Sự không tồn tại nghiệm dương của một số phương trình tích phân phi tuyến liên kết với bài toán NEUmann trong nửa không gian trên

u~0, va mQtsf)di~ukit%nph\!saudo. Phudngtrlnhtichphan(4.1)duQcthanhl~ptu bai loanNeumannphi tuye'nsau dayvoiN=n-l>2: TIm mQtham v Ia nghit%mcuabai loanNeumann (4.3) (4.4) ~v=O, xEIR: ={(xl,xn):xl EIRn-l,xn >O}, - vxn(Xl ,0) =g(XI, V(XI ,0)), Xl E IRn-l, thoacaetinhcha't: (8]) VEC2(IR:)nC(IR:), vxnEC(IR:), lim ( SUP I vex)I +R. sup ov (x) J =0, k--HOO Ixl=R,xn>O Ixl=R,xn>Ofun (82) d day g: IRn-1x[0,+00)~ [0,+00)chotru'octhoacacdi~ukit%nsau: (G]) (G2) g lahamlient\!e, 3a~0,3M>0: g(xl,v)~Mva, "dv~O, "dxlEIRn-l. va mQtsf)di~ukit%nph\!sed~tsau. Lu(jnvantotnghi~p Trang25 Khi do,n€u g 1ahamlient\lCvanghi~mv bai loan(4.3),(4.4)co cac tinhcha'"t(SI)' (S2)'thi v1anghi~mcuaphudngtrinhtichphansauday I - 2 f g(l,vel,0)dl I n(4.5) vex ,xn) - 2 (n-2)/2' VeX ,xn) E IRp (n-2)OJn Rn-I (1 I I I 2 )Y -x' +Xn trangdo OJn1adi~ntichcuam~tc~uddnvi trongIRn. Day 1ak€t qua trongph~nthi€t l~pphudngtrinhtich phan (chudng2, dinh1y2.1),trangdo co stfthayd6i cacky hi~utrangcachvi€t bangcachthay (a/,an)va (xl,xn)1~n1u'<!tbdix=(xl,xn)va Y=(/,Yn)' Taclinggiasaranggiatribien V(XI,0)cuanghi~mv cuabailoan(4.3), (4.4)thoatinhcha'"t: (s3)Tich phan f g(/, v(/ ,0)d/ /Rn-I I yl - xl In-2 t<3nt~i, VXI E IRn-l. Gia sa rangbai loan (4.3),(4.4)co nghi~mdudngv=V(XI,xn) thoacac di~uki~n (SI)- (S3)' Dungdinh1'9hQit\l bi ch~nLebesgue,cho Xn~ 0+trang phu'dngtrlnhtichphan(4.5),nhovao(S3)'tathuduQc: v(xl,0)= 2 f g(l, vel ,0)_~l , vxl EIRn-l. (n - 2)OJn /Rn-I Il - Xl In (4.6) Ta vi€t l~iphudngtrinhtichphan(4.6)bangcachthayl~icacky hi~u n-1 =N, Xl=x, l =Y,V(XI ,0)=U(XI), i.e., (4.7) u(x)= 2 f g(y,u(y»)dy(N -l)OJ ' I I N-I' '\IxE IRN. N+lIR' y-x Khi do,taphatbi~uk€t quachinhtrangph~nnaynhu sau: Djnh ly 4.1.Ntu g thoacaegia thitt (GJ, (Gz)vdi N >2 va 0~a ~N~l' Khi do,phl1angtrinhtickphdn(4.7)khongc6.nghi~mlien t~cdl1ang. Lu(inwin totnghifp Trang26 Ch6 thich4.1,K€t quanaym~nhhdnk€t quatfong[2],[8].Th~tv~y,vOi CY=N -1, d clingphu'dngtrlnhrichphan(4.7),caegiathi€t saudaydii sadt,mg trongcaebaibaa[2],[8]matrongehu'dngnaykhonge~nd€n: (G3)g(x,u)la hamkhonggiamd6ivdibi€n u, i.e., (g(x,u)-g(x,v))(u-v)~O VxEIRN, Vu~O,Vv~O. (G4) Tichphan J g(1,0;~-I t6nt~ivadu'dng.1/1' ( 1+ x ) Tru'deh€t tae~nmQtsO'ba'td&ngthuesauday: B6 d~4.1.VaimQiq~0,X E IRN, fadijt: (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'uq:::; 1, (4.10) A[q](x) hQifl;l va A[q](x)~ OJNN-I 111-I' ne'u q>1.