Luận văn Sự không tồn tại nghiệm dương của một số phương trình tích phân phi tuyến liên hệ với bài toán neumann

30 CHUONG4. Stj KHONG TON T~I NGHltM DUONG CUA PHUONG TRINH TICH PHAN PHI TUYEN YOlo <cr<y+N, cr<N, N~2 4.1.GIOI THItU Trongchuangnay,chungWixetslfkhongt6nt':linghi~mduongcuaphuong trlnhtichphanphituye'n (4.1) u(x)=bN f g(x,y,U~Y))dY,VxEIRN, IRN Iy - xl trong d6 bN =((N -1 )CON+lt voi CON+lIa di~ntich cua m~tc~udon vi trong IR n+l, N ~2; cr<N Ia mQts6 duongchotruocva g: IR2NXIR+~ IR la ham lientlJCchotruocthmldi~uki~n: T6nt':licach~ngs6 a,~,y~0 vaM >0 saGcho (4.2) g(x,y,u)~MlxIP.lyIYua,Vx,YEIRN, Vu~o, vamQts6di~uki~nb6sungthem. 4.2.nfNH LY VE sTjKHONG TONT~I NGHItM DUONG Khonglamma'tinht6ngquat,chungtac6thSgiasar~ngbN=I voivi~c thayd6ih~ngs6M tronggiathie't(4.2)cuag. Phuongtrlnhtichphan(4.1)duQcvie'tl':livoi bN =I (4.3) u(x)= fg(x,y,U~Y))dY=TU(X),VxEIRN, lRN Iy - xl trongd6 0<cr<y+N, cr<N, N ~2. Khi d6tac6ke'tquachinhnhusau. 31 Binh ly 4.1.Gidsit g:IR 2NXIR+~ IR fahamlient1:lCthoddiiu ki~n: (4.4) T8nt{licachlings6a,~,y~0 vaM>O,0<cr<y+N, cr<N, N ~2 saocho g(x,y, u)~ MlxlP.lylYua \lx,y E IR N, \lu ~O. Ne'u O~a~(N+y)/(cr-~) thi phTiO'ngtrinh tichphfin (4.1) khong c6 nghi~m lien t1:lCdTiO'ng. Trudelientaeffnb6dSsauday. B6 d~4.1.V{fim6ip~0, q~0, XEIRN tadi;it (4.5) Khi d6 (4.6) (4.7) A[P,qKx)= f lylP(1+Iyltq IRN Iy-xlcr dy. Ne'uq-p~N-cr, A[P,qlx)=+oo, Ne'uq -p >N -cr, A[P,qlx) hQit1:lva 1 1 ffiN !XIP+N-cr A[P,qKx)<:(N+P + q)20' (l+lxlt' trongd6 ffiNfadi~ntickcuami;itcdudO'nvi trangIRN. Chungminhb6d~4.1. a)Giasaq-p ~N-cr. (4.8) tasur ra (4.9) Dungba'td£ngthuetamgiae Iy- xl ~Iyl+lxi, \Ix E IR N, A[P,qlx) ~ f lylP(1+Iyltq . IRN Iy - xlcr dy +00 rP+N-ldr = ffiN f(1 +r)q(r +Ixlt = ffiNJp,q,cr. Tich phanJ p,q,crphanky khi q+cr- p- N +1~1 hay q- p ~N - cr vahQitl.1khi q- p >N - cr. Do d6 (4.10) 32 '\IxE IRN,A[P,qKx) phankykhi q- p S;N - cr. b)Gia sa q- p >N - cr.Ta sechungminhr~ngA[p, q](x)hQitlJ,vdimQix EIR N. (4.11) i) Gia sax =0,taco A[P,qKo)=f (1+lyltq IRN Iyl"-P dy +00 rN-ldr +00 dr = ffiN J (1 + r )qr"-P =ffiN J (1 +r)qr,,-p-N+l. +00 d VI the',tichphan f( ) r N 1 hQitlJ khivachIkhi1 q ,,-p- +0 +r r cr- p- N +1<1<q+cr- P - N +1 hay - p <N - cr<q- p hay N - cr<q- p va +00 d tichphan f( ) r N1 phanky khivachIkhi N - cr~q- p.1 q ,,-p- +0 +r r VI the', (4.12) va (4.13) A[P,qlo) hQitl;lkhivachIkhi N - cr<q- p, A[P,qlo) phankykhivachIkhi N - cr~q- p. ii) Giasax*-o vaR >31xl>O.Tavie'tl~iA[P,qKx)thanht6nghaitichphan A[P,qlx)= f IYIP(I+IYltqd f lyIP(I+lyl)-q 1 I " y+ dy ly-xlsR y - X ly-xl2RIy- xl" ==IR (x)+JR (x). * Danhgia IR(x)= f lylP(1+Iyltqd I I " y. ly-xlsR Y - X (4.14) Taco (4.15) IR(x)= f lylP(1+Iyltq 1 I " dy ly-xlsR Y- X s; suplyIP(I+lyltq f ~ ly-xl<R- ly-xlsRIy - xl" 33 R = sup lylP(1+!yltqffiNJ rN-cr-Idr ly-xl::;R 0 R N-cr = sup lylP (1+Iyl)-qffiN - < +00. ly-xl::;R N - 0' * Banhgia J R(x)= f lylP(1+Iyltq I I cr dy. ly-xl~R Y - X Ta co (4.16) JR(x)= J lyIP(I+lyltq ly-xl~R Iy- xlcr dy ~ f lylP(1+Iyl)-q lyl~R-lxlIy- xlcr dy ~ f iyiP (1+Iyltq lyl~R-NIlyl-Ixr dy =ffiN +J rP (1+rtqrN-Idr R-N Ir-Ixr -too dr =ffiN f . R-Ixllr-Ixr (1+r)Qr-p-N+l Cht1yding,tit R >31xl>0, taco r:;t:lxi, Vr ~R -Ixl. +00 dr Do do rich phan f cr hQi t1;l khi 0' +q- p- N +1>1 hay R-N Ir -Ixll (1+ r)q r-p-N+I N - 0'<q- p. Do do, (4.17) JR(x) hQit1;lkhi N -0' <q-p. Suyratit(4.12),(4.14),(4.15)va(4.17),taco (4.18) VxEIRN, A[P,qKx)hQit1;lkhiN-O'<q-p. Hannlla,vdi q- p>N - 0', tavie"tl~i (4.19) 34 Ixl p+N-ld +00 p+N-ld J p,q,cr=f(1 r)q( lit + f(1 r)q ( I I)cr0 + r +x Ixl +r r +x - (I) (2) \-I IR N- J p,q,cr+J p,q,cr, V X E . Chungta danhgia tungtichphan J~:q,crva J~~~,cr' (4.20) (4.21) . ) 1:"\' h "' J (I)J van gla p,q,cr 1 Ixl rP+N-Idr 1 Ix/P+N-cr J( ) > f - x . p,q,cr- 0(l+/xl)q~xl+lxlt (N +p)2cr (1+Ixl)q .. ) 1:"\' h "' J (2)JJ van gla p,q,cr +00 p+N-l d 1 +00 p+N-crd J(2) > f r r =- f r r p,q,cr- (1 ) q ( ) cr 2cr (1 ) q Ixl +r r +r Ixl r +r IxIP+N-cr+00 dr 1 IxIP+N-cr Z 2cr I~ (l+r)q+I =q2crx (l+lxJ)q. Vi the'tu(4.19),(4.20)va (4.21)tasuyradanhgia(4.7).Do d6b6dSduQcchung minhhoanloan. Chung minh dinh Iy 4.1.Ta chungminhbAngphanchung.Gia sudng tant<;li mQtnghi~mduonglientlJCu(x)cuaphuongtrinhtichphan(4.3).Gia sudng tan tO. Vi u lien tlJC,do d6 tant0saocho (4.22) u(x»!u(xo)=L, VxEIRN, Ix-xo/:::;ro.2 Tasuyratu(4.3),(4.4),(4.22)vatinhdondi~ucualoantutichphanfAng (4.23) u(x)=Tu(x)zMlxl~f lylY ua(y~dy IRN Iy-xl zM/xl~Cx fly/Y dy . Iy-xol~roIy - xlcr SudlJngbc1tdingthuc (4.24) 35 Iy - Xl :s;Iyl+Ixl :s;(1+Ixo1+foXl +Ixl), '\Ix, y E IR N, Iy - XoI :s;fo . Ta