Hous on Jou nal o Ma hema ics
c
2003 Uni e si y o Hous on
Volume 29, No. 3, 2003
POSITIVE SOLUTIONS FOR THE DEGENERATE LOGISTIC
INDEFINITE SUPERLINEAR PROBLEM: THE SLOW
DIFFUSION CASE
MANUEL DELGADO AND ANTONIO SU´
AREZ
Communica ed by Ha¨ım B ezis
Abs ac . In his wo k we s udy he exis ence, s abili y and mul iplici y
o he posi i e s eady-s a es solu ions o he degene a e logis ic inde ini e
supe linea p oblem. By an adequa e change o a iable, he p oblem is
ans o med in o an ellip ic equa ion wi h conca e and inde ini e con ex
nonlinea i ies. We use singula spec al heo y, he Le ay-Schaude deg ee,
bi u ca ion and mono ony me hods o ob ain he exis ence esul s, and ixed
poin index in cones and a Picone iden i y o show he mul iplici y esul s
and he exis ence o a unique posi i e solu ion linea ly asymp o ically s able.
1. In oduc ion
In his wo k we analyze he exis ence, s abili y and mul iplici y o nonnega i e
and non- i ial solu ions o he degene a e logis ic inde ini e supe linea model
(1) ºLwm=λw +a(x)w2in Ω,
w= 0 on ∂Ω,
whe e Ω is a bounded and egula domain o IRN,N≥1; m > 1; λ∈IR ha
i will be conside ed pa ame e , a∈Cα(Ω), α∈(0,1), changes sign and Lis a
2000 Ma hema ics Subjec Classi ica ion. 35B32; 35J25; 35K57; 92D25.
Key wo ds and ph ases. Degene a e logis ic inde ini e equa ion, singula eigen alue p ob-
lems, inde ini e supe linea p oblems, mul iplici y esul s.
The second au ho is g a e ul o P o esso s D. A coya and J. L´opez G´omez o hei help ul
commen s. The au ho s hank o CICYT and MCYT o Spain o esea ch suppo unde g an s
MAR98-0486 and BFM2000-0797 espec i ely.
801
802 M. DELGADO AND A. SU ´
AREZ
second o de uni o mly ellip ic ope a o o he o m
(2) Lu:= −
N
X
i,j=1
Di(aijDju) +
N
X
i=1
bi(x)Diu,
wi h aij =aji ∈C1(Ω) and bi∈C1(Ω).
We de ine a+(x) := max{a, 0},a−:= min{a, 0}, and so a=a++a−and
A+:= {x∈Ω : a+(x)>0}, A−:= {x∈Ω : a−(x)<0}, A0:= Ω (A+∪A−)
and assume ha A±a e open and su icien ly smoo h, and ha a±a e bounded
away om ze o on compac subse s o A±.
Equa ion (1) can be ega ded as a model o a s eady-s a e single species inhab-
i ing in Ω, so w(x) s ands o he popula ion densi y. The pa ame e λ ep esen s
he g ow h a e o he species and a(x) desc ibes he limi ing e ec s o c owding
in he species in A−and he in aspeci ic coope a ion in A+. Obse e ha in A0
he popula ion is ee om c owding and symbiosis e ec s. Finally, Lmeasu es
he di usi i y and he ex e nal anspo e ec s o he species. The e m m > 1
was in oduced in [18] by desc ibing he dynamics o biological popula ion whose
mobili y depends upon hei densi y. In his con ex , m > 1 means ha he
di usion, he a e o mo emen o he species om high densi y egions o low
densi y ones, is slowe han in he linea case (m= 1), which seems gi e mo e
ealis ic models, see [18].
