Positive solutions for the degenerate logistic indefinite superlinear problem the slow diffusion case

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). Re e ences [1] S. Alama, Semilinea ellip ic equa ions wi h sublinea inde ini e nonlinea i ies, Ad . in Di e en ial Equa ions, 4(1999), 813-842. [2] S. Alama and G. Ta an ello, On he semilinea ellip ic equa ions wi h inde ini e nonlinea - i ies, Calc. Va . Pa ial Di e en ial Equa ions, 1(1993), 439-475. [3] S. Alama and G. Ta an ello, Ellip ic p oblems wi h nonlinea i ies inde ini e sign, J. Func . Anal., 141 (1996), 159-215. [4] S. Alama and G. Ta an ello, On he sol abili y o a semilinea ellip ic equa ion ia an associa ed eigen alue p oblem, Ma h. Z.,221 (1996), 467-493. [5] H. Amann, Fixed poin equa ions and nonlinea eigen alue p oblems in o de ed Banach spaces, SIAM Re iew, 18 (1976), 620-709. [6] H. Amann and J. L´opez-G´omez, A p io i bounds and mul iple solu ions o supe linea inde ini e ellip ic p oblems, J. Di e en ial Equa ions, 146 (1998), 336-374. [7] A. Amb ose i, H. B ezis and G. Ce ami, Combined e ec s o conca e and con ex nonlin- ea i ies in some ellip ic p oblems, J. Func . Anal., 122 (1994), 519-543. [8] A. Amb ose i, J. Ga c´ıa-Azo e o and I. Pe al, Mul iplici y esul s o some nonlinea ellip ic equa ions, J. Func . Anal., 137 (1996), 219-242. [9] A. Amb ose i, and P. Hess, Posi i e solu ions o asymp o ically linea ellip ic eigen alue p oblems, J. Ma h. Anal. Appl., 73 (1980), 411-422. [10] D. A coya, J. Ca mona and B. Pellacci, Bi u ca ion o some quasi-linea ope a o s, P oc. Royal Soc. o Edin. Sec ion A., 131 (2001), 733-766. [11] H. Be es ycki, I. Capuzzo-Dolce a and L. Ni enbe g, Supe linea inde ini e p oblems and nonlinea Liou ille heo ems, Topol. Me hods Nonlinea Anal., 4(1994), 59-78. [12] H. Be es ycki, I. Capuzzo-Dolce a and L. Ni enbe g, Va ia ional me hods o inde ini e su- pe linea homogeneous ellip ic p oblems, NoDEA Nonlinea Di e en ial Equa ions Appl., 2(1995), 553-572. [13] H. B ezis and L. Ni enbe g, H1 e sus C1local minimize s, C. R. Acad. Sci. Pa is, S´e ie I, 317 (1993), 465-472. [14] W. Chen and C. Li, Inde ini e ellip ic p oblems in a domain, Disc e e Con in. Dynam. Sys ems, 3(1997), 333-340. [15] M. Delgado, J. L´opez-G´omez and A. Su´a ez, Non-linea e sus linea di usion. F om clas- sical solu ions o me asolu ions. Ad . Di e . Eq., 7(2002), 1101-1124. 820 M. DELGADO AND A. SU ´ AREZ [16] M. Delgado and A. Su´a ez, S abili y and uniqueness o coope a i e degene a e Lo ka- Vol e a model, Nonl. Anal., 49 (2002), 757-778. [17] R. G´omez-Re˜nasco and J. L´opez-G´omez, The e ec o a ying coe icien s on he dynamics o a class o supe linea inde ini e eac ion-di usion equa ions, J. Di e en ial Equa ions, 167 (2000), 36-72. [18] M. E. Gu in and R. C. MacCamy, On he di usion o biological popula ions, Ma h. Biosci., 33 (1977), 35-49. [19] J. He n´andez, F. Mancebo and J. M. Vega de P ada, On he linea iza ion o some singula nonlinea ellip ic p oblems and applica ions, Ann. Ins . H. Poinca e Anal. Non-Lineai e, 19 (2002), 777-813. [20] J. L´opez-G´omez, The maximum p inciple and he exis ence o p incipal eigen alues o some linea weigh ed bounda y alue p oblems, J. Di e en ial Equa ions,127 (1996), 263-294. [21] J. L´opez-G´omez, On he exis ence o posi i e solu ions o some inde ini e supe linea ellip ic p oblems, Comm. Pa ial Di e en ial Equa ions,22 (1997), 1787-1804. [22] J. L´opez-G´omez, Va ying bi u ca ion diag ams o posi i e solu ions o a class o inde ini e supe linea bounda y alue p oblems, T ans. Ame . Ma h. Soc., 352 (2000), 1825-1858. [23] L. Moschini, S. I. Pohozae and A. Tesei, Exis ence and nonexis ence o solu ions o non- linea Di ichle p oblems wi h i s o de e ms, J. Func . Anal., 177 (2000), 365-382. [24] T. Ouyang, On he posi i e solu ions o semilinea equa ions ∆u+λu+hup= 0 on compac mani ods. Pa II, Indiana Uni . Ma h. J., 40 (1991), 1083-1141. [25] P. Rabinowi z, Some global esul s o nonlinea eigen alue p oblems, J. Func . Anal.,7 (1971), 487-513. [26] N. Ta ulea, Posi i e solu ion o some nonlinea ellip ic equa ion wi h Neumann bounda y condi ions, P oc. Japan Acad. Se . A, 71 (1995), 161-163. 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]