Existence and nonexistence of radial positive solutions of superlinear elliptic systems

Publ. Ma . 45 (2001), 399–419
EXISTENCE AND NONEXISTENCE OF RADIAL
POSITIVE SOLUTIONS OF SUPERLINEAR ELLIPTIC
SYSTEMS
Abdelaziz Ahammou
Abs ac
The main goal in his pape is o p o e he exis ence o adial
posi i e solu ions o he quasilinea ellip ic sys em
(S+)


−∆pu= (x, u, )inΩ,
−∆q =g(x, u, )inΩ,
u= =0 on∂Ω,
whe e Ω is a ball in RNand ,ga e posi i e con inuous unc-
ions sa is ying (x, 0,0) = g(x, 0,0) = 0 and some g ow h con-
di ions which co espond, oughly speaking, o supe linea p ob-
lems. Two diffe en se s o condi ions, called s ongly and weakly
coupled, a e gi en in o de o ob ain exis ence. We use he opo-
logical deg ee heo y combined wi h he blow up me hod o Gidas
and Sp uck. When Ω = RN, we gi e some sufficien condi ions o
nonexis ence o adial posi i e solu ions o Liou ille sys ems.
1. In oduc ion and main esul s
We a e conce ned wi h he exis ence o adial posi i e solu ions o he
p oblem
(S+)




−∆pu=a(x)u|u|α−1+b(x) | |β−1in Ω,
−∆q =c(x)u|u|γ−1+d(x) | |δ−1in Ω,
u= =0 on∂Ω,
whe e Ω := BRis he ball in RNcen e ed a ze o and adius R>0. He e
as usual o m>1(m=p, q), ∆mdeno es he m-Laplacian ope a o .
2000 Ma hema ics Subjec Classifica ion. 35J25, 35J60.
Key wo ds. Blow up a gumen , opological deg ee heo y.
400 A. Ahammou
Du ing he las yea s he p oblem o exis ence o (S+) has been
s udied by many au ho s, see o example, [1], [2], [7], [8], [13], [14],
[15], [17]. In pa icula in [16], Sou o p o ed he exis ence o a posi i e
solu ion aking Ω a smoo h gene al bounded domain in RNand p=
q= 2. In he case p=q, we men ion he ecen esul s o Bocca do,
Fleckinge and de Th´elin [2] whe e he au ho s p o e he exis ence o
solu ions o he ollowing p oblem:





−∆pu=a(x)u|u|α−1+b(x) | |β−1+h1(x)inΩ,
−∆q =c(x)u|u|γ−1+d(x) | |δ−1+h2(x)inΩ,
u= =0 on∂Ω,
when
p−1≥α, q −1≥δ
and
(p−1)(q−1) >βγ
wi h Ω a smoo h gene al bounded domain in RN, and h1∈Lp(Ω),
h2∈Lq(Ω). We ema k ha i h1and h2a e iden ically ze o, he
solu ion (u, ) can be a i ial solu ion. Ou goal is o find sufficien
condi ions on exponen s α,β,γ, and δin o de o ha e a adial posi i e
solu ion o (S+) when Ω is a ball.
A his poin , we in oduce wo subclasses o sys ems (S+):
1) Sys em (S+) is s ongly coupled i
i) β≥qβ +p(q−1)
pγ +q(p−1)αand ii) γ≥pγ +q(p−1)
qβ +p(q−1)δ.
2) Sys em (S+) is weakly coupled i
j) β<qβ +p(q−1)
pγ +q(p−1)αo jj) γ<pγ +q(p−1)
qβ +p(q−1)δ.
Simila ly, he weakly coupled defini ion has been in oduced in De
Figuei edo [6] when j) and jj) a e bo h sa isfied. I is in e es ing o no e
ha ou defini ion includes clea ly his case.
The exis ence and nonexis ence o quasilinea sys ems ha e been
s udied by se e al au ho s by using diffe en app oaches; ecen esul s
can be seen in [4], [5], [16]. Fo he scala case see [3], [9] and [10].