The change o a iable u:= wm ans o ms (1) in o
(3) ºLu=λuq+a(x)upin Ω,
u= 0 on ∂Ω,
wi h q= 1/m and p= 2/m. Along his wo k we suppose
(H) 0 < q < 1< p
so, we a e assuming ha 1 <m<2, we call his case he slow di usion. When
q= 1, ha is m= 1, (3) has been s udied ex ensi ely in he las yea s, see o
example [2], [3], [4], [6], [11], [12], [13], [17], [21], [22] and [24]. Roughly speaking,
in hese wo ks i was p o ed ha om he i ial solu ion u= 0 a λ=σ1[L]
bi u ca es an unbounded con inuum o posi i e solu ions supe c i ically ( esp.
subc i ically) i
D:= ZΩ
aϕp
1ϕ∗
1<0 ( esp. >0,)
whe e σ1[L] = σ1[L∗], ϕ1and ϕ∗
1s and o he p incipal eigen alue and he
p incipal eigen unc ion o Land i s adjoin L∗in Ω unde homogeneous Di ichle
SLOW DIFFUSION AND SUPERLINEAR PROBLEM 803
bounda y condi ions. Mo eo e , in [6] and [11], assuming some es ic ions on p
and on he decay o a+nea ∂A+, he au ho s ob ained a p io i bounds o λin
compac in e al o IR. So, i o example D < 0, in [6] was shown he exis ence
o posi i e solu ion o λ∈(∞, λ∗] o some λ∗> σ1[L] and he exis ence o ,
a leas , wo posi i e solu ions o λ∈(σ1[L], λ∗). Recen ly, he exis ence o
a unique solu ion linea ly asymp o ically s able in (σ1[L], λ∗) and mul iplici y
esul s in such in e al ha e been showed in [17].
The esul s when q < 1 a e comple ely di e en . Indeed, in he speci ic case
L=−∆ and a(x) = 1, Amb ose i, B ezis and Ce ami, in he pionee wo k [7],
p o ed he exis ence o , a leas , wo posi i e solu ions o (3) in he in e al (0, λ∗)
i p < (N+ 2)/(N−2) and whe e λ∗is he sup emum o he se
Λ := {λ > 0 : (3) has a posi i e solu ion.}
To ob ain his esul , he au ho s used he sub-supe solu ion and a ia ional
me hods. Mo e ecen ly, in [10] he au ho s ha e p o ed ha om he i ial
solu ions u= 0 emana es an unbounded con inuum o posi i e solu ions a λ=
0. Unlike he case q= 1, his con inuum emana es supe c i ically independen
o he sign o D, in ac , he bi u ca ion di ec ion only depends on alue o p.
This has been p o ed in [10] e en when he ope a o Lis quasilinea . Then, i
a(x)≥a0>0 and p < (N+ 2)/(N−2) hey p o ed he exis ence o nonnega i e
solu ion in (−∞, λ∗) and o , a leas , wo posi i e solu ions in (0, λ∗). See also
simila esul s ob ained in [8] when he ope a o is he p-Laplacian and [1] when
he bounda y condi ions a e Neumann. In all hese wo ks, adoes no change sign.
When achanges sign, ecen ly in [23] he au ho s ha e p o ed he exis ence o a
weak nonnega i e solu ion i λ≤0 making use o a di ec a ia ional app oach,
see [26] o he case o Neumann bounda y condi ions.
In his wo k, we imp o e and gene alize he abo e esul s. We conside a non-
sel adjoin ope a o L, so i is well-known ha he a ia ional me hods do no
wo k, and a unc ion achanging sign. As in [10], we p o e ha an unbounded
con inuum o nonnega i e solu ions emana es om he i ial solu ion u= 0 a
λ= 0 supe c i ically. Mo eo e , we p o e he exis ence o a minimal solu ion
o λ∈(0, λ∗) and he exis ence o λ≥λ∗such ha o λ > λ, (3) does no
admi posi i e solu ion. Using he esul s o Sec ion 4 in [6], and assuming some
es ic ions on pand a+(see Theo em 5.1), we ob ain a p io i bounds o he pos-
i i e solu ions o (3) o compac in e als o λ. Finally, unde hese es ic ions,
we ob ain a unique posi i e solu ion linea ly asymp o ically s able in (0, λ∗), he
exis ence o , a leas , wo posi i e solu ion in (0, λ∗) and o he mul iplici y esul
in his in e al (see Theo em 6.9). These esul s we e s ongly mo i a ed by [17].
