Combining fast, linear and slow diffusion

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 .  Re e ences [1] S. Alama,Semilinea ellip ic equa ions wi h sublinea indefini e nonlinea i ies,Ad . Diffe en ial Equa ions 4(1999), 813–842. [2] H. Amann,Fixed poin equa ions and nonlinea eigen alue p oblems in o de ed Banach spaces,SIAMRe iew18 (1976), 620–709. [3] H. Amann and J. 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