Now, le us make he ollowing assump ions
max(p, q)<N,(H1)
a, b, c, d ∈C0([0,+∞[)(H2)
Radial Posi i e Solu ions 401
and sa is y
in
∈[0,+∞[(a( ),b( ),c( ),d( )) >0,
(p−1)(q−1) <βγ.(H3)
We suppose ha (S+) is a supe linea sys em, i.e.,
p−1<α and q−1<δ.(H4)
The main esul s a e he ollowing.
Theo em 1.1. We assume ha he sys em S+is s ongly coupled and
ha he hypo heses (H1),(H2),(H3)and (H4)hold. We suppose u -
he mo e ha
max qβ+p(q−1)
γβ−(p−1)(q−1) −N−p
p−1;pγ+q(p−1)
γβ−(p−1)(q−1) −N−q
q−1≥0,(Hs)
is sa isfied. Then, he p oblem (S+)has a solu ion (u, )in C1(BR)∩
C2(BR {0}), such ha u>0and >0in BR.
Theo em 1.2. We assume ha he sys em (S+)is weakly coupled and
ha he hypo heses (H1),(H2),(H3), and (H4)hold. We suppose u -
he mo e ha
N(p−1)
N−p>α and N(q−1)
N−q>δ,(Hw)
is sa isfied. Then he p oblem (S+)has a solu ion (u, )in C1(BR)∩
C2(BR {0}), such ha u>0and >0in BR.
We will adap a he classical echniques o he mapping deg ee: we
will conside he solu ion ope a o S1associa ed o a p oblem (S+) ac -
ing in a sui able unc ional space. Then, we will look o solu ions o
he p oblem as fixed poin s o S1. As i is usual in his se ing, he
main difficul y will be o ob ain exis ence o a p io i bounds o posi i e
solu ions.
This pape is o ganized as ollows. Sec ion 2 con ains no a ions and
some defini ions o unc ional spaces and o ope a o s Sλand Tτas-
socia ed o he p oblem (S+). In Sec ion 3 we ea he nonexis ence
o adial posi i e solu ion o he Liou ille p oblem (S+
∞) associa ed o
(S+). This is he goal o Theo em 3.1. In Sec ion 4 we ge a p io i es-
ima es o he solu ions o he sys em in he s ongly coupled case and
weakly coupled case. Finally in Sec ion 5 we apply ou esul s o ob ain
he p oo o Theo ems 1.1 and 1.2.
402 A. Ahammou
2. No a ions
Besides fixing no a ions, in his sec ion we ecall he esul s ha we
use h oughou he pape . Le Rbe a posi i e numbe ; we conside he
ollowing space:
χ:= {(u, )∈C0([0,R]) ×C0([0,R]) such ha u(R)= (R)=0}
endowed wi h he no m (u, )=u∞+ ∞, which makes i a
Banach space. Le Sλand Tτ:χ→χbe he ope a o s defined by
Sλ(u, )=(S1(u, ); S2(u, )) and Tτ(u, )=(T1(u, ); T2(u, )) such
ha
S1(u, )( ):=λ1
p−1R
 1−N
0
sN−1(a(s)|u(s)|α+b(s)| (s)|β)ds1
p−1
d ,
S2(u, )( ):=λ1
q−1R
 1−N
0
sN−1(c(s)|u(s)|γ+d(s)| (s)|δ)ds1
q−1
d
and
T1(u, )( ):=R
 1−N
0
sN−1(a(s)|u(s)|α+b(s)|( (s)+τ)|β)ds1
p−1
d ,
T2(u, )( ):=R
 1−N
0
sN−1(c(s)|u(s)|γ+d(s)| (s)|δ)ds1
q−1
d .