(q-I)2 (1+x)q Chungminhb6d~4.1. a)Gia sa q :::;1.Chti Y d€n ba'td&ngthuetamgiae (4.11) Iy - xl :::;Iyl+Ix! vdi mQi x,y E IRN , ta suyfa tueongthue(4.8) ding A[q](x)= J (1+lyl)-:-~y [RN Iy - x I > J (1+Iylrq d =+Joo (1+rrq d J d:S- 1III N1Y II Nlr r'1/' ( Y + x ) - 0 ( r +x ) - lyl=r (4.12) trongd6 J dSr la richphanm~trenm~te~u,tam0, bankinhr trongIRN. Iyl=r Tich phann~yehinhla dit%nricheuam~ttrenm~te~uIyl=r, tuela: (4.13) J N-l dSr =r OJN' Iyl=r LucJnvantotnghifp Trang27 Dodo,tasuytu (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 rll N-I r philokykhi q~1va hQit1;1khi q>1.0 (r+ x) (1+r)q Do do, richphilo (4.15) A[q](x)philoky khi q~1. a)Gia sa q >1. i) Xet t~ix =0,taco (4.16) - f (1+Iylrq dy - +foo(1+rrq rl-Ndr=w +foo~ .A[q](O)- I I N-I -wN N-I N (I+r)qm~ y 0 r, 0 / / A +00 dr A' , Do do, hch Phan f hOI tu VI q >1. 0 (1+r)q . . V~y,richphilo (4.17) A[q](0) hQi t1;1khi q >1. ii) Xet t~ix =F0, chQnR >31xJ>O.Ta vie'tl~i A[q](x)thanht6nghaitichphilo A[q](x)= f (1+IYI)~q_~y+ f (1+IYI)~q_~y=J~I>CX)+J~2)(X). IY-Xl$/?Iy - xl Jy-xl"/? Iy - xl (4.18) U)Banhgia J~I)(X)= f (1+lylrqdy I N 1 . IY-Xl$/? Y - xl - Taco: (4.19) J (l)() = f (1+lylrqdy< (I II) -q f ~Ii X N-I - sup +Y N-I IY-XI$R Iy - xl ly-xl:>R ly-xl:SRIy - xl d R N-Id = sup(1+!ylrq f :-1 =sup(1+!ylrqwNrN-/ IY-XI$R Izl:SRIzi ly-xl:SR 0 r = sup(1+Iylrq wNR<+00. ly-xl:SR Lugnwlntotnghi~p Trang28 OJ) Danhgia J~2)(X)= f (1+lyl)-qdy I N I . ly-4~1I Y - xl - Ta co: (4.20) (21 = f (1+lylrqdy< f (1+lylrqdy< f (1+lylrqdyJII (x) NI - NI - NI ly-xl~RIy-xl - lyl~R-lxlIy-xl - IYI~R-Ixillyl-Ixil - +00 (1 ) -'1 N-Id +00 N-I d f +r r r f r r =OJN N 1 =OJN N I - . II-Ixl Ir-Ixll - R-Ixllr-Ixll - (1+r)q Chu y rang, do R>3Ixl>O,tacolr-lxll=r-lxl:=::R-2Ixl>lxl>O,voi mQi r:=::R-Ixl. +00 N-] d D d' ' h hA f r r hA' ~. 10 0, tIc p an N I 'I Q1tl,l VOl q> . R-Ixl I r -Ixll - (1+r) V~y,tichphan (4.21) J~2)(x) hQi W khi q>1. T6 h<;5pl(;li(4.17),(4.18),(4.19)va (4.21)tathudu<;5c (4.22) \IxE JRN, A[q](x)hQitl,lkhi q>1. Hdnnua,voi q>1,tavie"t (4.23) +00 N-l d +00 N-I dJ = f r r :=::f r r 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 . Dodob6d~4.1du<;5cchungminh.