804 M. DELGADO AND A. SU ´
AREZ
In o de o ob ain his esul , we ha e used he Picone iden i y, so we assume
ha Lis sel adjoin . Finally, we would like o poin ou ha he s abili y esul s
a e ob ained by linea izing a a posi i e solu ion. Obse e ha , since q < 1, he
linea ized p oblem is a p oblem wi h a po en ial blowing up nea ∂Ω. So, we need
some spec al heo y wi h singula po en ial. Fo ha , we ha e included some
esul s ob ained in [16] and [19].
An ou line o his pape is as ollows: In Sec ion 2 we ha e collec ed some
spec al heo y wi h singula po en ial. In Sec ion 3 we s udy p oblem (3) in he
case a+= 0. These esul s come om [15] and will be used in he nex sec ions.
In Sec ion 4 we apply he Le ay-Schaude deg ee and bi u ca ion heo y o show
he exis ence o an unbounded con inuum o nonnega i e solu ion emana ing su-
pe c i ically a λ= 0 om he i ial solu ion u= 0. In Sec ion 5 we ob ain a
p io i bounds o he posi i e solu ions o (3) o compac in e als o λ. Finally,
in Sec ion 6 we ob ain mul iplici y and s abili y esul s.
2. Singula eigen alue p oblem
In his sec ion we collec some esul s abou he exis ence o p incipal eigen-
alue o a singula linea eigen alue p oblem o he o m
(4) º(L+M(x))u=σu in Ω,
u= 0 on ∂Ω,
whe e M∈C1(Ω) bu i can blow-up nea ∂Ω a a con olled way. The nex
esul was p o ed in [19].
Theo em 2.1. Suppose M∈C1(Ω) and he e exis wo cons an s K > 0and
ε > 0 o which
(5) |M(x)| ≤ K
[dis (x, ∂Ω)]2−εx∈Ω.
Then, he e exis s a unique alue o σ, deno ed by σΩ
1[L+M]and called p incipal
eigen alue o (4), o which (4) possesses a weak posi i e solu ion (in H1
0(Ω) ∩
L∞(Ω)), unique up o mul iplica i e cons an s, deno ed by ϕΩ
1and called p incipal
eigen unc ion o (4).
Mo eo e , by ellip ic egula i y, ϕΩ
1∈C1
0(Ω),ϕΩ
1(x)>0 o each x∈Ωand
∂ϕΩ
1
∂n (x)<0 o each x∈∂Ω, whe e ns ands o he ou wa d uni no mal o Ωa
x.
SLOW DIFFUSION AND SUPERLINEAR PROBLEM 805
Fu he mo e, σΩ
1[L+M]is inc easing wi h espec o Mand dec easing wi h
espec o Ω, and i σΩ
1[L+M]>0 hen u= 0 is he unique weak solu ion o
º(L+M(x))u= 0 in Ω,
u= 0 on ∂Ω.
The ollowing esul was shown in [20] when M∈L∞(Ω), and in [16] when M
sa is ies (5).
De ini ion 2.2. A unc ion ϕ∈C2(Ω) ∩C1(Ω) is said a supe solu ion o L+M
i (L+M)ϕ≥0in Ωand ϕ≥0on ∂Ω.I in addi ion, (L+M)ϕ > 0in Ωo
ϕ > 0on ∂Ω, hen i is said ha ϕis a s ic supe solu ion.
Theo em 2.3. Assume ha Msa is ies (5). Then:
(1) σΩ
1[L+M]>0i , and only i , L+Madmi s a posi i e s ic supe solu ion.
(2) I he e exis s ϕ∈C2(Ω) ∩C1(Ω) wi h ϕ > 0in Ωsuch ha ϕ= 0 on
∂Ωand (L+M)ϕ < 0in Ω, hen
σΩ
1[L+M]<0.