I is well known ha , o all λ∈[0,1] and o all τ∈[0,∞[, Sλand Tτ
a e comple ely con inuous ope a o s on χ. F om he Maximum P inciple
his implies ha Sλ(χ)⊂χand ha he p oblem (S+) is equi alen o
find some non i ial posi i e fixed poin (u, )∈χo he ope a o (S1)
(by aking λ= 1) such ha u(0) = (0) = 0. The main difficul y
will be o ob ain sui able a p io i es ima es o gua an ee ha we a e in
he condi ions o he fixed poin heo em. To s udy his ques ion, we
use a Blow up a gumen like in Gidas-Sp uck pape [9]. This echnique
ans o ms he p oblem (S+) in o a p oblem
(S+
∞)




−∆pu=A∞(y)u|u|α−1+B∞(y) | |β−1in RN,
−∆q =C∞(y)u|u|γ−1+D∞(y) | |δ−1in RN,
u>0 and >0inRN,
whe e A∞,B∞,C∞and D∞a e posi i e con inuous unc ions on
[0,+∞[.
Radial Posi i e Solu ions 403
A Liou ille ype Theo em p o es ha (S+
∞) has no adial posi i e
solu ions. Fo he case A∞=D∞=0,p=q=2,β>1 and γ>1 his
kind o heo ems a e ob ained in [12] unde he hypo hesis
1
β+1+1
γ+1 >N−2
2.
3. Liou ille’s Theo em
In his sec ion we p o e he nonexis ence o adial posi i e solu ions
o he ollowing Liou ille’s sys em (S+
∞):
−∆pu≥a(x)u|u|α−1+b(x) | |β−1in RN,(3.1)
−∆q ≥c(x)u|u|γ−1+d(x) | |δ−1in RN.(3.2)
We will need he ollowing undamen al lemmas gi en in [5].
Lemma 3.1. Le w∈C1([0,R]) ∩C2(]0,R]),w≥0, sa is ying
−d
d  N−1
dw
d ( )
m−2dw
d ( )≥0on [0,R],(3.3)
wi h N>m>1. Then, o any ∈]0,R
2[we ha e:
w( )≥CN,m 
dw
d ( )
(3.4)
whe e
CN,m =m−1
N−m(1 −2m−N
m−1).(3.5)
Lemma 3.2. Le be a posi i e unc ion w∈C0([0,+∞[) ∩C2(]0,+∞[)
sa is ying
−d
d  N−1
dw
d ( )
m−2dw
d ( )≥0in ]0,+∞[(3.6)
wi h N>m>1. We suppose ha o some eal 0≥0we ha e
w( 0)>0. Then we ob ain, w( )>0 o all ≥ 0and he e exis s
CN,m >0such ha
N−m
m−1w( )≥CN,m o any ≥ 0.(3.7)

404 A. Ahammou
Theo em 3.1. We assume (H1),(H3),(H4)and one o he ollowing
assump ions:
max qβ+p(q−1)
βγ−(p−1)(q−1) −N−p
p−1;pγ+q(p−1)
βγ−(p−1)(q−1) −N−q
q−1≥0(Hs)
N(p−1)
N−p>α o N(q−1)
N−q>δ.(Hw)
Then he only adial posi i e solu ion o (S+
∞)is (0,0).
P oo : We assume ha he hypo heses (H1), (H3) and (H4) hold:
1s case: I (Hs) is sa isfied, in [5] and [11] he au ho s p o e he nonex-
is ence o adial posi i e solu ions o
−∆pu≥b(x) | |β−1in RN,(3.8)
−∆q ≥c(x)u|u|γ−1in RN.(3.9)
Applying his esul , we deduce he nonexis ence o adial posi i e solu-
ions o (S+
∞).
2nd case: Le u, be a adial posi i e solu ion o he sys em (S+
∞). We
can ew i e i as
−( N−1|u( )|p−2u( ))= N−1a( )|u( )|α+b( )| ( )|β,(3.10)
−( N−1| ( )|q−2 ( ))= N−1c( )|u( )|γ+d( )| ( )|δ
(3.11)
u(0) = (0) = 0.(3.12)
In eg a ing (3.10) and (3.11) on (0, ) and aking in o accoun ha u( ),
( )<0, o >0, we find
−u( )≥a(0)
N1
p−1
1
p−1(u( )) α
p−1
(3.13)
− ( )≥d(0)
N1
q−1
1
q−1( ( )) δ
q−1.(3.14)
Thus, om Lemma 3.1 we deduce ha he e exis s some cons an C>0
depending only o (N, p, q, a, d) such ha
u( )≥−CN,p u( )≥C p
p−1(u( )) α
p−1
(3.15)
( )≥−CN,q ( )≥C q
q−1( ( )) δ
q−1
(3.16)
o all ∈]0,+∞[.