- Chungminhdinhly 4.1. Bangcachthayhamg(x,u)bdi gI(x,u)=bNg(x,u)vahangs6 M trong (4.2)thaybdi bNM,taco th~giasarangbN=1makhonglamm!t tinht6ng quat. LucJnvantotnghifp Trang29 (4.24) trongdo (4.25) Ta vie'tphuongtrlnhtichphan(4.7) voi bN=1theod~ng u(x)=Tu(x)=A[g(y,u(y))](x),\/xE IRN, A [w(y)](x)= J w(y) d~-I' XE IRN. iii' I y - x I Ta chungmintb~ngphanchung.Gia su u Ia nghi~mlient\lCvaduong cua(4.24).Khi dot6nt~iXoE IRNsaochou(xo)>o.VI u lient\lcnent6nt~i ro>0 saocho: u(x»~u(xo)=L \/xEIRN, Ix-xol:::;;ro.2 Ta suytugiathie't(G2),(4.24)-(4.26)r~ng (4.26) (4.27) u(x)=A[g(y,u(y))](x)~MA[ua(y)](x) 2::MLa J dyN-l' \/x E IRN. Iy-xol:s:roI y - x I Sud\lngba'td~ngthucsau (4.28) I y - x I :::;;Iyl + Ixl :::;;(1 + Ixl)(1 + Iyl) =(1+Ixl)(1+Iyl- Xo+xo) :::;;(1+Ixl)( 1+jxoI+Iy - XoI ) :::;;(1+lxl)(1+lxol+ro)'\/x,YEIRN, Iy-xo I:::;;ro' tasuytu (4.27), (4.28)dng (4.29) u(x) 2::MLa J ~ N-l Iy-xol:s:ro I y - x I Ta vie'tl~i (4.30) trongdo > MLa 1 -(1+lxol+ro)N-lx(1+lxl)N-l J dyIy-xol:s:ro = MLa 1 OJ N X NrO (l+lxol+ro)N-l (1+lxl)N-l N ' \/xEIRN. u(x) 2::u1(x) =m](1+Ixlrq), \/x E IRN, Lugnwintotnghifp Trang30 (4.31) a N M L ())NrO ql = N -1, m] = N(1+lxo!+ro)N-I' Sa dl;lngffiQtl~nnii'ad&ngthuc(4.24),tasur tITghlthi~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. Baygidtaxetcactru'dnghQpkhacnhaucuagiatti a. 1 O::;a::;-. N-1 Ta sur ratU(4.9),(4.32)voi q =aql =a(N -1)::;1, dng Truong hQ'p1: (4.33) u(x) =+00\::IxE IRN. D6 ladi~uvo19. Truonghdp2: ~ <a <~.. N-1 N-1 Sa dl;lng(4.10)voi q =aq]=a(N -1) >1,tasur ratIT(4.32)r[tng: (4.34) u(x)2 Mm~A[(1+Iylraq,](x)=Mm~A[aql](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, trongd6 (4.36) q2=aq ] -1 m - M())N ma , 2 - I 2N-l .q2 Giasadng (4.37) u(x)2 Uk-I(X) =mk-I(1+!X!rqk-l, \::IXEIRN. N€u aqk-I>1,khid6 tadungba"td&ngthuc(4.10)voi q=aqk-I>1,tathudu'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) Luc7nvantotnghi~p Trang31 =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, trongd6 cacdtiy {qk},{mk}duQCxacd~nhbdicaccongthucquin~p sau: (4.39) a M())N mk-I k =2,3,.,1 m = N I ' qk=aqH-' k 2 qk Tli (4.31),(4.39)tathuduQc (4.40) { N - k, ntu a =1, qk= k I I-a k-I A'" 1 N (N-l)a - - , neu -<a<-, a=t:l, I-a N-l N-l Ta suytli (G2),(4.10)va(4.24)ding (4.41) U(x)2 Mm:A [(1+Iylraqk](x), '\IxE IRN. Nhuv~ytachIcftnchQnffiQts6t1,1'nhien k saGcho: (4.42) 0 <aqk ::;1. Do (4.40),tachQnffiQts6t1,1'nhien k nhusau: i) N€u a=1, tachQn k=N-1,khid6: aqk=a(N-k)=a(N-N+l)=a=1, ii) N€u ~<a<~ va a=t:1,tachQnk thoako:=;;k<ko+l,N-l N-l voi k() =21n[N -(N -1)a].1na N Tni<tnghtjp3: a =N -1 . Ta vi€t l~i(4.20) (4.43) u(x)2 M A[ua(y)](x) 2 Mm~A[(1+Iylraql](x) =Mm~A[(1+lylrN](x), '\IxEIRN, M~Hkhac,voiffiQixEIRN, IxI21,tac6. (4.44) A[(1+lylrN](x)= f (1+IYI~~INdy RN Iy- xl Lugnvantotnghifp Trang32 > f (1+lylrN d >+f'" (1+rrN d IdS- IIII NIY- II Nlr rIi \ ( y + x ) - 0 ( r +x ) - lyl=r +"'(1+rrNrN-I 1\I+rrNrN-I =OJv f II dr ~OJN f II dr. 0 (r + x )N-I I (r + x )N-I Ixl rN-Idr ~OJN [(1+r)N(r+lxj)N-I. Chuyr~ngvoimQir saocho 1~r ~Ix!taco (4.45) ( ) N r 1, 1 1 1+r ~2N va r +Ixl~ 21xJ. V~y,tacota(4.45)dug Ixl rN-Idr 1 1 Ixl dr !