We do no w i e he supe index Ω, when no con usion a ises.
3. The sublinea case: a+≡0.
In his sec ion we s udy he sublinea case, ha is, when a+≡0. The ollowing
esul cha ac e izes he exis ence, uniqueness and linea s abili y in his case.
Theo em 3.1. Assume a+≡0. Then, he e exis s a unique posi i e solu ion o
(3) i , and only i , λ > 0. Mo eo e , i we deno e i by θ[λ,a−], hen
(6) lim
λ↓0kθ[λ,a−]k∞= 0.
Fu he mo e, i λ > 0 hen θ[λ,a−]is linea ly asymp o ically s able, ha is,
(7) σ1[L+Mλ(x)] >0,
whe e Mλ:= −λqθq−1
[λ,a−]−pa−θp−1
[λ,a−]
P oo . Excep (7), he esul ollows by a simila a gumen o Theo em 4.2 in
[15] whe e he esul was p o ed when L=−∆. We a e going o show (7). Fi s ly,
obse e ha o λ > 0, (3) sa is ies he s ong maximum p inciple, so he e exis s
C > 0 such ha
Cdis (x, ∂Ω) ≤θ[λ,a−](x), o all x∈Ω,

806 M. DELGADO AND A. SU ´
AREZ
and so, Mλsa is ies (5). On he o he hand, by (H) and Theo em 2.1, i ollows
0 = σ1[L − λθq−1
[λ,a−]−a−θp−1
[λ,a−]]< σ1[L+Mλ].
This comple es he p oo . £
The ollowing esul will be used in he nex sec ions.
Lemma 3.2. Assume a+≡0. Then,
(8) lim
λ↓0σ1[L+Mλ] = σ1[L − qzq−1]>0,
whe e zis he unique posi i e solu ion o
(9) ºLz=zqin Ω,
z= 0 on ∂Ω.
P oo . The exis ence o a unique posi i e solu ion o (9) ollows by he sub-
supe solu ion me hod, see [15] o de ails. Mo eo e , again by he s ong maxi-
mum p inciple, zq−1sa is ies (5) and so i is well-de ined σ1[L − zq−1]. By (H)
and Theo em 2.1, we ha e
(10) 0 = σ1[L − zq−1]< σ1[L − qzq−1].
In o de o p o e (8), by (6) i is su icien o show ha
(11) ξλ:= λ1/(q−1)θ[λ,a−]→zas λ↓0.
I is no ha d o p o e ha ξλsa is ies
Lξλ=ξq
λ+a−λ(p−1)/(1−q)ξp
λin Ω, ξλ= 0 on ∂Ω.
By (H), i ollows (11), and hanks o (10) we ob ain he esul . £
4. Bi u ca ion om he i ial solu ion
In his sec ion we will show ha a bi u ca ion om he i ial solu ion o (3)
occu s a λ= 0. Fo ha , we conside he Banach space X:= C0(Ω), deno e
Bρ:= {u∈X:kuk∞< ρ}and ake K > 0 su icien ly la ge. We ex end he
unc ion (λ, x, s) := λsq+a(x)sp+Ks by aking (λ, x, s) := 0 i s < 0. No e
ha can ake nega i e alues. Finally, we de ine he map
Kλ:X7→ X;Kλ(u) := u−(L+K)−1( (λ, x, u))
whe e (L+K)−1is he in e se o he ope a o L+Kunde homogeneous Di ichle
bounda y condi ions, which is well-de ined since σ1[L+K]>0. Indeed, since
posi i e cons an s a e supe solu ions o L, hen
σ1[L]>0,
SLOW DIFFUSION AND SUPERLINEAR PROBLEM 807
whence i ollows ha σ1[L+K]>0. Now, we can p o e ha uis a nonneg-
a i e solu ion o (3) i , and only i , uis a ze o o he map Kλ. I is clea ha
e e y nonnega i e solu ion is a ze o o Kλ; con e sely, i uis a ze o o Kλ hen,
mul iplying (3) by u−, we ob ain
(12) ZΩ
N
X
i,j=1
aijDi(u−)Dj(u−) + ZΩ
(K−1
2
N
X
i=1
Dibi)(u−)2≤0,
and so, since Lis a second uni o mly ellip ic ope a o , i ollows ha u−≡0.
