Topological Me hods in Nonlinea Analysis
Jou nal o he Juliusz Schaude Cen e
Volume 23, 2004, 275–300
COMBINING FAST, LINEAR AND SLOW DIFFUSION
Juli´
an L´
opez-G´
omez — An onio Su´
a ez
Abs ac . Al hough he pionee ing s udies o G. I. Ba enbla ([8]) and
A. G. A onson and L. A. Pele ie ([7]) did esul in o a huge indus y
a ound he po ous media equa ion, none u he s udy analyzed he effec
o combining as , slow, and linea diffusion simul aneously, in a spa ially
he e ogeneous po ous medium. Ac ually, i migh be his is he fi s wo k
whe e such a p oblem has been add essed. Ou main findings show how
he he e ogeneous model possesses wo diffe en egimes in he p esence o
a p io i bounds. The minimal s eady-s a e o he model exhibi s a genuine
as diffusion beha io , whe eas he emaining s a es a e a he eminiscen
o he pu ely slow diffusion model. The ma hema ical ea men o hese
he e ogeneous p oblems should dese e a huge in e es om he poin o
iew o i s applica ions in fluid dynamics and popula ion e olu ion.
1. In oduc ion
In his pape we s udy he posi i e solu ions o he bounda y alue p oblem
(1.1) −∆(wm(x))=λw in Ω,
w=0 on∂Ω,
whe e Ω ⊂RN,N≥1, is a bounded domain o class C2,λ∈R,and
m=1+pχΩ+−qχΩ−
2000 Ma hema ics Subjec Classifica ion. 35B32, 35J25, 35J60, 35K57.
Key wo ds and ph ases. He e ogeneous nonlinea diffusion, as , slow and linea diffusion.
The esea ch o he fi s named au ho suppo ed by he Spanish Minis y o Science and
Technology unde G an s BFM2000-0797 and BFM2003-06466.
The esea ch o he second named au ho was as well suppo ed by REN2003-00707.
c
2004 Juliusz Schaude Cen e o Nonlinea S udies
275
276 J. L´
opez-G´
omez — A. Su´
a ez
whe e Ω+and Ω−a e wo subdomains o Ω o class C2such ha
(1.2) Ω+⊂Ω,Ω+∩Ω−=∅,
and p∈L∞(Ω+)∩C(Ω+), q∈L∞(Ω−)∩C(Ω−)sa is y
(1.3) p(x)>0and0<q(y)<1 o each(x, y)∈Ω+×Ω−.
Th oughou his pape , o any measu able se M⊂Ω, we deno e by χM he
cha ac e is ic unc ion o M, i.e. χM(x)=1i x∈M,andχM(x)=0 o each
x∈Ω M.Also,wese
Ω1:= Ω (Ω+∪Ω−),
he open se whe e m= 1, and suppose, by simplici y, ha Ω1is connec ed.
Though we allow Ω1,Ω
+,o Ω
− o be emp y, Figu e 1.1 shows one o he
admissible configu a ions deal wi h in his wo k.
Ω
Ω
Ω
+
−
1
Figu e 1.1.An admissible configu a ion
Th oughou his pape we deno e
(1.4) m+:= m|Ω+=1+pχΩ+,m
−:= m|Ω−=1−qχΩ−.
Then, m+(x)>1 o eachx∈Ω+and 0 <m
−(x)<1 o eachx∈Ω−,and
hence (1.1) p o ides us wi h he s eady s a es o a po ous medium equa ion
whe e diffusion is linea in Ω1and nonlinea in Ω+∪Ω−(slow in Ω+and as in
Ω−). The analysis o hese kind o bounda y alue p oblems gene a ed a huge
indus y since he pionee ing s udies o G. I. Ba enbla ([8]) and A. G. A onson,
L. A. Pele ie ([7]), al hough mos o he li e a u e ea ed he e y special case
when mis cons an .
Up o he bes o ou knowledge, he fi s wo k whe e mhas been allowed
o a y is M. Delgado e al. in [12], whe e he special case when m−=0was
ea ed. The p esen pape seems o be he fi s wo k whe e he gene al p oblem
o analyzing he in e play be ween slow, as and linea diffusion, simul aneously,
Combining Fas , Linea and Slow Di usion 277
has been add essed. The e o e, mos o he esul s ound in his pape a e com-
ple ely new and, undoub edly, open new esea ch di ec ions ha migh be o
g ea ele ance om he poin o iew o he applica ions o he unde lying ab-
s ac ma hema ical heo y o popula ion dynamics and po ous media dynamics.
To summa ize ou main esul s we need o in oduce some basic concep s and
no a ions.
As he change o a iable u=wm ans o ms (1.1) in o
(1.5) −∆u=λu1/min Ω,
u=0 on∂Ω,
ou effo s will be ocused in o he p oblem o analyzing he exis ence and mul-
iplici y o posi i e solu ions o (1.5). A unc ion u∈H1
0(Ω) ∩L∞(Ω) is said
o be a solu ion o (1.5) i u1/m∈L2N/(N+2)(Ω) and i sa isfies he equa ion in
he classical weak sense. By ellip ic egula i y, any weak non-nega i e solu ion
u= 0 p o ides us wi h an s ong solu ion almos e e ywhe e wice diffe en iable
in Ω and, as a esul o he s ong maximum p inciple, u(x)>0 o eachx∈Ω
and ∂u(x)/∂n < 0 o eachx∈∂Ω, whe e ns ands o he ou wa d no mal
ec o -field o Ω. In he emaining o his pape , i should be kep in mind ha ,
as a esul o he maximum p inciple, (1.5) canno admi a posi i e solu ion i
λ≤0.
Th oughou he es o his pape , o any po en ial V∈L∞(Ω) we deno e
by σ[−∆+V; Ω] he p incipal eigen alue o −∆+Vin Ω unde homogeneous
Di ichle bounda y condi ions. No e ha i
Ω+=Ω
−=∅,
hen (1.5) becomes linea and, hence, i possesses a posi i e solu ion i , and only
i , λ=σ[−∆; Ω]. The e o e, we subsequen ly assume
(1.6) Ω+∪Ω−=∅.