Radial Posi i e Solu ions 405
Thus, by Lemma 3.2, he e exis s some cons an C>0 depending
only o (N, p, q, a, d) and 0>0 such ha o all ≥ 0,weha e
1≥C p
p−1−N−p
p−1
α−p+1
p−1
(3.17)
and
1≥C q
q−1−N−q
q−1
δ−q+1
q−1.(3.18)
Consequen ly i (Hw) is sa isfied, we ge
p
p−1−N−p
p−1
α−p+1
p−1>0,(3.19)
o
q
q−1−N−q
q−1
δ−q+1
q−1>0.(3.20)
Hence a con adic ion.
4. A p io i bounds o posi i e solu ions o (S+)
In his sec ion we will s udy a p io i bounds o adial posi i e solu-
ions o he sys em (S+), in he s ongly coupled and weakly coupled
cases. Fo ha , we need he ollowing lemma:
Lemma 4.1. We assume ha he e is a sequence {(˜un,˜ n)}in
(C1([0.Rn]) ∩C2(]0,R
n]))2o posi i e solu ions o he ollowing sys-
em (˜
Sn):
−d
dy yN−1
d˜un
dy (y)
p−2d˜un
dy (y)=yN−1Fn(˜un(y),˜ n(y)),(4.1)
−d
dy yN−1
d˜ n
dy (y)
q−2d˜ n
dy (y)=yN−1Gn(˜un(y),˜ n(y)),(4.2)
d˜un
dy (0) = d˜ n
dy (0) = ˜un(Rn)=˜ n(Rn)=0(4.3)
whe e
Fn(˜un(y),˜ n(y)) = An(|y|)|˜un(y)|α+Bn(|y|)|˜ n(y)+τ
n|β,
Gn(˜un(y),˜ n(y)) = Cn(|y|)|˜un(y)|γ+Dn(|y|)|˜ n(y)|δ,
406 A. Ahammou
whe e {(An,B
n,C
n,D
n)}a e sequences o posi i e unc ions in
(C0
loc
([0,+∞[))4which con e ge o (A∞,B
∞,C
∞,D
∞)in (C0
loc
([0,+∞[))4
.
I ,
lim
n→∞ τ
n=0,lim
n→∞ Rn=+∞,(4.4)
and 0<(˜un(0),˜ n(0))≤1 o all n∈N, hen he e exis s a sub-
sequence {(˜unk,˜ nk)}o {(˜un,˜ n)}con e ging in (C0
loc([0,+∞[))2and
whose limi (˜u, ˜ )is a posi i e adially symme ic solu ion o he ollow-
ing Liou ille’s sys em
(S+
∞)−∆pu=A∞(|y|)u|u|α−1+B∞(|y|) | |β−1in RN,
−∆q =C∞(|y|)u|u|γ−1+D∞(|y|) | |δ−1in RN.
P oo : We a gue as in [5]. We fi s no e ha , since (˜un(0),˜ n(0))≤1
he e exis s a subsequence {(˜unk(0),˜ nk(0))}which con e ges in R2 o
(˜u0,˜ 0) such ha (˜u0,˜ 0)≤1.
Nex we will p o e ha he es ic ion o {(˜un,˜ n)} o [0,R
]is
equicon inuous in (C0([0,R
]))2.
In ac , mul iplying (4.1) by d˜un
dy and (4.2) by d˜ n
dy we ob ain he
ollowing equa ions:
p−1
p
d
dy 
d˜un
dy (y)
p+N−1
y
d˜un
dy (y)
p
(4.5)
+Fn(˜un(y),˜ n(y))d˜un
dy (y)=0
q−1
q
d
dy 
d˜ n
dy (y)
q+N−1
y
d˜ n
dy (y)
q
(4.6)
+Gn(˜un(y),˜ n(y))d˜ n
dy (y)=0.