thudu<jctu(4.23),(4.24)ding (4.25) trongd6 (4.26) u(x)~u](x)=m]lxIPl(I+lxltql'\IxEIRN, fP] =~, q] =cr, lm] =MU(l+lxo l+fof"ly-!lr:IYdy. Dungbfftd£ngthuc(4.3)mQtI~nmIa,tasuyfa tu(4.25)dug (4.27) u(x)=Tu(x)~Mlxl~flylY ua(y~dy IRN Iy - xl ~ Mlxl~ f lylY u~(y~dy IRN Iy-xl =Mm~Ixl~f lylY Iylapl (1+lYltaqldy IRN Iy - xl ~ lyly+aPI(1+Iyltaql =Mm~lxl f cr dy IRN Iy - xl =Mm~lxl~A[y +apI,aqIKx) '\IxE IR N. Ta xetcactfuongh<jpkhaccuaa. Tru'ong hqp 1 : O:s;a:S;(N-cr+y)/(cr-~). Ta thu du<jctu (4.7), (4.27)voi P =Y+api=Y+a~,q=aql=acr,q- P=a(cr-~)-y:s;N - cr,f~ng (4.28) u(x) =+00, '\Ix E IR N. Di€u nay matithuan. Tru'ong hqp 2 : (y+N-cr)/(cr-~)<a«N+y)/(cr-~). Dung (4.7) voi p=y+ap] =y+a~, q=aq] =acr,q-p=a(cr-~)-y>N-cr, ta suy tu (4.27) f~ng 36 ( 1 1 J I IP+y+aPI +N-cr (4.29)u(x)ZU2(x)=Mm~ . +- .roN.X() VxEIRNN +y+apl aql 2cr l+lxl aql trongd6 =m21xlPz(1+Ixj)-QZ Vx E IR N, PZ =api +P+y+N - cr, (4.30) ~ qz =aql a ( 1 1 J ruN mz =Mml N +y+api +aql . 2" . Giiisading (4.31) u(x)Zuk-l(x)=mk-llxIPk-1 (1+IxltQk-l, Vx E IRN. NSu aqk-I-y-apk-I >N -cr, khid6,sadl;lng(4.3),(4.7)va(4.31),tathuduQc (4.32) u(x)=Tu(x)zMlxlPf lylY ua(y~dy IRN Jy - xl zMJxlP f lylYU~-I(y] dy IRN Iy-xl =Mm~-llxIPf lylY lylaPk-l (1+lyltaQk-1dy IRN Iy- xl" =Mm~-llxIPf lylaPk-l+Y(1+lyltaQk-ldy IRN Iy-xl" =Mm~-llxIPA[apk-l +y,aqk-llx) zMm~-l [ 1 +~ ] . ro~ .lxIP+N+aPk-l+Y-"(1 + IxltaQk-l N +aPk-l +y aqk-l 2 zuk(x)=mklxIPk(l+lxltQk, VxEIRN, trongd6cacday {Pk}'{qk}'{mk}duQcxacdinhbdicaccongthti'cquin<;lpsail (4.33) 37 Pk =aPk-l +~+y+N - cr, qk =aqk-I =Mm~-I(J)N [ 1 ~ ] k >3mk + - . 2" N +y+apk-I aqk-I Tu (4.30),(4.32)tathuduQc f (k-lXY+N +~-cr)+~, a=l, (4.34)Pk= ( 1- a k-l J( R N ) R k-l N - cr+y N +y 1Y+I-'+ -cr +I-'a , <a<-, a:l:- . I-a cr-~ cr-~ (4.35) qk =aq =cra k-I k-l . Ta suyfa fU (4.3)va(4.32)ding (4.36) u(x)2 Mm~lxlPA[y +aPk,aQkKx) \7'xE IR N. Vi v~y,tu(4.35),(4.36),tachic~nchQnsf)tl!nhienk 2 3 saDcho (4.37) aqk- ark - Y~N - cr<aqk-l- aPk-l- Y bdivi A[y+apk'aqkIx) =+00. Do (4.34),(4.35)bfftding thuc(4.37)tuongduongvdi (4.38) hay (4.39) cr k -1 < ~k, a =1 N -cr+y+~ k 1 1 I ( - a~- (N- cr+y) ) k- <- n < Ina a(cr-~)-(N-y) - , K N -cr+y N +yneu <a<-, a:l:-l. cr-~ cr-~ Do (4.34),(4.35),(4.37)-(4.39)tachQnk nhusau i) Ne'u a =1, ta chQn k thoa cr/(N - cr+~+y)~k <1+cr/(N - cr+~+y). ii) Ne'u (N- cr+y)/(cr-~)<a <(N+y)/(cr-~),a:l:-1 ta chQnk thoa ,. 