Al hough mos o ou findings a e comple ely new e en in he special case when
m−1 does no change o sign, he mos in e es ing esul s o his pape a e hose
ound o he gene al case when m−1 changes sign, whe e one mus assume
m+and m− o be cons an o ge op imal esul s. Unde hese assump ions ou
main esul is Theo em 4.1, which can be ew i en as ollows.
Theo em 1.1. Suppose Ω+and Ω−a e non-emp y and m+,m−a e con-
s an . Then, he e exis λ∗>0and an unbounded componen , C,o hese o
posi i e solu ions (λ, u)o (1.5) such ha :
(a) (λ, u)=(0,0) ∈C,andΛ:=PλC∈{(0,λ
∗],(0,λ
∗)}, o Pλ(λ, u):=λ.
(b) P oblem (1.5) does no admi a posi i e solu ion i λ∈(−∞,0]∪(λ∗,∞).
278 J. L´
opez-G´
omez — A. Su´
a ez
(c) Fo each λ∈Λ,(1.5) possesses a minimal posi i e solu ion, deno ed
by θλ,and hemapλ→ θλis smoo h and inc easing. Mo eo e ,
σ[−∆−(λ/m)θ1/m−1
λ;Ω]>0i λ∈Λ {λ∗},
i.e. θλis linea ly asymp o ically s able wi h espec o he pa abolic coun-
e pa o (1.5),while
σ[−∆−(λ∗/m)θ1/m−1
λ∗;Ω]=0 i λ∗∈Λ,
i.e. θλ∗is linea ly neu ally s able.
(d) The componen Ccon ains he a c o diffe en iable cu e Γ:={(λ, θλ):
λ∈Λ {λ∗}},limλ↓0θλC0(Ω) =0,andlimλ↑λ∗θλ=θλ∗i Λ=
(0,λ
∗],whilelimλ↑λ∗θλC0(Ω) =∞i Λ=(0,λ
∗).Ac ually,C=Γi
Λ=(0,λ
∗).
(e) I Λ=(0,λ
∗], hen he e exis s λω∈[0,λ
∗)such ha (1.5) has wo
posi i e solu ions, a leas , o each λ∈(λω,λ
∗). Ac ually, i ei he
N∈{1,2},o N≥3and m−>(N−2)/(N+2), henΛ=(0,λ
∗]and
λω=0.
( ) Fo each λ∈Λ,θλp o ides us wi h he unique linea ly s able posi i e
solu ion o (1.5).
The dis ibu ion o his pape is he ollowing: Sec ion 2 analyzes he case
when Ω+=∅, Sec ion 3 analyzes he case when Ω−=∅, and, hen, in Sec ion 4,
we p o e Theo em 1.1. Th oughou he manusc ip we sho ly desc ibe some
special pe u ba ion esul s connec ing each o hese cases wi h he emaining
one, hough we ha e e ained o include he de ails o all hei p oo s o keep
he leng h o he manusc ip wi hin a easonable le el. All hose esul s will be
deeply discussed and collec ed elsewhe e.
2. The case Ω+=∅
As we a e assuming (1.6), we ha e Ω−=∅and, hence, (1.5) is supe linea
wi hin Ω−. The ollowing esul holds in he special case when Ω1=∅.
Theo em 2.1. Suppose Ω+=Ω
1=∅. Then, he ollowing asse ions a e
ue:
(a) Unde he ollowing condi ion
(2.1) in
Ω−
m−>N−2
N+2 i N≥3,
p oblem (1.5) possesses a posi i e solu ion o each λ>0. Mo eo e ,
i (λn,u
n),n≥1, is a sequence o posi i e solu ions o (1.5) such ha
Combining Fas , Linea and Slow Di usion 279
limn→∞ λn=0, hen
(2.2) lim sup
n→∞
unC0(Ω) =∞.
(b) I m−is cons an , hen uis a posi i e solu ion o (1.5) i , and only i ,
u=λ−m−/(1−m−)
o some posi i e solu ion o
(2.3) −∆ = 1/m−in Ω,
u=0 on ∂Ω.
In pa icula , he numbe o posi i e solu ions o (1.5), o eachλ>0,
equals he numbe o posi i e solu ions o (2.3) and, he e o e, he ol-
lowing holds:
(b1) Suppose m−>(N−2)/(N+2) i N≥3.Then(2.3) possesses
a posi i e solu ion, a leas , and, ac ually, each posi i e solu ion
o (2.3) p o ides us wi h a cu e
λ→ uλ:= λ−m−/(1−m−) , λ > 0,
o posi i e solu ions o (1.5). Mo eo e ,
lim
λ↓0uλ=∞and lim
λ↑∞ uλ=0
uni o mly in compac subse s o Ω.
(b2) Suppose N≥3,m−≤(N−2)/(N+2),andΩis s a -shaped.
Then, (1.5) canno admi a posi i e solu ion.
Subsequen ly, we shall deno e by Pρ:R×C
0(Ω) →C
0(Ω) he ρ-p ojec ion
ope a o , i.e.
Pρ(ρ, u)=ρ o each (ρ, u)∈R×C
0(Ω).
P oo o Theo em 2.1. Suppose (2.1) and conside , o each λ>0, he
auxilia y p oblem
(2.4) −∆u=µu +λu1/m−in Ω,
u=0 on∂Ω,
whe e µ∈Ris ega ded as a bi u ca ion pa ame e . Thanks o (2.1), he
blowing-up a gumen o B. Gidas and J. Sp ¨uck (see [14]) can be easily adap ed o
show ha he posi i e solu ions o (2.4) possess L∞(Ω) a p io i bounds uni o m
in compac in e als o µ∈R. Mo eo e , hanks o local bi u ca ion esul o
M. G. C andall and P. H. Rabinowi z ([10]), µ:= σ[−∆; Ω] is a bi u ca ion alue
o posi i e solu ions o (2.4) om he i ial s a e (µ, u)=(µ, 0). Ac ually, by he
global unila e al heo em o P. H. Rabinowi z ([22]), he componen o posi i e
solu ions o (2.4) emana ing om (µ, 0) a µ=σ[−∆; Ω], subsequen ly deno ed

280 J. L´
opez-G´
omez — A. Su´
a ez
by C, mus be unbounded in R×C
0(Ω) (c . E. N. Dance [11], as well as [19,
Chap e s 6, 7], o a comple e de elopmen o he necessa y abs ac heo y, as
he o iginal pape o P. H. Rabinowi z [22] con ains some se ious gaps). Suppose
(2.4) possesses a posi i e solu ion. Then,
(−∆−λu1/m−−1)u=µu
and, hence, by he uniqueness o he p incipal eigen alue,
µ=σ[−∆−λu1/m−−1;Ω].