Obse e ha a nonnega i e solu ion u∈Xo (3), i belongs o C1+ν(Ω) ∩C1
0(Ω)
o ν:= min{α, q}.
The main esul o his sec ion is:
Theo em 4.1. The alue λ= 0 is he only bi u ca ion poin om he i ial
solu ions o (3). Mo eo e , he e exis s a con inuum C0o nonnega i e solu ions
o (3) unbounded in IR ×Xemana ing om (0,0). In addi ion, C0bi u ca es o
he igh o λ= 0, i.e., i is supe c i ical.
In o de o p o e his esul we use he Le ay-Schaude deg ee o Kλon Bρ
wi h espec o ze o, deno ed by deg(Kλ, Bρ), and he index o he isola ed ze o
uo Kλ, deno ed by i(Kλ, u). In he ollowing esul s, we use homo opies which
we e used in [10], see also [9].
Lemma 4.2. I λ < 0, hen i(Kλ,0) = 1.
P oo . Fix λ < 0. De ine he map
H1: [0,1] ×X7→ X;H1( , u) := (L+K)−1( (λ, x, u)).
We claim ha he e exis s δ > 0 such ha
u6=H1( , u)
o u∈Bδ,u6= 0 and ∈[0,1]. Indeed, suppose ha he e exis sequences
un∈X {0}wi h kunk∞→0 and n∈[0,1] such ha
un=H1( n, un).
We know ha un≥0. Since kunk∞→0 and λ < 0, he e exis s n0∈IN such
ha o n≥n0, i holds
Lun≤0 in Ω,
which is impossible.
808 M. DELGADO AND A. SU ´
AREZ
Taking now ε∈(0, δ], he homo opy de ined by H1is admissible and so,
i(Kλ,0) = deg(Kλ, Bε) = deg(I− H1(1,·), Bε) = deg(I− H1(0,·), Bε) =
= deg(I, Bε)=1.
£
Lemma 4.3. I λ > 0, hen i(Kλ,0) = 0.
P oo . Fix λ > 0 and φ∈X, φ > 0. We de ine he map
H2: [0,1] ×X7→ X;H2( , u) := (L+K)−1( (λ, x, u) + φ).
We will show ha he e exis s δ > 0 such ha u6=H2( , u) o all u∈Bδ,u6= 0
and ∈[0,1]. Indeed, suppose he con a y: he e exis sequences un∈X {0}
wi h kunk∞→0 and n∈[0,1] such ha
un=H2( n, un).
Since nφ≥0, mul iplying by u−, and by a simila a gumen o he used in (12),
we ob ain ha un≥0. Mo eo e since λ > 0, by he s ong maximum p inciple
un>0. We ix M≥σ1[L]. Since kunk∞→0 and λ > 0, he e exis s n0∈IN
such ha o n≥n0we ge
Lun=λuq
n+a(x)up
n+ nφ > Mun+ nφ,
and so,
(L − M)un>0.
So, unis a posi i e s ic supe solu ion o L − M, and by Theo em 2.3, we ge
σ1[L − M]>0, and so M < σ1[L]. This is impossible.
This p o es ha he homo opy de ined by H2is admissible. Then, i we ake
ε∈(0, δ] we ha e
i(Kλ,0) = deg(Kλ, Bε) = deg(I− H2(0,·), Bε) = deg(I− H2(1,·), Bε)=0.
