Bay gid ta xet cac tru'dng hQpkhac nhau cua gia tti a.
Trang 1LU{J.nvan tot nghi~p Trang 24
CHUONG 4 st; KHONG TON T~I NGHIEM DUONG
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 )=b N f g(y,u(y)) N-l' dy "dx E IRN ,
IRN Iy- xl
trang do bN = 2((N -l)lUN+ltl voi lUN+1la dit%ntich 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)
(4.4)
- vxn (Xl ,0) = g(XI, V(XI ,0)), Xl E IRn-l,
thoa cae tinh cha't:
lim
Ivex)I + R sup ov (x)
J= 0,
k HOO Ixl=R,xn>O Ixl=R,xn>O fun
(82 )
d day g: IRn-1x [0,+00)~ [0,+00)cho tru'oc thoa cac di~u kit%nsau:
(G])
(G2)
g la ham lien t\!e,
3a~0,3M>0: g(xl,v)~Mva, "dv~O, "dxl EIRn-l.
va mQt sf) di~u kit%nph\! se d~t sau
Trang 2Lu(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 phudngtrinh tich phan sau day
f g(l,vel ,0) ) dl I n
(n-2)OJn Rn-I
(1
I
2
)
Y -x' +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'<!tbdix=(xl,xn) va Y=(/,Yn)'
Ta cling gia sa rang gia tri bien V(XI,0) cua nghi~m v cua bai loan (4.3),
(4.4) thoa tinh cha'"t:
(s3)Tich phan f g(/, v(/ ,0)) d/
/Rn-I I yl - xl In-2
t<3n t~i, VXI E IRn-l.
Gia sa rang bai loan (4.3), (4.4) co nghi~m dudng v = V(XI,xn) thoa cac di~u ki~n (SI)- (S3)' Dung dinh 1'9hQi t\l bi ch~n Lebesgue, cho Xn ~ 0+ trang
phu'dngtrlnh tich phan (4.5), nho vao (S3)' ta thu duQc:
v(xl ,0) = 2 f g(l, vel ,0))_~l , vxl E IRn-l.
(n - 2)OJn /Rn-I Il - Xl In
(4.6)
Ta vi€t l~i phudng trinh tich phan (4.6) bang cach thay l~i cac ky hi~u
n -1 = N, Xl = x, l =Y, V(XI ,0) = U(XI), i.e.,
(4.7) u(x) = (N -l)OJ2 f'g(y,u(y») dy
N-I' '\Ix E IRN.
N+l IR' y-x
Khi do, ta phat bi~u k€t qua chinh trang ph~n nay nhu sau:
Djnh ly 4.1 Ntu g thoa cae gia thitt (GJ, (Gz) vdi N > 2 va 0 ~ a ~ N~l' Khi
do, phl1ang trinh tick phdn (4.7) khong c6.nghi~m lien t~c dl1ang.
Trang 3Lu(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 1/1'J ( 1+ xg (1,0;~-I ) 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:
lRN Iy - x I
Khi an
(4.10)
A[q](x) hQifl;l va A[q](x)~ (q-I)2 OJNN-I (1+ x)q111 -I' ne'u q>1.
Chung minh b6 d~ 4.1
a) Gia sa q :::;1 Chti 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
A[q](x)= J (1+lyl )-:-~y
[RN Iy - x I
>
J (1 + Iylrq d =+
J
oo (1+ rrq d
Jd:S
II Nlr r'
1/' ( Y + x ) - 0( r + x ) - lyl=r
(4.12)
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:
Iyl=r
Trang 4LucJn van tot nghifp Trang 27
Do do, ta suy tu (4.12), (4.13) ding
Tich philo Jq =0f(r+rllx) N-I (1+r)q r philo ky khi q~ 1va hQi t1;1khi q > 1.