Thus, since λ>0, i is appa en , om he mono onici y o he p incipal eigen-
alue wi h espec o he po en ial, ha µ<σ[−∆; Ω], and, hence,
PµC⊂(−∞,σ[−∆; Ω]).
Ac ually, hanks o he exis ence o uni o m a p io i bounds,
PµC=(−∞,σ[−∆; Ω])
and, he e o e, 0 ∈P
µC. In pa icula , (1.5) possesses a posi i e solu ion.
Now, le (λn,u
n), n≥1, be a sequence o posi i e solu ions o (1.5) wi h
limn→∞ λn= 0. I he e exis s a cons an M>0 such ha
unC0(Ω) ≤M, n ≥1,
hen, by he compac ness (−∆)−1( he in e se o he ope a o −∆ in Ω unde ho-
mogeneous Di ichle bounda y condi ions), along some subsequence o (λn,u
n),
labeled again by n,
lim
n→∞ un−u∞C0(Ω) =0,
o some s ong solu ion u∞o he p oblem
(2.5) −∆u=0 inΩ,
u=0 on∂Ω.
Necessa ily u∞= 0 and, hence,
lim
n→∞ unC0(Ω) =0.
Now, se
n:= un
unC0(Ω)
,n≥1.
Then, o each n≥1, we ha e ha
n=(−∆)−1(λn nu1/m−−1
n),
and, hence, along some subsequence, labeled again by n,weha e ha
lim
n→∞  n− ∞C0(Ω) =0.
Combining Fas , Linea and Slow Di usion 281
Necessa ily,  ∞C0(Ω) =1, ∞>0, and ∞sol es (2.5). This is impossible,
since u= 0 is he unique solu ion o (2.5). This con adic ion shows (2.2) and
concludes he p oo o (a).
(b1) is an easy consequence om (a), and (b2) ollows eadily om a cele-
b a ed iden i y by S. I. Pohozae ([21]). 
E enin hecasewhenm−is a cons an sa is ying (2.1), i is well known
ha he numbe o posi i e solu ions o (1.5) is s ongly dependen upon he
geome y o he domain Ω. Indeed, i Ω consis s o n≥2 sepa a ed balls joined
by n−1 na ow co ido s, hen (2.3) has 2n−1 posi i e solu ions and, he e o e,
(1.5) possesses 2n−1 global cu es o posi i e solu ions. E en ually, e en o he
simples domain geome ies, he numbe o solu ions o (1.5) migh be s ongly
dependen upon he local oscilla ion p ope ies o he unc ion m−(x) (c . [15],
as well as he e e ences he e in, o simila closely ela ed discussions).
In he gene al case when N≥3 and he auxilia y unc ion
s(x):=m−(x)−N−2
N+2,x∈Ω,
changes o sign, he p oblem o cha ac e izing he exis ence o posi i e solu-
ions o (1.5) inc eases in complexi y. The co esponding esul s will be gi en
elsewhe e, as hey a e s ill in p og ess.
In he mos gene al case when Ω1=∅ he ollowing esul is sa isfied.
Theo em 2.2. Suppose Ω+=∅,Ω1=∅, and conside he unc ion
σ(λ):=σ[−∆−λχΩ1;Ω],λ≥0.
Then, he exis s a unique λ0=λ0(Ω1)>0sa is ying
σ−1(0) ∩[0,∞)={λ0}.
Mo eo e , (1.5) canno admi a posi i e solu ion i λ≥λ0. Suppose, in addi ion,
ha
sup
Ω−
m−<1
and ega d o λas a bi u ca ion pa ame e . Then λ=λ0is a bi u ca ion alue
om (λ, u)=(λ, 0) o an unbounded con inuum C⊂(0,λ
0)×C
0(Ω) o posi i e
solu ions o (1.5). Mo eo e , PλC=(0,λ
0)i condi ion (2.1) is sa isfied, hough,
in gene al, PλCmigh be a p ope subin e al o (0,λ
0).
Figu e 2.1 shows h ee admissible si ua ions wi hin he se ing o Theo-
em 2.2.
Figu e 2.1(a) ep esen s Cunde assump ion (2.1), while Figu e 2.1(b), (c)
ep esen wo admissible C’s whe e (2.1) ails. In case (b), PλC=[λ∗,λ
0), o
some λ∗∈(0,λ
0), while, in case (c), PλC=(λ∗,λ
0). In all cases he p oblem
282 J. L´
opez-G´
omez — A. Su´
a ez
0λ
λ
0
u
0λ
λ
0
u
0λ
λ
0
u
λλ
∗∗
(a) (b) (c)
CCC
Figu e 2.1.Th ee admissible bi u ca ion diag ams
migh ha e an a bi a y numbe o solu ions as a esul o he geome y o Ω
and he local p ope ies o m−.
A c ucial ea u e, diffe en ia ing he case when Ω1=∅ om he case de-
sc ibed by Theo em 2.2, is he ac he e exis s ε>0 such ha [λ0−ε, λ0)⊂P
λC
i Ω1=∅, and, he e o e, (1.5) always possesses a posi i e solu ions o each
λ<λ
0sufficien ly close o λ0, independen ly o he size o m−; in s ong con-
as wi h he si ua ion desc ibed by Theo em 2.1, whe e (1.5) canno admi
a posi i e solu ion i Ω is s a -shaped, N≥3andm−≤(N−2)/(N+2).