This las equali y is ue because he p oblem Lu=λuq+a(x)up+φhas no
solu ion in Bεbecause we ha e shown ha u6=H2(1, u) o all u∈Bδ,u6= 0. £
P oo o Theo em 4.1: The ac ha λ= 0 is a bi u ca ion poin ollows
by Lemma 4.2 and Lemma 4.3. Mo eo e , om Lemma 4.2, (3) does no ha e
bi u ca ion poin s in (−∞,0). Assume ha he e exis s a sequence o solu ions
(λn, un) such ha λn→λ0>0 and kunk∞→0. We ake M≥σ1[L], so he e
exis s n0∈IN such ha
λnuq
n+a(x)up
n> Mun o all n≥n0.
As in he p oo o Lemma 4.3, we ob ain ha σ1[L − M]>0, a con adic ion.
SLOW DIFFUSION AND SUPERLINEAR PROBLEM 809
Now, e en hough ou map Kλdoes no sa is y exac ly he hypo heses o The-
o em 1.3 in [25], he p oo can be modi ied o ob ain he esul , see Theo em 3.1
in [1] and Theo em 4.4 in [10], and we can conclude he exis ence o a con inuum
o solu ions o (3) such ha mee s (0,0) ei he in ini y o (λ0,0) wi h λ06= 0. We
can disca d he las possibili y by he abo e easoning, and so he exis ence o an
unbounded con inuum o solu ions o (3) ollows.
We a e going o p o e ha he bi u ca ion is supe c i ical, o which plays an
essen ial ole ha p > 1. Indeed, assume ha he e exis s a sequence (λn, un)
o solu ions o (3) such ha λn≤0 and un≥0, un6= 0 wi h λn→0 and
kunk∞→0. Since σ1[L]>0, he e exis s a su icien ly small ε > 0 such ha
(13) σ1[L − ε]>0.
Fo such ε > 0, he e exis s n0(ε)∈IN such ha o n≥n0, we ge
Lun=λnuq
n+a(x)up
n≤a(x)up
n< εun
whence by (13) we ob ain a con adic ion. £
The nex esul shows ha o λla ge, (3) has no solu ion.
P oposi ion 4.4. The e exis s λ > 0such ha o λ > λ,(3) has no solu ion.
P oo . We ix δ > 0 su icien ly small and de ine he se
Dδ:= {x∈A+:dis (x, ∂A+)> δ} 6=∅.
Then, he e exis s a posi i e cons an c+(δ)>0 such ha
a+(x)≥c+(δ)>0 in Dδ.
Obse e ha Dδhas only ini ely many connec ed componen s, say Dδ
i,i=
1,..., .
Since λ > 0, he s ong maximun p inciple assu es ha any nonnega i e and
non i ial solu ion o (3) is in ac s ic ly posi i e. So, by Theo em 2.1 is well-
de ined σ1[L − λuq−1−a(x)up−1], and we ha e
(14) 0 = σ1[L − λuq−1−a(x)up−1]< σDδ
1
1[L − λuq−1−a(x)up−1].
Le ϕDδ
1
1 he p incipal eigen unc ion associa ed wi h Lin Dδ
1. We claim ha
he e exis s λ > 0 such ha o λ > λ,ϕDδ
1
1is a s ic subsolu ion o L1:=
L − λuq−1−a(x)up−1in Dδ
1, and so by Theo em 2.3
σDδ
1
1[L1]<0
816 M. DELGADO AND A. SU ´
AREZ
P oposi ion 6.7. Assume Lsel adjoin (bi= 0 in (2)) and le (λ0, u0)be a
posi i e solu ion o (3) wi h λ=λ0>0, such ha σ1[L+Rλ0]=0.Then,
λ2<0,
whe e λ2is de ined (18).
P oo . By Lemma 6.6, o s∈J, we ha e
L(u0+sΦ0+s2Ψ0+O(s3)) = (λ0+s2λ2+O(s3))(u0+sΦ0+s2Ψ0+O(s3))q+
+a(x)(u0+sΦ0+s2Ψ0+O(s3))p.