F om (4.5) and (4.6) i ollows ha
p−1
p
d
dy 
d˜un
dy (y)
p+an
d˜un
dy (y)≤0(4.7)
q−1
q
d
dy 
d˜ n
dy (y)
q+bn
d˜ n
dy (y)≤0(4.8)
whe e
an= 2 max
s∈[0,R]{An;Bn}and bn= 2 max
s∈[0,R]{Cn;Dn}.(4.9)
Radial Posi i e Solu ions 407
Since he sequence {(An,B
n,C
n,D
n)}con e ges o (A∞,B
∞,C
∞,D
∞)
in (C0([0,R
]))4, hen he e exis a1>0 and b1>0 such ha an<a
1
and bn<b
1 o all n∈N. Hence, om (4.7), (4.8) and by in eg a ing
om 0 o y∈[0,R
] we find ha
p−1
p
d˜un
dy (y)
p
+a1y
0
d˜un
dy (s)ds ≤0(4.10)
q−1
q
d˜ n
dy (y)
q
+b1y
0
d˜ n
dy (s)ds ≤0(4.11)
and hence ha

d˜un
dy (y)
≤C1
(4.12)

d˜ n
dy (y)
≤C2
(4.13)
uni o mly in n, which imply he equicon inui y o he sequence {(˜un,˜ n)}.
Thus, by he Ascoli-A zel`a Theo em, he e exis s a subsequen-
ce {(˜unk,˜ nk)}, such ha {(˜unk,˜ nk)}con e ges o (˜u, ˜ ) when k→∞,
in (C0([0,R
]))2. Now om (4.1) and (4.2) we ha e ha {(˜unk,˜ nk)}
sa is y
˜unk(0)−˜unk(y)=
y
01
N−1
0
sN−1Fnk(˜unk(s),˜ nk(s))ds1
p−1
d (4.14)
˜ nk(0)−˜ nk(y)=
y
01
N−1
0
sN−1Gnk(˜unk(s),˜ nk(s))ds1
q−1
d .(4.15)
By he Domina ed Con e gence Theo em we ob ain ha (˜u, ˜ ) sa is y
˜u0−˜u(y)=y
01
N−1
0
sN−1F∞(˜u(s),˜ (s))ds1
p−1
d (4.16)
˜ 0−˜ (y)=y
01
N−1
0
sN−1G∞(˜u(s),˜ (s))ds1
q−1
d (4.17)
whe e
F∞(˜u(s),˜ (s)) = A∞(s)|˜u(s)|α+B∞(s)|˜ (s)|β,
G∞(˜u(s),˜ (s)) = C∞(s)|˜u(s)|γ+D∞(s)|˜ (s)|δ,
414 A. Ahammou
whe e
Fn(¯un(y),¯ n(y)) = An(y)|¯un(y)|α+Bn(y)
¯ n(y)+ τn
σk
n
β,(4.65)
Gn(¯un(y),¯ n(y)) = Cn(y)|¯un(y)|γ+Dn(y)|¯ n(y)|δ
(4.66)
and
An(y)=ay n
σn p
nσn−p+l(α−p+1)
Bn(y)=by n
σn p
nσn−p−l(p−1)+kβ,
(4.67)
Cn(y)=cy n
σn q
nσn−q+k(q−1)+lγ
Dn(y)=dy n
σn q
nσn−q+k(δ−q+1),
(4.68)
Rn=σn
n
R.(4.69)
By choosing he posi i e numbe s land kas in (4.37) we ob ain
An(y)=ay n
σn p
nσnlα−kβ,B
n(y)=by n
σn p
n,(4.70)
Cn(y)=cy n
σn q
n,D
n(y)=dy n
σn q
nσnkδ−lγ .(4.71)
1s case: lim
n→+∞¯un(0) >0.
By choosing n=σ
−lα+kβ
p
n, as in Lemma 4.1, we ob ain ha he fi s
equa ion o he sys em (S+
n) con e ges o he equa ion
−∆p¯u=a(0)¯u|¯u|β−1in RN,(4.72)
whe e ¯u>0 is he limi o ¯un.