(1+r)N ( r +Ixl)N-I ~ 2N ( 21xl)N-2 !r( r +Ixl) (4.46) 1 1 1+Ixl N =4N-I x IxlN-I x In( 2)' "Ix E IR , Ix!~1. Ta (4.43),(4.44),(4.46)tasuyrading (4.47) 0, Ixl~1, u(x) ~V2(x) =~~ ( In 1+Ixl ) PZ, Ixl~1, IxIN-I 2 voi (4.48) PZ =1, Cz=MOJNm~ 4N-I Giasur~ng (4.49) 0, Ixl~1, u(x) ~vk-l(x) =~ Ck-l ( In1+ixi J Pk-l, Ixl ~1, IxlN-l 2 trongdo Pk-l>Ck-llacaeh~ngs6dtiong. Sud\lnggiathie't(G2)va(4.49),tasuyradug (4.50) u(x)~M A[ua(y)](x) Lwjn vantotnghi~p Trang33 ~M A[v:-1(y)](x) =M J V:-J~~Idy 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 J (r +Ixl)N II Iyl=r ) a Pk-II+r ( In(- ) +w 2 dr =M OJNC:-1J r(r +Ixl)N I1 Ta xettru'onghcJpIxl~I, taco (4.51) ( 1+r ) a Pk-l ( 1+r ) a Pk-I +00 In( -) +00 In( -) J 2 J 2dr~ dr I r(r+lxl)N-1 Ixl r(r+lxl)N-l ( 1+Ixl J a Pk-I +00 dr ~ In(-) J ( I I) N-l 2 Ixl r r +x [ II J a Pk-I +00 d ;, In(l: x). I~r(r +:)N-I - 1 ( - 1+x aPk-I (N -1)2N-Ilxt-1 In-fl) . Tli (4.50), (4.51),ta suy ra r~ng 0, Ixl~1, U(X)~Vk(X)=~ Ck ( 1+lxl ) Pk II- In- , x:2:1, IxlN-l 2 (4.52) trongdo Pk>Ckla caeh~ngs6du'dngxacdinhb~ngcaecongthuquin(;lpnhu' sau: (4.53) Pk =apk-I' C MOJ Ca k = N k I (N -1)2N-I' k =3,4,... Ta tinhfa cDngthuchiSncua Pk>Cknhovao(4.48),(4.53),nhu'sau Lu4n vantotnghi~p Trang34 (4.54) k-2 l-N N-I ak-2Pk =a , Ck =dN (dN C2) , k=3,4,... tronga6 (4.55) MOJN dN =(N -1)2N-J . Ta vie'tI~i(4.52)voi Ixi ~1,tac6 I-N 1 ( N-I 1+Ixi J a k-2 (4.56) u(x)~vk(x)=dN IX!N-I dN C21n(2) . ChQnXl saGcho (4.57) dZ-iC21n(I+lxll»I,2 Do (4.56),tasuyrarang u(xi)~ limVk(Xl)=+00.k->+oo EHylaai~uva19. Dinh194.2au'<;1cchungminhhoanta't. Chti thich4.2. i) Trong tru'ongh<;1pcua g(XI,U) chungta chu'aco ke'tlu~nv~ tru'ongh<;1p a>(n-l)/(n-2), n~3. Tuy nhien,khi g(XI,U)=Ua, n~3, (n-l)/(n-2):::;a< n/(n - 2), B. Hu trong[6]ail chungminhrAngbaitmin(4.1),(4.2),(1.7)khang c6 nghi~mdu'ong.Trong tru'ongh<;1p"gidi hf:lna =n/(n- 2)", nghi~mdu'ong khangt6nt~i(Xem [4-6]). ii) Voi a =n/(n- 2), caclacgiatrong[4]ailmatata'tcacaenghi~mkhangam khangt~mthu'ongUEc2(IR;)n C(IR;) cuabailoan { -!J.u =au(n+2)/(n-2) trong IR; , - uxn(xl,0)=bua(Xl,0) tren xn=0 trongcactru'ongh<;1psau: (j) a>0 hay a:::;0,b>B =~a(2- n)/n, (jj) a=b=0, (jjj) a=O,b<O, (4j) a <O,b=B. ._.