I Ωδ
1,δ∈[0,1], s ands o an inc easing amily o smoo h domains such ha
Ω1
1=Ω
1and limδ↓0Ωδ
1=∅, hen, limδ↓0λ0(Ωδ
1)=∞(c . he de ails o he
p oo o [13, Theo em 12]). Ac ually, he co esponding bi u ca ion diag ams
app oxima e, as δ↓0, o he bi u ca ion diag am o he p oblem in case Ω1=∅,
hough, being ou side he gene al scope o his wo k, his sha pe analysis will
appea elsewhe e.
P oo o Theo em 2.2. By he mono onici y o he p incipal eigen alue
wi h espec o he po en ial, he unc ion σ(λ) is dec easing wi h λ.Mo eo e ,
σ(0) = σ[−∆; Ω] >0, and, o any ball B⊂Ω1and λ>0, we ha e ha
σ(λ)<σ[−∆−λ;B]=σ[−∆; B]−λ,
and, hence, limλ↑∞ σ(λ)=−∞. This shows he exis ence and he unique-
ness o λ0.
Suppose (1.5) possesses a posi i e solu ion u. Then,
(−∆−λχΩ1)u=λχΩ−u1/m−>0
and, hence, uis a s ic posi i e supe solu ion o −∆−λχΩ1in Ω unde homo-
geneous Di ichle bounda y condi ions. Thus, hanks o [17, Theo em 2.5],
σ[−∆−λχΩ1;Ω]>0
and, he e o e, λ<λ
0. Now, we ega d o λas he main bi u ca ion pa ame e
and conside he nonlinea ope a o F:R×C
0(Ω) →C
0(Ω) defined by
(2.6) F(λ, u):=u−(−∆)−1(λχΩ1u+λχΩ−|u|1/m−),
Combining Fas , Linea and Slow Di usion 283
whose posi i e fixed poin s p o ide us wi h he posi i e solu ions o (1.5). Fo
each λ∈R,F(λ, 0) = 0. Mo eo e , Fis con inuous and admi s he decomposi ion
F(λ, u)=L(λ)u−λ(−∆)−1(χΩ−|u|1/m−),
whe e
L(λ)u:= u−λ(−∆)−1(χΩ1u),u∈C
0(Ω).
The e o e, i adjus s o he abs ac se ing o [19, Chap e 6]. I should be
no ed ha condi ion supΩ−m−<1 canno be elaxed, because o he wise he
nonlinea i y would no be o(uC0(Ω)).
Le ϕ0>0 deno e a p incipal eigen unc ion associa ed o σ[−∆−λ0χΩ1;Ω].
Then,
(2.7) N[L(λ0)] = span[ϕ0]and d
dλL(λ0)ϕ0∈ R[L(λ0)],
whe e, gi en any linea con inuous ope a o L,N[L]andR[L] s and o he null
space and he ange o L, espec i ely. Indeed, he fi s iden i y o (2.7) is ue
by cons uc ion. Fo he second, suppose
(2.8) −(−∆)−1(χΩ1ϕ0)=u−λ0(−∆)−1(χΩ1u)
o some u∈C
0(Ω). Then,
(−∆−λ0χΩ1)u=−χΩ1ϕ0
and mul iplying his iden i y by ϕ0and in eg a ing by pa s in Ω gi es
Ω1
ϕ2
0=0,
which is impossible, since ϕ0(x)>0 o eachx∈Ω. The e o e, since L(λ)is
a F edholm ope a o o index ze o, λ0is a 1- ans e sal eigen alue o he amily
L(λ) and, hence, he gene alized algeb aic mul iplici y χ[L;λ0] in oduced in [19,
Chap e 4] equals 1. The e o e, hanks o [19, Theo em 4.2.4], λ0is a nonlinea
eigen alue o L(λ). Ac ually, his ac is a di ec consequence om he main
local bi u ca ion heo em o M. G. C andall and P. H. Rabinowi z ([10]). I
should be no ed ha he main heo em o [10] does no apply in o de o ge
he exis ence o a cu e o posi i e solu ions o (1.5) emana ing om u=0a
λ=λ0, because ou nonlinea i y does no ha e he equi ed egula i y. Bu
his is a om being a ouble, since, due o [19, Theo em 5.6.2], he index
– local opological deg ee – o L(λ) a ze o, Ind(L(λ),0), λ∼λ0,λ=λ0,
mus change as λc osses λ0, because χ[L;λ0] = 1. The e o e, hanks o [19,
Theo em 6.2.1] he e is a componen , C, o he se o non i ial solu ions o (1.5)
such ha (λ0,0) ∈C. Finally, he p oo o [19, Theo em 6.5.5] ca ies o e
mu a is mu andis o show he exis ence o an unbounded subcomponen o C,
C, en i ely consis ing o posi i e solu ions o (1.5) and such ha (λ0,0) ∈C.I
290 J. L´
opez-G´
omez — A. Su´
a ez
Thus, he e exis s a con inuum o posi i e solu ions o (1.5) emana ing om
u=0a λ=0. The maximal con inuum, o he inclusion, p o ides us wi h he
componen C.
P oo . The p oo o (4.9) and (4.10) is based upon some homo opies coming
om A. Amb ose i and P. Hess ([5]), and D. A coya e al. ([6]).
Fix λ<0 and conside he map H1:[0,1] ×C
0(Ω) →C
0(Ω) defined by
H1( , u):=u−(−∆)−1( (λ, ·,u)).
Since he non i ial ze oes o H1( , ·) a e he posi i e solu ions o
(4.11) −∆u= λu1/min Ω,
u=0 on∂Ω,
and λ ≤0, we ob ain ha H1( , u)=0i ∈[0,1] and u=0. Thus, o each
R>0, he homo opy in a iance o he opological deg ee gi es
Ind(K(λ, ·),0) = Deg(K(λ, ·),B
R)=Deg(H1(1,·),B
R)
=Deg(H1(0,·),B
R)=Deg(I,BR)=1,
which concludes he p oo o (4.9).