Now, di e en ia ing wice wi h espec s, aking accoun ha
(L+Rλ0)Φ0= 0,
we ob ain
(L+Rλ0)Ψ0=λ2uq
0+1
2Φ2
0(q(q−1)λ0uq−2
0+p(p−1)a(x)up−2
0),
and so, by he F edholm al e na i e
λ2=1
2
ZΩ
Φ3
0uq−2
0(q(1 −q)λ0+p(1 −p)a(x)up−q
0)
ZΩ
uq
0Φ0
.
Obse e ha , since u0and Φ0a e s ic ly posi i e, he e exis Ci>0, i= 1,2,
such ha
Φ3
0uq−2
0≤C1dis (x, ∂Ω)3−2+q≤C1,
and so λ2is well-de ined.
To p o e ha λ2<0, he basic ool is a Picone iden i y (see Sec ion 4 in [12]
and Lemma 4.1 in [21], o ins ance). Le u, ∈C2(Ω) ∩C1
0(Ω) be such ha
/u ∈C(Ω) ∩C1(Ω) and Υ : [0,∞)7→ IR o class C1. Then
(20) ZΩ
Υ(
u)( Lu−uL ) = −ZΩ
Υ0(
u)u2
N
X
i,j=1
aijDi(
u)Dj(
u).
We ake Υ( ) = 2, = Φ0and u=u0. Obse e ha /u ∈C(Ω) ∩C1(Ω) by he
s ong maximum p inciple. Hence, by (20) and since ucanno be a mul iple o ,
we ob ain ZΩ
Φ3
0uq−2
0(λ0(1 −q) + a(x)(1 −p)up−q
0)<0,
and so, since q < 1
0< λ0(1 −q)ZΩ
Φ3
0uq−2
0<(p−1) ZΩ
a(x)Φ3
0up−2
0

SLOW DIFFUSION AND SUPERLINEAR PROBLEM 817
now, by (H)
λ0q(1 −q)ZΩ
Φ3
0uq−2
0< p(p−1) ZΩ
a(x)Φ3
0up−2
0,
and he e o e λ2<0. £
As an easy consequence o Lemma 6.6, ela ion (19) and P oposi ion 6.7, we
ob ain:
Co olla y 6.8. Assume Lsel adjoin and le (λ0, u0)be a posi i e solu ion o
(3) wi h λ=λ0>0, such ha σ1[L+Rλ0]=0.Then, he e exis s ε > 0such
ha o each λ∈(λ0−ε, λ0),(3) has wo posi i e solu ions, one o hem linea ly
asymp o ically s able and he o he one linea ly uns able. Mo eo e , he e exis
a neighbo hood No (λ0, u0)in IR ×Psuch ha (3) does no ha e a posi i e
solu ion in N o λ>λ0.
We a e eady o p o e he main esul o his sec ion.
Theo em 6.9. Assume Lsel adjoin and ha he hypo heses o Theo em 5.1 a e
sa is ied. Then,
(1)
Λ=(−∞, λ∗],
(2) The e exis , a leas , wo posi i e solu ion in (0, λ∗),
(3) The e exis s a unique posi i e solu ion in (0, λ∗)linea ly asymp o ically
s able,
(4) I we assume ha (3) has a ini e numbe o non-degene a e posi i e so-
lu ions, say u1,...,u , hen = 2k o some k≥1, and exac ly kamong
hem ha e index −1, and he o he kha e index 1.
P oo . To show he i s pa ag aph i emains o p o e ha he e exis s solu ion
o λ=λ∗. Le (λn, un) a sequence o solu ions wi h 0 < λn< λ∗and λn→λ∗.
By Theo em 5.1 and a s anda d compac ness a gumen , we ob ain ha un→u∗,
wi h u∗solu ion o (3) o λ=λ∗. Mo eo e , u∗6= 0 because o λ= 0 is he
unique bi u ca ion alue om he i ial solu ion, hence u∗>0.
We will show ha he minimal solu ion uλis he unique linea ly asymp o ically
s able. Indeed, we ake λ1>0 su icien ly small such ha σ1[L+Rλ1]>0, o
(λ1, uλ1). This is possible by Lemma 6.3, P oposi ion 6.4 and Co olla y 6.8. By
con inua ion o he le o λ1, and hanks o P oposi ion 6.4 and P oposi ion 6.7,
we ob ain ha uλis asymp o ically s able o 0 < λ ≤λ1.
