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'dngcua phu'dng
trlnh tich phan phi tuye'n sau day
(4.1)
U
(
x
)
=b
f
g(y,u(y)) d
y
"dx E IRN
N N-l' ,
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 xIR+ ~ 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, "dxE 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)

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 EIRn-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) = 2
f
g(y,u(y») dy
(N -l)OJ '
I I
N-I' '\Ix E IRN.
N+l IR' y-x

J
(1 + Iyl rq
d
= +
J
oo (1+rrq d
J
d:S
-
1III
N1Y
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:
(4.13)
J
N-l
dSr = r OJN'
Iyl=r
LucJnvan tot nghifp
Trang27
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'

(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$/?Iy - xl Jy-xl"/? Iy - xl
(4.18)
U)Banhgia J~I)(X)=
f
(1+lylrqdy
I
N 1
.
IY-Xl$/? Y - xl -
Ta co:
(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 + !ylrq wN rN-/

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,ta colr-lxll=r-lxl:=::R-2Ixl>lxl>O, voi mQi
r:=::R-Ixl.
+00 N-]
d
D d
' '
h h
A
f
r r
h
A' ~.
1
0 0, tIc p an N I 'I Q1 tl,l VOl q> .
R-Ixl I r -Ixll - (1+ r)
V~y, tich phan
(4.21 )
J~2)(x) hQi W khi q > 1.
T6 h<;5pl(;li(4.17), (4.18), (4.19) va (4.21) ta thu du<;5c
(4.22)
\Ix E JRN, A[q](x) hQi tl,lkhi q > 1.
Hdn nua, voi q > 1, ta vie"t
(4.23)
+00 N-l
d
+00 N-I

(4.26)
(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)
:::;;(1 + Ixl)( 1+ jxoI+ Iy - Xo 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 ~ N-l
Iy-xol:s:ro I y - x I
Ta vie't l~i
(4.30)
trong do
> MLa 1
-(1+lxol+ro)N-lx(1+lx l
)N-l J dy
Iy-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,
Lugn win tot nghifp Trang30
(4.31 )
a N
M L ())NrO

N
ma
, 2 - I
2N-l .
q2
Giasa 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&ngthuc (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:
(4.39)
a
M())N mk-I k = 2,3,.,
1 m = N I '
qk=aqH-' k 2 qk
Tli (4.31), (4.39) ta thu duQc
(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
dS
-
IIII
NIY-
II
Nlr 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
r 1, 1 1
1+ r ~ 2N va r + Ixl~ 21xJ.
V~y, ta co ta (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 Ixl N-I x In( 2)' "Ix E IR , Ix!~ 1.

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 I
I Iyl=r
)
a Pk-I
I+r
(
In(- )
+w 2 dr
= M OJNC:-1J r(r + Ixl)N I
1
Ta xet tru'ong hcJp Ixl~ I, ta co
(4.51)
(
1+ r
)
a Pk-l
(
1+ r
)
a Pk-I
+00 In( -) +00 In( -)
J
2
J
2
dr~ dr

sau:
(4.53)
Pk =apk-I'
C
MOJ
C
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
Lu4n van tot nghi~p
Trang 34
(4.54)
k-2 l-N N-I ak-2
Pk
=a , Ck =dN (dN C2) , k=3,4,
trong a6
(4.55)
MOJN
dN = (N -1)2N-J .
Ta vie'tI~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
(4.57) dZ-iC21n(I+lxll»I,
2