Now, fix λ>0, φ∈C
0(Ω), φ>0, and conside he map H2:[0,1] ×C
0(Ω) →
C0(Ω) defined by
H2( , u):=u−(−∆)−1( (λ, ·,u)+ φ).
We claim ha he e exis s δ>0 such ha H2( , u)=0 o each ∈[0,1] and
u∈Bδ {0}. No e ha , in pa icula , his shows ha u= 0 is an isola ed solu ion
o (1.5). We shall p oceed by con adic ion. Fi s , no e ha i H2( , u)=0 o
some ∈[0,1] and u=0, hen
−∆u=λ (λ, ·,u)+ φ
and, hence, Ω|∇u−|2=0,since φ ≥0. Consequen ly, u>0. Now, suppose
he e is a sequence
( n,u
n)∈[0,1] ×(C0(Ω) {0}),n≥1,
such ha limn→∞ un=0andH2( n,u
n)=0 o eachn≥1. Then, o each
n≥1, we ha e ha un>0and
−∆un=λu1/m+
n+ nφ≥λun1/m+−1
C(Ω+)un+ nφin Ω+.
Mo eo e , un>0on∂Ω+.Thus,un|Ω+p o ides us wi h a s ic posi i e
supe solu ion o
−∆−λun1/m+−1
C(Ω+)

Combining Fas , Linea and Slow Di usion 291
in Ω+, unde homogeneous Di ichle bounda y condi ions. Thus, hanks o [17,
Theo em 3.2],
σ[−∆−λun1/m+−1
C(Ω+);Ω
+]=σ[−∆; Ω+]−λun1/m+−1
C(Ω+)>0.
This is impossible, since
lim
n→∞ λun1/m+−1
C(Ω+)=∞.
This con adic ion shows he claim abo e. Now, hanks o he homo opy in a i-
ance o he opological deg ee, we ob ain ha
Ind(K(λ, ·),0) = Deg(K(λ, ·),B
δ)
=Deg(H2(0,·),B
δ)=Deg(H2(1,·),B
δ)=0,
since H2(1,0) = −(−∆)−1φ<0, and, hence, H2(1,u)=0 o eachu∈Bδ.This
concludes he p oo o (4.10).
Now, fix λ1<0<λ
2,pickε>0 such ha K(λj,u)=0 o eachj∈{1,2}
and u∈Bε {0}, and conside he cylinde s
Qη:= [λ1,λ
2]×Bη⊂R×C
0(Ω),η∈(0,ε].
Fix η∈(0,ε]. We claim ha he e exis λη∈[λ1,λ
2]anduη∈∂Bηsuch ha
K(λη,u
η)=0.
No e ha , necessa ily, λη>0. Indeed, hanks o (4.9) and (4.10), i his we e
no ue, hen, by he homo opy in a iance o he deg ee, we would ge
1=Deg(K(λ1,·),B
η)=Deg(K(λ2,·),B
η)=0,
which is a con adic ion. By he compac ness o K, i ollows ha he e exis s a
sequence ηn∈(0,ε), n≥1, such ha
lim
n→∞ ηn= 0 and lim
n→∞(ληn,u
ηn)=(0,0).
Ac ually, hanks o a celeb a ed esul by G. T. Whybu n ([24]), he e is a con-
inuum o non- i ial ze oes o Kconnec ing (0,0) wi h uC0(Ω) =η.As he
echnical de ails o he p oo ha e been al eady gi en in he p oo o [19, Theo-
em 6.2.1], we will omi hem he e in (c . [1, Theo em 3.1] and [6, Theo em 4.4]
as well). This concludes he p oo . 
4.3. The exis ence and linea s abili y o he minimal solu ion. The
main esul o his sec ion is he ollowing.
292 J. L´
opez-G´
omez — A. Su´
a ez
P oposi ion 4.4. Suppose (1.5) possesses a posi i e solu ion. Then, i pos-
sesses a minimal posi i e solu ion, deno ed by θλ. By minimal i is mean ha
θλ<u o any o he posi i e solu ion uo (1.5). Mo eo e , θλis linea ly s able,
i.e.
(4.12) σ−∆−λ
mθ1/m−1
λ≥0.
P oo . Suppose (1.5) has a posi i e solu ion, say u. Necessa ily, λ>0. Le
Bbe any ball such ha B⊂Ω+,deno ebyψ he unique posi i e eigen unc ion
associa ed o σ[−∆; B], no malized so ha ψC0(B)=1,andse
Ψ:=ψin B,
0inΩ B.
Then, o sufficien ly small ε>0, he unc ion εΨ p o ides us wi h a subsolu ion
o (1.5) such ha εΨ<u. As a consequence, (1.5) possesses a minimal posi i e
solu ion in he o de in e al [εΨ,u]o C0(Ω). Thus, i possesses a minimal
posi i e solu ion in he o de in e al [0,u], since λcanno be a bi u ca ion alue
o posi i e solu ions om u= 0, because o P oposi ion 4.2. Le θu
λdeno e he
minimal posi i e solu ion in [0,u]andle u(x, ;εΨ) be he unique solu ion o he
pa abolic coun e pa o (1.5) s a ing a εΨ<θ
u
λ≤u. Thanks o he heo y o
D. Sa inge [23], u(·, ;εΨ) is inc easing in ime and i app oaches θu
λas ↑∞.
Suppose is ano he posi i e solu ion o (1.5) and sho en ε, i necessa y, so
ha εΨ< . Then, by he uniqueness o he limi lim ↑∞ u(·, ;εΨ), we find
ha θu
λ=θ
λand, he e o e, θu
λis independen o he posi i e solu ion u.Thus,
i p o ides us wi h he minimal posi i e solu ion θλo (1.5). Rela ion (4.12)
ollows om [2, P oposi ion 20.4] (c . [4, Lemma 3.5] as well). 
4.4. Solu ion cu es h ough linea ly s able solu ions. The main e-
sul o his sec ion eads as ollows. No e ha , hanks o P oposi ion 4.4, i
e eals some c ucial p ope ies sa isfied by all minimal solu ions θλo (1.5).