Do do, rich philo
a) Gia sa q > 1.
i) Xet t~i x= 0, ta co
f
oo(1+ rrq rl-Ndr= w +foo~
/ / A +00 dr A' ,
Do do, hch Phan f hOI tu VI q > 1
V~y, rich philo
(4.17) A [q](0) hQi t1;1khi q > 1.
ii) Xet t~i x =F0, chQn R > 31xJ> O Ta vie't l~i A[q](x) thanh t6ng hai tich philo
A[q](x)= f (1+IYI)~q_~y+ f (1+IYI)~q_~y =J~I>CX)+J~2)(X).
IY-Xl$/?Iy - xl Jy-xl"/? Iy - xl
(4.18)
I
N 1
IY-Xl$/? Y - xl
-Ta co:
(4.19) J (l) () =
f (1+lylrqdy< (I II)-q f ~
IY-XI$R Iy - xl ly-xl:>R ly-xl:SRIy - xl
= sup (1+ !ylrq f :-1 = sup (1 + !ylrq wNr N-/
= sup (1 + Iylrq wNR < +00.
ly-xl:SR
Trang 5Lugn wln tot nghi~p Trang 28
OJ) Danhgia J~2)(X)= f (1+lyl)-qdy
I
N I
ly-4~1I Y xl
-Ta co:
f (1+lylrqdy < f (1+lylrqdy < f (1+lylrqdy
ly-xl~R Iy-xl - lyl~R-lxl Iy-xl - IYI~R-Ixillyl-Ixil
-+00
d
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 N-]
d
D0 d0, tIc' ' h p anhA f r N I r 'I hQ1 tl,l VOl q>A' ~. 1
R-Ixl Ir -Ixll - (1+ r)
V~y, tich phan
T6 h<;5pl(;li(4.17), (4.18), (4.19) va (4.21) ta thu du<;5c
Hdn nua, voi q > 1, ta vie"t
(4.23)
q o(r+lxl)N-I(1+r)q Ixl(r+lxl)N-I(1+r)q
:=::J( r+r )N-I(1+r)q =2N-I J(1+r)q
Do do b6 d~ 4.1 du<;5cchung
minh.-Chung minh dinh ly 4.1.
Bang cach thay ham g(x,u) bdi gI(x,u) = bNg(x,u) va hang s6 M trong (4.2) thay bdi bNM, ta co th~ gia sa rang bN= 1 ma khong lam m!t tinh t6ng
quat
Trang 6LucJn van tot nghifp Trang29
(4.24)
trong do
(4.25)
Ta vie't phuong trlnh tich phan (4.7) voi bN = 1 theo d~ng
u(x) = Tu(x) = A [g(y,u(y))](x), \/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:
Ta suy tu gia thie't (G2),(4.24)-(4.26) r~ng
(4.26)
(4.27) u(x) = A[g(y,u(y))](x) ~ MA[ua(y)](x)
Iy-xol:s:ro I y - xI
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)
:::;;(1 + Ixl)( 1+ jxo I+ Iy- XoI)
:::;;(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 ~ N-l
Iy-xol:s:ro I y- xI
Ta vie't l~i
(4.30)
trong do
-(1+lxol+ro)N-lx(1+lx l )N-l Iy-xol:s:ro J dy
N
u(x) 2::u1(x) = m](1 + Ixlrq), \/x E IRN,
Trang 7Lugn win tot nghifp Trang 30
(4.31 )
M L ())NrO
Sa dl;lng ffiQt l~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
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: ~ < a <~.
. N-1 N-1
Sa dl;lng (4.10) voi q = a q] = a(N -1) > 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
trong d6
(4.36) q2 =aq ] -1 , m2 -- M()) N maI
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"t d&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)
Trang 8Luc7nvan 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} duQC xac d~nh bdi cac cong thuc qui n~p sau:
(4.39)
a M())N mk-I k = 2,3,.,
Tli (4.31), (4.39) ta thu duQc
(4.40)
{
Ta suy tli (G2),(4.10) va (4.24) ding
(4.41) U(x) 2 Mm: A [(1 + Iylraqk ](x), '\Ix E IRN.
Nhu v~y ta chI cftn chQn ffiQt s6 t1,1'nhien k saG cho:
(4.42) 0 < aqk ::;1.
