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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)

~v=O, xEIR: ={(xl,xn):xl EIRn-l,xn >O},
- 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:),
lim
(
SUP I vex) 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.
Lu(jn van tot nghi~p
Trang25
Khi do, n€u g 1a ham lien t\lCva nghi~m v bai loan (4.3), (4.4) co cac
tinh cha'"t(SI)' (S2)'
thi v 1anghi~mcua phudngtrinhtichphan sau day
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
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

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.
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'dngtrlnh 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 t6n t~iva du'dng.
1/1' ( 1+ x )
Tru'de h€t ta e~n mQt sO'ba't d&ngthue 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)~ OJNN-I 111 -I' ne'u q>1.
(q-I)2 (1+ x)q
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 + 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'

+00 N-I d
Tich philo Jq = f rll N-I r philo ky khi q ~ 1 va hQi t1;1khi q > 1.
0 (r+ x) (1+r)q
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
(4.16)
-
f
(1+ Iylrq dy - +
f
oo(1+ rrq rl-Ndr =w +
f
oo~ .
A[q](O)-
I I
N-I -wN N-I N (I+r)q
m~ y 0 r, 0
/ / A +00 dr A' ,
Do do, hch
P
han
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 =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-/

IY-XI$R Izl:SRIzi ly-xl:SR 0 r
= sup (1 + Iylrq wNR < +00.
ly-xl:SR
Lugn 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:
(4.20)
(21 =
f
(1+lylrqdy <
f
(1+lylrqdy <
f
(1+lylrqdy
JII (x) NI - NI - NI
ly-xl~R Iy-xl - lyl~R-lxl Iy-xl - IYI~R-Ixillyl-Ixil -
+00
(1 )
-'1 N-I
d
+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,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

d
J =
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 .
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.
LucJnvan tot nghifp Trang 29
(4.24)
trongdo
(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 EIRN,
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 XoEIRN sao cho u(xo)> o. VI u lien t\lc nen t6n t~i
ro > 0 sao cho:
u(x»~u(xo)=L \/xEIRN, Ix-xol:::;;ro.
2
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)
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

ql = N -1, m] = N(1+lxo!+ro)N-I'
Sa dl;lngffiQtl~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
Truong hQ'p1:
(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
(4.35)
U(X)2u2(x)=m2(1+lxlrq2, \::IxEIRN,
trong d6
(4.36)
q2 =aq ]
-1 m
- M())

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-a N-l N-l
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 ffiQts6 t1,1'nhien k saGcho:
(4.42)
0 < aqk ::;1.
Do (4.40), ta chQn ffiQts6 t1,1'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 thoa ko :=;;k<ko+l,
N-l N-l
voi k() = 21n[N -(N -1)a].
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)
=Mm~A[(1+lylrN](x), '\IxEIRN,
M~Hkhac,voiffiQi xEIRN, IxI21,tac6.
(4.44)
A[(1+lylrN](x)= f (1+IYI~~IN dy
RN Iy - xl
Lugn van tot nghifp Trang 32
>
f
(1+lylrN
d
> +
f
'" (1+rrN d

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.

Ta (4.43), (4.44), (4.46) ta suy ra ding
(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
Gia su r~ng
(4.49)
0, Ixl~1,
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)
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 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

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)
trong do Pk>Ckla cae h~ng s6 du'dngxac dinh b~ng cae cong thu qui n(;lpnhu'

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

Do (4.56), ta suy ra rang u(xi) ~ lim Vk(Xl) =+00.
k->+oo
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<
n/(n - 2), B. Hu trong [6] ail chung minh rAngbai tmin (4.1), (4.2), (1.7) khang
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 UEc2 (IR;) n C(IR;) cua bai loan
{
-!J.u
=au (n+2)/(n-2) trong IR; ,
- 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,
(4j) a < O,b= B.