Thus, om (Hw) and Liou ille’s Theo em 3.1, he equa ion (4.72) has
no adial posi i e solu ion. Hence he con adic ion ollows.

Radial Posi i e Solu ions 415
2nd case: lim
n→+∞¯un(0) = 0.
We ecall ha ¯un(0) = ˜un(0) and ¯ n(0) = ˜ n(0) whe e (˜un,˜ n) a e
he unc ions defined in he p oo o P oposi ion 4.2. Then om ou
claim, we deduce ha
lim
n→+∞¯ n(0) >0 and γ<pγ +q(p−1)
qβ +p(q−1)δ.
Consequen ly, in his case i suffices o choose n=σ
−kδ+lγ
q
n. Then, we
ob ain ha {¯ n}con e ges o ¯ and ¯ sa isfies
−∆q¯ =d(0)¯ |¯ |δ−1in RN.
Hence, as in he 1s case, we ge a con adic ion.
P oposi ion 4.3. Assume ha (H1),(H2),(H3), and (H4)hold. Mo e-
o e , assume ha one o he assump ions (Hs)o (Hw)holds. Then,
he amily o ope a o s (Tτ)τ≥0sa isfies he ollowing p ope ies:
(P1)∃τ0>0such ha , i Tτhas a fixed poin (u, )τ o some τ≥0
hen τ≤τ0.
(P2)∃C0>0such ha , i (u, )τis a fixed poin o Tτwi h τ≤τ0+1,
hen (u, )τ≤C0.
P oo : Fi s , om he Maximum P inciple, i ollows ha he p oblem
(u, )=Tτ((u, ))(4.73)
is equi alen o find posi i e solu ions u, o he ollowing sys em
−( N−1|u( )|p−2u( ))= N−1a( )|u( )|α+b( )| ( )+τ|β,(4.74)
−( N−1| ( )|q−2 ( ))= N−1c( )|u( )|γ+d( )| ( )|δ,(4.75)
u(0) = (0) = u(R)= (R)=0.(4.76)
I ollows ha 0 ≤u( ), 0 ≤ ( ) and by in eg a ing on [0, ] we ob ain
−u( )≥C 1
p−1( ( )+τ)β
p−1,(4.77)
− ( )≥C 1
q−1(u( )) δ
q−1.(4.78)
Thus om (4.77)
−u( )≥C 1
p−1τβ
p−1.(4.79)
416 A. Ahammou
By in eg a ing (4.79) om 0 o R, we ob ain ha
u(0) ≥CR p
p−1τβ
p−1.(4.80)
Now we suppose ha (P1) is no ue. Then he e is a sequence {τn}
such ha
lim
n−→ +∞τn=+∞(4.81)
and such ha o each τn he e exis s a solu ion (un,
n) o (4.74)–(4.76).
Then om (4.80) and (4.81), we ha e
lim
n−→ +∞(un,
n)=+∞.(4.82)
Hence, espec i ely i (Hs) [o (Hw)] holds hen om P oposi ion 4.1
[P oposi ion 4.2] we ob ain a con adic ion.
We a gue simila ly i (P2) is no ue.
Then o all n∈N he e exis s τn≤τ0and a solu ion (un,
n)o
(4.74)–(4.76) such ha (un,
n)>n, his implies ha
lim
n−→ +∞(un,
n)=+∞.
Hence, om P oposi ion 4.1, and P oposi ion 4.2 we deduce a con a-
dic ion and P oposi ion 4.3 is p o ed.
5. P oo o he heo ems
P oposi ion 5.1. We assume (H1),(H2)and (H3). Then he e exis s
aρ1>0such ha :
∀ρ∈[0,ρ
1[and ∀λ∈[0,1], we ha e ha (0,0) is he only fixed poin
o Sλ,inB(0,ρ).