Theo em 4.5. Suppose (λ0,u
0)is a posi i e solu ion o (1.5).
(a) I
(4.13) σ−∆−λ0
mu1/m−1
0;Ω
>0,
hen, he e exis ε>0and a eal analy ic map U:(λ0−ε, λ0+ε)→
C1+α
0(Ω),0<α<1, such ha U(λ0)=u0and (λ, U(λ)) is a posi i e
solu ion o (1.5) o each λ∈(λ0−ε, λ0+ε). Mo eo e , he map
λ→ U(λ)is poin -wise inc easing and he e exis s a neighbou hood N
o (λ0,u
0)in (0,∞)×C
0(Ω) such ha i (λ, u)∈N sol es (1.5), hen
u=U(λ).
Combining Fas , Linea and Slow Di usion 293
(b) I
(4.14) σ−∆−λ0
mu1/m−1
0;Ω
=0,
hen, he e exis ε>0and a eal analy ic map (Λ,U): (−ε, ε)→(0,∞)×
C1+α
0(Ω),0<α<1, such ha (Λ(0),U(0)) = (λ0,u
0)and o each
s∈(−ε, ε),(Λ(s),U(s)) is a posi i e solu ion o (1.5). Mo eo e , he e
exis s a neighbou hood No (λ0,u
0)in (0,∞)×C
0(Ω) such ha i
(λ, u)∈N sol es (1.5), hen(λ, u)=(Λ(s),U(s)) o some s∈(−ε, ε).
Fu he mo e, i Φ>0deno es a p incipal eigen unc ion associa ed wi h
he p incipal eigen alue (4.14), hen he unc ion U(s)can be chosen so
ha he auxilia y map s→ V(s)defined by
(4.15) V(s):=U(s)−u0−sΦ,|s|<ε,
sa is y ΩV(s)Ψ = 0 and V(s)=O(s2),ass→0. Also, o his choice,
(4.16)
Λ(s)=λ0+s2λ2+O(s3),
λ2:= λ0
2ΩΦ3u1/m−2
0
1
m1−1
mΩ
u1/m
0Φ<0,
and, o each s∈(−ε, ε),
(4.17) sign dΛ
ds (s)=signσ−∆−Λ(s)
mU(s)1/m−1;Ω
.
Summa izing, a ound any linea ly asymp o ically s able posi i e solu ion he
se o solu ions o (1.5) consis s o a smoo h cu e o linea ly asymp o ically
s able solu ions, while a ound any linea ly neu ally s able posi i e solu ion he
se o solu ions consis s o a second o de sub-c i ical u ning poin whose uppe
cu e is filled in by linea ly uns able posi i e solu ions, whe eas i s lowe cu e is
filled in by linea ly asymp o ically s able posi i e solu ions. Fo a mo e de ailed
discussion we send o he in e es ed eade o [15] and [16], whe e he linea
diffusion case was ea ed.
P oo o Theo em 4.5. Pa (a) is an easy consequence om he implici
unc ion heo em applied o he ope a o Kdefined in Sec ion 4.2. As any non-
i ial solu ion pai (λ, u) mus ha e he second componen , u, in he in e io o
he cone o posi i e unc ions o C0(Ω) and we a e assuming ha Ω+⊂Ω, he
map u→K(λ, u)isanaly ic o eachλ>0. Thus, he implici unc ion heo em
p o ides us wi h an analy ic solu ion cu e.
The exis ence and he uniqueness o he cu e (Λ(s),U(s)) in Pa (b), as well
as (4.17), ha e been al eady shown in [2, P oposi ion 20.8]. Ac ually, hey can be
ob ained by applying he implici unc ion heo em o a ce ain ope a o ela ed
o K h ough a Lyapuno –Schmid decomposi ion pa allel o span[Φ]. I should
294 J. L´
opez-G´
omez — A. Su´
a ez
be no ed ha , hanks o (4.14), Λ(0) = 0, whe e s ands o diffe en ia ion
wi h espec o he pseudo-leng h o a c o cu e s. Consequen ly, he p oo will
be comple ed i we show ha λ2=Λ
(0)/2 sa isfies (4.16). Indeed, o each
s∈(−ε, ε)weha e ha
(4.18) −∆[u0+sΦ+V(s)] = [λ0+s2λ2+O(s3)][u0+sΦ+V(s)]1/m,
and, hence, diffe en ia ing (4.18) wice wi h espec s, pa icula izing he esul -
ing exp ession a s= 0 and ea anging e ms gi es
(4.19) −∆−λ0
mu1/m−1
0V(0) = 2λ2u1/m
0+λ0
m1
m−1u1/m−2
0Φ2.
I should be no ed ha he second e m in he igh hand side o (4.19) makes
sense since u−2
0Φ2∈C(Ω). Now, mul iplying (4.19) by Φ, in eg a ing in Ω and
applying he o mula o in eg a ion by pa s gi es
λ2=λ0
2ΩΦ3u1/m−2
0
1
m1−1
mΩ
u1/m
0Φ.
Thus, o conclude he p oo , i emains o show ha
(4.20) ΩΦ3u1/m−2
0
1
m1−1
m<0.
As in [15] and [16], his inequali y will be ob ained om a celeb a ed a ia ional
iden i y a ibu ed o M. Picone [20] (c . e.g. [9, Sec ion 4] and [18, Lemma
4.1]). Fo any u, ∈C
1
0(Ω) wice diffe en iable a.e. in Ω and such ha /u ∈
C(Ω)∩C1(Ω), and e e y Υ ∈C
1([0,∞); R), he ollowing iden i y, usually e e ed
o as Picone’s iden i y, holds
(4.21) Ω
Υ
u(− ∆u+u∆ )=−Ω
Υ
uu2∇
u
2
.
Choosing
Υ( )= 2, =Φ,u=u0,
iden i y (4.21) gi es
(4.22) ΩΦ3u1/m−2
01−1
m=ΩΦ
u02
(−Φ∆u0+u0∆Φ)<0,
since Φ canno be a mul iple o u0. Clea ly, (4.22) implies
ΩΦ3u1/m−2
01−1
m1
m≤ΩΦ3u1/m−2
01−1
m<0,
since (1−x)x≤1−x o each x∈R. This shows (4.20) and concludes he p oo
o he heo em. 