Do (4.40), ta chQn ffiQt s6 t1,1'nhien k nhusau:
ii) N€u ~<a<~ va a=t:1, tachQnk thoa ko :=;;k<ko+l,
1na
N
Tni<tng htjp 3: 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)
M~Hkhac,voiffiQi xEIRN, IxI21,tac6
RN Iy - xl
Trang 9Lugn van tot nghifp Trang 32
>
f (1+lyl rN d >+
f
'" (1+rrN d IdS
-IIII NIY- IINlr r
Ii\ ( 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.
Chu yr~ng voi mQi r sao cho 1 ~ r ~ Ix!ta co
(4.45)
N
1+ r ~ 2 N va r + Ixl ~ 21xJ.
V~y, ta co ta (4.45) dug
!(1 + r)N ( r + Ixl)N-I ~ 2N ( 21xl)N-2 !r( r + Ixl)
(4.46)
=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
(4.47)
u(x) ~ V2(x) =~~
(
In 1+ Ixl
)PZ, Ixl ~ 1,
IxIN-I 2
voi
4N-I
Gia su r~ng
(4.49)
u(x) ~ vk-l (x) = ~ Ck-l
(
In 1+ ixi
J Pk-l, Ixl~ 1,
IxlN-l 2
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)
Trang 10Lwjn van tot nghi~p Trang 33
~ M A[v:-1(y)](x) = M J V:-J~~I dy
RN Iy - xl
- I?' (lyl+lxl)N-1 Y - lyl~1(lyl+lxl)N-1 Y
+W V:-I (y) dSr
= M Jdr J ( r + Ixl)I Iyl=r N I
)
a Pk-I
I+r
(In(- )
= M OJNC:-1J1 r(r + Ixl)N I
Ta xet tru'ong hcJp Ixl~ I, ta co
1+ r
)
a Pk-l
(
1+ r
)
a Pk-I
J
2
J
2
I r(r+lxl)N-1 Ixl r(r+lxl)N-l
(
1+Ixl J
a Pk-I +00 dr
N-l
[
II J
a Pk-I +00 d
(
(N -1)2N-Ilxt-1 In-fl) .
Tli (4.50), (4.51), ta suy ra r~ng
(
1+lxl )
Pk
II
(4.52)
trong do Pk>Ck la cae h~ng s6 du'dng xac dinh b~ng cae cong thu qui n(;lpnhu'
sau:
a
k = 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
Trang 11Lu4n van tot nghi~p Trang 34
(4.54) Pk =a k-2 , Ck =dN l-N (dN N-I C2) ak-2 , k=3,4,
trong a6
dN =(N -1)2N-J .
Ta vie't I~i (4.52) voi Ixi ~ 1, ta c6
I-N 1
(
N-I 1+Ixi
J
a k-2
(4.56) u(x)~vk(x)=dN IX!N-I dN C21n(2)
ChQn Xl saGcho
2
Do (4.56), ta suy ra rang u(xi) ~ lim Vk(Xl) k->+oo =+00.
EHy la ai~u va 19
Dinh 194.2 au'<;1cchung minh hoan ta't
Chti thich 4.2
i) Trong tru'ong h<;1pcua g(XI,U) chung ta chu'a co ke't lu~n v~ tru'ong h<;1p a>(n-l)/(n-2), n~3 Tuy nhien, khi g(XI,U)=Ua, n~3, (n-l)/(n-2):::;a<
c6 nghi~m du'ong Trong tru'ong h<;1p"gidi hf:ln a = n/(n - 2) ", nghi~m du'ong khang t6n t~i (Xem [4-6])
ii) Voi a = n/(n - 2), cac lac gia trong [4] ail ma ta ta't ca cae nghi~m khang am khang t~m thu'ong UE c2 (IR;) n C(IR;) cua bai loan
{
- uxn(xl ,0) = bua (Xl,0) tren xn = 0
trong cac tru'ong h<;1psau:
(j) a> 0 hay a:::;0, b > B = ~a(2 - n)/n,
(jj) a = b = 0,
(jjj) a=O,b<O,