P oo : Le us ake λ∈[0,1] and (u, )∈χsuch ha
(u, )=Sλ(u, )(5.1)
wi h (u, )=ρ>0. No ice ha by he defini ion o Sλwe ge u≤0,
≤0in[0,R]. By in eg a ing on [0,R]weha e
u(0) =λ1
p−1R
0 1−N
0
sN−1(a(s)|u(s)|α+b(s)| (s)|β)ds1
p−1
d ,(5.2)
(0) =λ1
q−1R
0 1−N
0
sN−1(c(s)|u(s)|γ+d(s)| (s)|δ)ds1
q−1
d .(5.3)
Radial Posi i e Solu ions 417
Hence (u, )=u(0) + (0). Thus, om (H3) he e exis wo num-
be s l>0 and k>0 such ha
β
p−1>l
k>q−1
γ.(5.4)
Deno e
σ=(u(0))1
l+( (0)) 1
k,(5.5)
hen (u, )<σ
l+σkand om (5.2) and (5.3) we deduce,
(u(0))1
l≤Cλ 1
l(p−1) [σlα +σkβ]1
l(p−1) ,(5.6)
( (0))1
k≤Cλ 1
k(q−1) [σlγ +σkδ]1
k(q−1) .(5.7)
Summing up (5.6) and (5.7), we deduce ha σsa isfies
(5.8) 1 ≤Cλ 1
l(p−1) [σl(α−p+1) +σkβ−l(p−1)]1
l(p−1)
+Cλ 1
k(q−1) [σlγ−k(q−1) +σk(δ−q+1)]1
k(q−1) .
Hence, he e exis m1>0, m2>0 and C>0 such ha
C<λ
m1σm2≤σm2.(5.9)
On he o he hand, by defini ion o σ, he e exis s m3>0 such ha
σ≤2ρm3.(5.10)
Then, om (5.9) and (5.10) i suffices o choose ρ1such ha 21+ 1
m3ρ1=
C1
m2m3, o comple e he p oo o P oposi ion 5.1.
P oo o Theo em 1.1: F om P oposi ion 4.3, i ollows ha o ρ2=
C0+ 1, he equa ion (u, )=Tτ((u, )) wi h (u, )∈∂B(0,ρ
2) has no
solu ion o τ∈[0,τ
0+ 1]. Then, deg(I−Tτ,B(0,ρ
2),0) is well defined
and by he p ope y o opological deg ee we ge ha
deg(I−Tτ,B(0,ρ
2),0) = cons an ,∀τ∈[0,τ
0+1].(5.11)
Mo eo e , since o τ1=τ0+ 1, he ope a o Tτ1has no a fixed poin in
B(0,ρ
2), we ha e
deg(I−Tτ1,B(0,ρ
2),0) = 0.(5.12)
Then, i ollows om (5.11), ha
deg(I−T0,B(0,ρ
2),0) = deg(I−Tτ1,B(0,ρ
2),0) = 0.(5.13)
Mo eo e , om P oposi ion 5.1, o ρ1>0 sufficien ly small we ha e
deg(I−Sλ,B(0,ρ
1),0) = cons an ∀λ∈[0,1].(5.14)
418 A. Ahammou
Hence
deg(I−S1,B(0,ρ
1),0) = deg(I−S0,B(0,ρ
1),0) = +1.(5.15)
Then, since S1=T0and om (5.13) and (5.15), by he excision p ope y
we ob ain
deg(I−S1,B(0,ρ
2) B(0,ρ
1),0) =0.(5.16)
So, he e is a fixed poin (u, )o S1. Hence we ob ain he conclusions
in Theo em 1.1.
P oo o Theo em 1.2: Using P oposi ion 4.3, he p oo o Theo em 1.2
is simila as he p oo o Theo em 1.1.
Acknowledgemen . The au ho hanks p o esso F an¸cois de Th´elin
o sugges ing he s udy o his p oblem and o help ul commen s.
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D´epa emen des Ma h´ema iques e In o ma ique
Facul ´e des Sciences
Uni e si ´e Cadi Ayyad
El Jadida, BP20
Ma oc
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 28 de se emb e de 2000,
da e a e si´o ebuda el 23 de eb e de 2001.