As an immedia e consequence om Theo em 4.5, he ollowing esul holds.
Combining Fas , Linea and Slow Di usion 295
Co olla y 4.6. Le (λ0,u
0)be a posi i e solu ion o (1.5) sa is ying (4.14).
Then, he e exis s ε>0such ha o each λ∈[λ0−ε, λ0),(1.5) has, a leas ,
wo posi i e solu ions; one o hem linea ly asymp o ically s able and he o he
linea ly uns able. Mo eo e , he e exis s a neighbou hood No (λ0,u
0)in R×
C0(Ω) such ha (1.5) canno admi a posi i e solu ion in Ni λ>λ
0.
4.5. Local s uc u e o Ca (λ, u)=(0,0).The main esul o his sec ion
eads as ollows.
P oposi ion 4.7. The e exis ε>0and β>0such ha , o each λ∈(0,ε],
he minimal posi i e solu ion θλis he unique posi i e solu ion o (1.5) in Bβ.
In pa icula ,
C∩[(0,ε]×Bβ]={(λ, θλ):0<λ≤ε}.
Ac ually, hanks o Co olla y 4.6, o eachλ∈(0,ε], he ollowing holds
σ−∆−λ
mθ1/m−1
λ;Ω
>0
and, he e o e, hanks o Theo em 4.5(a),C∩[(0,ε]×Bβ]is a compac a c o
analy ic cu e.
P oo . Thanks o P oposi ion 4.2 and Theo em 4.3, he e exis s R>0
such ha (1.5) has a posi i e solu ion, a leas , o each λ∈(0,R], because
PλCis a connec ed in e al o (0,∞). Ac ually, due o P oposi ion 4.4, (1.5)
possesses a minimal solu ion, θλ, o eachλ∈(0,R]. Thus, θλis well defined o
any sufficien ly small λ>0.
Suppose (1.5) possesses, o some λ∈(0,R], a u he solu ion uλ. Then,
uλ>θ
λand, hence,
(−∆−λχΩ1)(uλ−θλ)=λχΩ+(u1/m+
λ−θ1/m+
λ)+λχΩ−(u1/m−
λ−θ1/m−
λ)
≤λ
m+
χΩ+θ1/m+−1
λ(uλ−θλ)+ λ
m−
χΩ−u1/m−−1
λ(uλ−θλ).
Thus,
−∆−λχΩ1−λ
m+
χΩ+θ1/m+−1
λ−λ
m−
χΩ−u1/m−−1
λ(uλ−θλ)≤0,
and, he e o e, hanks o he s ong maximum p inciple,
(4.23) σ−∆−λχΩ1−λ
m+
χΩ+θ1/m+−1
λ−λ
m−
χΩ−u1/m−−1
λ;Ω
≤0.
The p oo o he p oposi ion will ollow om (4.23), a guing by con adic ion.
Suppose he e exis s a sequence (λn,u
λn), n≥1, o posi i e solu ions o (1.5)
such ha
lim
n→∞(λn,u
λn)=(0,0),u
λn>θ
λn>0,n≥1.

296 J. L´
opez-G´
omez — A. Su´
a ez
Then, hanks o (4.23),
(4.24) σ−∆−λnχΩ1−λn
m+
χΩ+θ1/m+−1
λn−λn
m−
χΩ−u1/m−−1
λn;Ω
≤0,n≥1.
Since m−<1,
(4.25) lim
n→∞
λn
m−
χΩ−u1/m−−1
λn=0.
Mo eo e , hanks o he es ima e (4.7), we ha e ha
θλn≥λm+/(m+−1)
n 1,n≥1,
and, hence,
−λn
m+
χΩ+θ1/m+−1
λn≥− 1
m+
χΩ+ 1/m+−1
1,n≥1.
Thus, hanks o (4.24) and (4.25), passing o he limi as n→∞gi es
σ−∆−1
m+
χΩ+ 1/m+−1
1;Ω
≤0,
which is impossible, since 1is a non-degene a e solu ion o (4.6) wi h λ=1.
This con adic ion concludes he p oo o he p oposi ion. 
4.6. The componen Cis unbounded. The main esul o his sec ion is
he ollowing.
P oposi ion 4.8. The componen Cis unbounded in R×C
0(Ω).
P oo . We will a gue by con adic ion. Suppose Cis bounded. Then, he
ex ended componen
C0:= C∪{(0,0)}
is bounded in X:= R×C0(Ω), and, hence, i is compac , since i consis s o fixed
poin s o he compac ope a o Kdefined in Sec ion 4.2. Thus, since
K−1(0) ∩({0}×C
0(Ω)) = {(0,0)},
i is appa en , om P oposi ion 4.7, ha he e exis s η∈(0,ε] such ha
(4.26) C0∩([0,η]×C
0(Ω)) = {(λ, θλ):0≤λ≤η}.
Subsequen ly, we use he no a ions in oduced in he s a emen o P oposi-
ion 4.7. Se δ:= β/2 and conside he open neighbo hood o C0defined by
U:= C0+[(−η/2,η/2) ×Bδ],
as well as he se o non- i ial ze oes o K
S:= {(λ, u)∈X:K(λ, u)=0,u=0}∪{(0,0)}.
Combining Fas , Linea and Slow Di usion 297
Subsequen ly, a bounded open se O⊂Xis said o be an open isola ing neigh-
bo hood o C0in Xi C0⊂Oand
(4.27) ∂O∩S=∅.
I ∂U ∩S=∅, henUp o ides us wi h an open isola ing neighbo hood o he
componen C0, bu , in gene al, ∂U ∩S=∅. When his is he case, Whybu n’s
Lemma [24] uses he ac ha C0is a maximal compac and connec ed subse o
S o show he exis ence o an open isola ing neighbou hood Oo C0such ha
C0⊂O⊂U
(c . e.g. he p oo o [19, Theo em 6.3.1]). Now, o each λ>0wese
Oλ:= {u∈X:(λ, u)∈O}.