1 ( - a~- (N- cr+Y) Jko~k <ko+1,VOlko=-In ( )a ( ) .Ina cr-~ - N +y 38 Tru'dng h(jp 3 : a =(N+y)/(cr- ~). Vdi Y+aPI =Y+a~,aql=N +y+a~,tavie'tI~i(4.27)nhusan (4.40) I I y+apl(1 I I) -aql u(x)?:: Mm~lxlJ3 f y +: dy IRN Iy-xl J3 lyly+aJ3(1+lyltN-y-aJ3 =Mm~lxl f "dy IRN Iy-xl =Mmnxl~A[y+a~,N+y+a~Ix) \Ix E IRN. V di m6i x E IR N,Ixl ?::1, ta c6 (4.41) A[y+a~,N +y+a~Ix)?:: f lyly+aJ3(1+lyltN-y-aJ3d IRN ~yl+lxlt y +00 = ffiN f ry+aJ3+N-l 0(1+r)Y+aJ3+N(r+lxltdr Ixl > f ry+aJ3+N-l - ffiN 1(I+r)Y+aJ3+N(r+lxlt dr=ffiNB(x) Vdi m6i r thoa 1::::;r ::::;Ixl ta c6 (4.42) Dod6 (4.43) ( r ) y+ap+N 1 1 mill {I,21-a} 1+r ?::2y+ap+N;(r +Ixl)a-I?:: Ixla-I' IXI ( r ) y+aJ3+N I dr B(x)= - . . ! 1+r (r+Ixlt-l r(r+Ixl) > 1 .min{I,21-,,}.lxldr - 2y+aJ3+N Ixl" !r(r+Ixl) = I .mill{I,21-aL In ( l +Ixl ] . 2y+(X~+N Ixla 2 Tli (4.40),(4.41),(4.43)taduQc (4.44) trongd6 (4.45) Gii sar~ng (4.46) 39. fO, Ixl:::;l, u(x)~v,(x)~lC,lxl>-<H 1+;xl)r Ixl ~1, sz =1,Cz =Mm~coN2Y+~~+Nmin{1,21-o}. 0, IxJ:::;1, u(x)~v,-.(x)~jc,-"x,P-H1+2Ixl))"', Ixl~1, Ck-h Sk-lla nhfi'ngh~ngsf). Sadvng(4.3),(4.4),(4.46)tadu'Qc (4.47) u(x)~Mlxl~J lylY V~-l(y)dy IRN Iy - xl<J ~Mlxl~ J lylY . V~-l(Y) d IRN ~yl+Ixlt y Iyly+a(p-cr) [ In ( 1+Iyl ]J aSk-l =MlxIPc~-lJ 2 ~yl+Ixl)cr dy +00ry+a(p-cr)+N-l [ ln ( l +r )] aSk-l =Mco I x l PCa f 2 N k-l 1 (r+lxlt dr. Vdi Ixl~1, ta du'Qc (4.48) Y+a(p-cr)+N-l [ ln ( ~ )] aSk-l [ ( IIJ] aSk-l +00 y+a(p-cr)+N-l +00r 2 1+x f r dr ! (r+lxlt dr~ In 2 Ixl (r+rt 40 [ ( I IJ] ask-l +00 =~ In I +x fry+a~+N-aY-O"-ldr 20" 2 Ixl [ ( I IJ] ask-l +00 =~ In 1+x fr-O"-ldr 20" 2 Ixl =~ .~ [ In ( ~ J] aSk-l . cr20" IxlO" 2 Tit (4.47),(4.48)taduQc (4.49) 0 Ixl~l,, u(x)~ Vk(X)~jCklxl~-crHl~lxl)r, [xld, trongd6 (4.50) C Ia Sk=ask-I' k =-MCONCk-l' k~3. cr20" Tit (4.45),(4.50)taduQc (4.51) ( ) k-2 - k-2 - N +y C -! [dC] ak-2 sk- a - cr- p , k - d 2 , trongd6 (4.52) 1 d= ( ~MCON ) a-l, a= N +y. cr20" cr- P Do d6 voi Ixl~1, ta viSt I<.ti(4.49) (4.53) k-2 u(x);,Vk(X)=~IxIP~cr(dC2lnC :Ixl) r \fL 2. ChQnXo E IR N vdi IxoI ;" 1 saocho dC, InC +~xoI) >1. Ap d\lllg b3 dE (4.2), ta co k-2 U(Xo)20V(Xo)=~lxolP-cr(dC21f +~xolJr -H<'O khi k-Hoo. 41 Suy fa U(Xo)=+00 DiSunaymallthu§.n. Dinhly (4.1)duQcchungminhhoantfft.