818 M. DELGADO AND A. SU ´
AREZ
Now, we p olonga e o he igh o λ1 o each a alue λ2≤λ∗whe e σ1[L+Rλ]>
0 o 0 <λ<λ2and
σ1[L+Rλ2]=0.
I λ2=λ∗, we ha e jus p o ed he exis ence o a linea ly asymp o ically s able
posi i e solu ion o λ∈(0, λ∗). So, assume λ2< λ∗and ake λ3∈(λ2, λ∗)
and conside (λ3, uλ3). In any case, σ1[L+Rλ3] = 0 o σ1[L+Rλ3]>0, by
Co olla y 6.8 we can ake λ4∈(λ2, λ3] such ha
σ1[L+Rλ4]>0.
We can p olonga e o he le o λ4by a b anch, say uλ, o linea ly asymp o ically
s able posi i e solu ion, see P oposi ion 6.7. This b anch can no degene a e in
he b anch uλdue o he uniqueness o posi i e solu ion a ound he minimal so-
lu ion uλ. Nei he , i can degene a e o 0 in λ= 0, because o P oposi ion 6.4.
Hence, he e exis s a posi i e linea ly asymp o ically s able o λ= 0, say u0,
which is impossible by Lemma 6.5.
A simila a gumen can be used o show he uniqueness o posi i e linea ly asymp-
o ically s able solu ion. Mo eo e , Theo em 5.1 and Theo em 4.1 show he sec-
ond pa ag aph.
Now, we ake Γ := [0, b] wi h b>λ∗. By Theo em 5.1, he e exis s a posi i e
cons an C(independen om λ) such ha kuk∞≤C o all λ∈Γ. We ake
R:= C+ 1 and hen o all λ∈Γ
(21) iP(K, PR)=0.
Indeed, we can conside he homo opy
H: [0,1] ×P7→ P, H( , u) := (L+K)−1((λ(1 − ) + b)uq+a(x)up+Ku).
Then,
iP(K, PR) = iP(H(0,·), PR) = iP(H(1,·), PR) = iP(H(1,·),0) = 0,
because u= 0 is he only solu ion o λ>λ∗and by Lemma 4.3. Mo eo e , by
pa ag aph 3 and he Le ay-Schaude o mula we ha e
iP(K, uλ)=1.
Wi hou los o gene ali y we can suppose ha u1=uλ. Now, o each nonde-
gene a e solu ions u2,...,u we ha e
iP(K, ui)=(−1)ni
SLOW DIFFUSION AND SUPERLINEAR PROBLEM 819
whe e niis he sum o he algeb aic mul iplici ies o all he eigen alues g ea e
han one o he linea ized o Ka ui. Since u= 0 has index ze o by Lemma 4.3
and (21), we ob ain
0 = 0 + 1 +
X
i=2
(−1)ni
om whe e he esul ollows. £
Rema k 6.10. Obse e ha we only ha e used ha Lis sel adjoin in P opo-
si ion 6.7 in o de o apply he Picone iden i y. So, pa ag aphs (1) and (2) o
Theo em 6.9 a e ue i Lis a gene al ope a o as (2).
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Recei ed Ap il 10, 2001
Re ised e sion ecei ed Decembe 14, 2001
(M. Delgado) Dp o. Ecuaciones Di e enciales y An´
alisis Num´
e ico, Fac. Ma em´
a icas,
C/ Ta ia s/n, C.P. 41012, Uni . Se illa, Spain.
E-mail add ess:[email p o ec ed]
(A. Su´a ez) Dp o. Ecuaciones Di e enciales y An´
alisis Num´
e ico, Fac. Ma em´
a icas,
C/ Ta ia s/n, C.P. 41012, Uni . Se illa, Spain.
E-mail add ess:[email p o ec ed]