By cons uc ion,
Oη/3∩S={θη/3}.
Thus, combining Le ay–Schaude ’s o mula wi h P oposi ion 4.7 gi es
Deg(K(η/3,·),Oη/3) = Ind(K(η/3,·),θ
η/3)=1
and, hence, by homo opy in a iance, Deg(K(λ, ·),Oλ) = 1 o all λ>0.
On he o he hand, o sufficien ly la ge λwe ha e ha Oλ=∅and, hence,
Deg(K(λ, ·),Oλ) = 0. This con adic ion concludes he p oo . 
4.7. P oo o Theo em 4.1. Suppose he e exis 
λ>0andu

λ=θ

λsuch
ha (
λ, u

λ) is linea ly s able (ei he neu ally s able, o asymp o ically s able).
Then, hanks o P oposi ion 4.4 and Theo em 4.5 (c . Co olla y 4.6), by global
con inua ion o he le o 
λ, (1.5) mus admi wo linea ly asymp o ically s able
solu ions o each λ∈(0,
λ). As he solu ions in each o he co esponding
cu es a e inc easing wi h λ, hanks o P oposi ion 4.7, (1.5) mus admi a
posi i e solu ion o λ= 0. This is impossible. The e o e, o each λ>0, θλis
he unique linea ly s able posi i e solu ion o (1.5) i i admi s a solu ion. This
shows ( ). I should be no ed ha , hanks o P oposi ion 4.2, λ= 0 is he unique
bi u ca ion alue o posi i e solu ions om u=0.
Le λ∗be he maximal λ>0 sa is ying he ollowing condi ion
(4.28) σ−∆−λ
mθ1/m−1
λ;Ω
>0,λ∈(0,λ
∗).
Thanks o P oposi ions 4.2, P oposi ion 4.7, λ∗is well defined. Mo eo e , since
Cis he maximal connec ed se such ha (0,0) ∈C,
(4.29) γ:= {(λ, θλ):λ∈(0,λ
∗)}⊂C,
because γis connec ed.
298 J. L´
opez-G´
omez — A. Su´
a ez
Ei he γis bounded in R×C0(Ω), o i is unbounded. Suppose γis bounded.
Then,
uλ∗:= lim
λ↑λ∗θλ
p o ides us wi h a solu ion o (1.5) o λ=λ∗. Mo eo e , by he con inuous de-
pendence o he p incipal eigen alue wi h espec o he po en ial, (4.28) implies
σ−∆−λ∗
mu1/m−1
λ∗;Ω
=0,
because o he maximali y o λ∗.Asθλ∗is he unique linea ly s able solu ion,
necessa ily
uλ∗=θλ∗.
Ac ually, hanks o Co olla y 4.6, a ound (λ∗,θ
λ∗), Cconsis s o a second o de
sub-c i ical u ning poin . In pa icula , he e exis s λω∈[0,λ
∗) such ha C
possesses wo solu ions, a leas , o each λ∈(λω,λ
∗); his shows he fi s claim
o Pa (e). No e ha he e exis s an open se Osuch ha :
(1) {(λ, θλ):λ∈(0,λ
∗]}⊂O.
(2) Any solu ion o (1.5) in Olies in C.
(3) Any posi i e solu ion o (1.5) in ∂Ois linea ly uns able.
Clea ly,
(0,λ
∗]⊂Λ:=PλC.
We claim ha Λ = (0,λ
∗]. Indeed, suppose he e exis s 
λ>λ
∗such ha 
λ∈Λ.
Then, by global con inua ion om (
λ, θ

λ) o hele o 
λone can cons uc
a linea ly s able posi i e solu ion o (1.5), ou side O,e.g. o λ=λ∗.This
con adic s he uniqueness o he s able solu ion, and, he e o e,
Λ=(0,λ
∗].
To comple e he p oo o he heo em when γis bounded i emains o show
ha Cpossesses wo posi i e solu ions o each λ∈(0,λ
∗)i ei he N∈{1,2},
o N≥3andm−>(N−2)/(N+ 2). I suffices o show ha , unde hese
condi ions, he componen Cis bounded in [ε, λ∗]×C
0(Ω) o any ε∈(0,λ
∗).
Pick one o hose ε’s. Then, he blowing-up a gumen o B. Gidas and J. Sp ¨uck
([14]) ca ies o e mu a is mu andis o show he exis ence o a posi i e cons an
M>0 such ha
uλC(Ω−)≤M
o any posi i e solu ion (λ, uλ) o (1.5) wi h λ∈[ε, λ∗]. Thus, uλ|Ω1∪Ω+is a
subsolu ion o
(4.30) 




−∆u=λu1/min Ω1∪Ω+,
u=0 on∂Ω,
u=Mon ∂Ω−.
Combining Fas , Linea and Slow Di usion 299
Now, we ha e o dis inguish wo diffe en cases. Assume Ω1=∅. Then (4.30)
possesses a unique posi i e solu ion o each λ>0, say λ, and, as an easy
consequence om he s ong maximum p inciple,
uλ|Ω+≤ λin Ω+,
o each λ∈[ε, λ∗], which p o ides us wi h he desi ed a p io i bounds. I Ω1=∅,
hen, hanks o P oposi ion 4.2, λ∗<λ
+
0, and, simila ly, uλ|Ω1∪Ω+is bounded
abo e by he unique posi i e solu ion o (4.30). The exis ence and he uniqueness
o he posi i e solu ion o (4.30) ollows wi h he same a gumen used in [13] o
ea he case o homogeneous Di ichle bounda y condi ions. This concludes
he p oo o he heo em when γis bounded.
Now, suppose γis unbounded (c . (4.29)). Then, necessa ily, (4.5) holds.
Indeed, i (1.5) possesses a posi i e solu ion (λ∗,u
∗), hen i possesses a minimal
solu ion (λ∗,θ
λ∗) and, consequen ly, i possesses wo s able posi i e solu ions o
some ange λ<λ
∗, which is impossible. Ac ually, in his case C=γ.This
concludes he p oo . 
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