhgia J~I)(X)= f N 1 . IY-Xl$/? IY - xl - Ta co: J Ii ()X = f ~ (l) (1+lylrqdy< -q (4.19) f N-I - sup (I + II) Y N-I IY-XI$R Iy - xl ly-xl:>R ly-xl:SR - xl Iy = sup (1+ !ylrq IY-XI$R f Izl:SRIzi d :-1 = sup (1 + !ylrq wN ly-xl:SR r R N-Id 0 r N-/ = sup (1 + Iylrq wNR < +00. ly-xl:SR
Lugn wln tot nghi~p Trang 28 (1+lyl)-qdy OJ) Danhgia J~2)(X)= f N I . ly-4~1I IY - xl - Ta co: (21 = (1+lylrqdy < (1+lylrqdy < (1+lylrqdy (4.20) JII (x) f NI - f NI - f NI ly-xl~R Iy-xl - lyl~R-lxl Iy-xl - IYI~R-Ixillyl-Ixil - +00 (1 +r ) -'1 r N-I dr +00 r N-I dr = OJN f N 1 = OJN f N I- . II-Ixl Ir-Ixll - R-Ixllr-Ixll - (1+r)q Chu y rang, do R>3Ixl>O,ta colr-lxll=r-lxl:=::R-2Ixl>lxl>O, voi mQi r:=::R-Ixl. +00 ' ' A r N-] dr A' ~. D d h h h Q1 tl,l 0 0, tIc p an f N I 'I VOl q> 1. R-Ixl I r -Ixll - (1+ r) V~y, tich phan (4.21 ) J~2) (x) hQi W khi q > 1. T6 h 1, ta vie"t +00 N-l +00 d r N-I d r J = (4.23) f r r :=:: f q o(r+lxl)N-I(1+r)q Ixl(r+lxl)N-I(1+r)q +00 rN-Idr 1 +00 dr :=:: J( r+r )N-I(1+r)q = 2N-I J(1+r)q = 1 1 \Ix E JRN (q-l)2N-l (1+lxl)q-l . Do do b6 d~ 4.1 du
LucJn van tot nghifp Trang 29 Ta vie't phuong trlnh tich phan (4.7) voi bN = 1 theo d~ng (4.24) u(x) = Tu(x) = A [g(y,u(y))](x), \/x E IRN, trong do (4.25) A [w(y)](x) = J w(y) d~-I' X E IRN. iii' I y- x I Ta chung mint b~ng phan chung. Gia su u Ia nghi~m lien t\lCva duong cua (4.24). Khi do t6n t~i XoE IRN sao cho u(xo) > o. VI u lien t\lc nen t6n t~i ro > 0 sao cho: (4.26) u(x»~u(xo)=L \/xEIRN, Ix-xol:::;;ro. 2 Ta suy tu gia thie't (G2),(4.24)-(4.26) r~ng (4.27) u(x) = A[g(y,u(y))](x) ~ MA[ua(y)](x) 2::MLa J dy N-l' \/x E IRN. Iy-xol:s:ro I y-x I Su d\lng ba't d~ng thuc sau (4.28) I y - x I :::;;Iyl + Ixl :::;; 1 + Ixl)(1 + Iyl) ( = (1+ Ixl)(1+ Iyl- Xo + xo) :::;; + Ixl)( 1+ jxo I+ Iy - Xo (1 I ) :::;;(1+lxl)(1+lxol+ro)' \/x,YEIRN, Iy-xo I:::;;ro' ta suy tu (4.27), (4.28) dng (4.29) u(x) 2:: MLa J Iy-xol:s:ro ~- x yI I N-l > MLa 1 J dy -(1+lxol+ro)N-lx(1+lx l )N-l Iy-xol:s:ro N OJ = MLa X 1 NrO (l+lxol+ro)N-l (1+lxl)N-l N ' \/xEIRN. Ta vie't l~i (4.30) u(x) 2::u1(x) = m](1 + Ixlrq), \/x E IRN, trong do
Lugn win tot nghifp Trang30 a N M L ())NrO (4.31 ) ql = N -1, m] = N(1+lxo!+ro)N-I' Sa dl;lng ffiQtl~n nii'a d&ng thuc (4.24), ta sur tITghl thi~t (G2), (4.27) r[tng (4.32) u(x) 2 MA[ua (y)](x) 2 M4[u~ (y)](x) = Mm~A[(1 + Iylraq, ](x) \::IxE IRN. Bay gid ta xet cac tru'dng hQpkhac nhau cua gia tti a. 1 Truong hQ'p1: O::;a::;-. N-1 Ta sur ra tU (4.9), (4.32) voi q = a ql = a(N -1)::; 1, dng (4.33) u(x) = +00 \::IxE IRN. D6 la di~u vo 19. Truong hdp 2: . ~ N-1 < a 1,ta sur ra tIT(4.32) r[tng: (4.34) u(x) 2 Mm~A[(1+ Iylraq, ](x) = Mm~A[a ql ](x) ()) 2 Mmla N N-I(1+lx!)I-aq" \::IxEIRN. (aql -1)2 hay (4.35) U(X)2u2(x)=m2(1+lxlrq2, \::IxEIRN, trong d6 m2 - M()) N ma (4.36) q2 =aq ] -1 , - I 2N-l q2 . Gia sa dng (4.37) u(x) 2 Uk-I (X) =mk-I(1+!X!rqk-l, \::IXEIRN. N€u aqk-I > 1, khi d6 ta dung ba"td&ng thuc (4.10) voi q = aqk-I > 1, ta thu du'Qc tITgia thi€t (G2), (4.24), (4.37), r[tng (4.38) u(X) 2 M4[ua (y)](x) 2 M m:_]A[ (1 + Iylraqk-' ](x)
Luc7nvan tot nghi~p Trang 31 = M m:-lA[a qk-I ](x) 2 M ma k-l ())N (aqk-I -1)2N-l (1+IXI)I-aqk-1 2mk(I+lxlrqk = Uk(X), '\IxEIRN, trong d6 cac dtiy {qk},{mk} duQCxac d~nh bdi cac cong thuc qui n~p sau: a (4.39) 1m = M())N N I mk-I ' k = 2,3,., qk=aqH-' k 2 qk Tli (4.31), (4.39) ta thu duQc N - k, ntu a = 1, (4.40) qk = k I I-a k-I A'" 1 N (N-l)a - - , neu -
Lugn van tot nghifp Trang 32 (1+lyl rN > - f IIII ( y + x Ii \ ) d NIY- - > +'" (1+rrN d f 0 ( r + x ) -II Nlr lyl=r IdS r +"'(1+rrNrN-I 1\I+rrNrN-I = OJv . f 0 (r + II )N-I x f dr ~ OJN I (r + II )N-I dr x Ixl rN-Idr ~OJN [(1+r)N(r+lxj)N-I. Chu y r~ng voi mQi r sao cho 1 ~ r ~ Ix!ta co N r 1, 1 1 (4.45) ( 1+ r ) ~ 2N va r + Ixl ~ 21xJ. V~y, ta co ta (4.45) dug Ixl rN-Idr 1 1 Ixl dr (4.46) ! (1 + r)N ( r + Ixl)N-I ~ 2N ( 21xl)N-2 ! r( r + Ixl) 1 1 1+ Ixl N = 4N-I x Ixl N-I x In( 2)' "Ix E IR , Ix! ~ 1. Ta (4.43), (4.44), (4.46) ta suy ra ding 0, Ixl~1, (4.47) u(x) ~ V2(x) = ~ ~ In 1+ Ixl PZ, Ixl ~ 1, IxIN-I ( 2 ) voi (4.48) PZ = 1, Cz = MOJNm~ 4N-I Gia su r~ng 0, Ixl~1, (4.49) u(x) ~ vk-l (x) = ~ Ck-l In 1+ ixi Pk-l, Ixl ~ 1, IxlN-l ( 2 J trong do Pk-l>Ck-lla cae h~ng s6 dtiong. Su d\lng gia thie't (G2) va (4.49), ta suy ra dug (4.50) u(x) ~ M A[ua(y)](x)
Lwjn van tot nghi~p Trang 33 ~ M A[v:-1(y)](x) = M J V:-J~~I dy RN Iy - xl >M J v:-I(y) d >M J v:-I(y) d - I?' (lyl+lxl)N-1 Y - lyl~1 (lyl+lxl)N-1 Y +W V:-I (y) dSr = M Jdr Iyl=r ( r + Ixl) N I J I a Pk-I I+r In(- ) +w ( 2 ) dr 1 = M OJNC:-1J r(r + Ixl)N I Ta xet tru'ong hcJp Ixl~ I, ta co 1+ r a Pk-l 1+ r a Pk-I +00 In( -) +00 In( -) 2 ) 2 ) (4.51) J ( dr~ J ( dr I r(r+lxl)N-1 Ixl r(r+lxl)N-l 1+ Ixl a Pk-I +00 dr N-l ~ ( In(-)2 J J r (r + Ixl I x I) II a Pk-I +00 d ;, [ In(l: x). J I~ r(r +:)N-I - 1 1+ x a Pk-I (N -1)2N-Ilxt-1 ( In-fl) . Tli (4.50), (4.51), ta suy ra r~ng 0, Ixl~ 1, Pk (4.52) U(X)~Vk(X)=~ Ck 1+lxl - In- , II x:2:1, IxlN-l ( 2 ) trong do Pk>Ck la cae h~ng s6 du'dng xac dinh b~ng cae cong thu qui n(;lpnhu' sau: a MOJ C (4.53 ) Pk =apk-I' Ck = N k I (N -1)2N-I' k = 3,4,... Ta tinh fa cDng thuc hiSn cua Pk>Ck nho vao (4.48), (4.53), nhu'sau
Lu4n van tot nghi~p Trang 34 k-2 l-N N-I ak-2 (4.54) Pk =a , Ck =dN (dN C2) , k=3,4,... trong a6 MOJN (4.55) dN = (N -1)2N-J . Ta vie't I~i (4.52) voi Ixi ~ 1, ta c6 a k-2 I-N 1 N-I 1+ Ixi (4.56) u(x)~vk(x)=dN IX!N-I ( dN C21n(2) J . ChQn Xl saGcho (4.57) dZ-iC21n(I+lxll»I, 2 Do (4.56), ta suy ra rang u(xi) ~ k->+ooVk(Xl) = +00. lim EHy la ai~u va 19. Dinh 194.2 au'