Combining fast, linear and slow diffusion

Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 23, 2004, 275–300 COMBINING FAST, LINEAR AND SLOW DIFFUSION Juli´ an L´ opez-G´ omez — Antonio Su´ arez Abstract. Although the pioneering studies of G. I. Barenblatt ([8]) and A. G. Aronson and L. A. Peletier ([7]) did result into a huge industry around the porous media equation, none further study analyzed the effect of combining fast, slow, and linear diffusion simultaneously, in a spatially heterogeneous porous medium. Actually, it might be this is the first work where such a problem has been addressed. Our main findings show how the heterogeneous model possesses two different regimes in the presence of a priori bounds. The minimal steady-state of the model exhibits a genuine fast diffusion behavior, whereas the remaining states are rather reminiscent of the purely slow diffusion model. The mathematical treatment of these heterogeneous problems should deserve a huge interest from the point of view of its applications in fluid dynamics and population evolution. 1. Introduction In this paper we study the positive solutions of the boundary value problem (1.1) −∆(wm(x))=λw in Ω, w=0 on∂Ω, where Ω ⊂RN,N≥1, is a bounded domain of class C2,λ∈R,and m=1+pχΩ+−qχΩ− 2000 Mathematics Subject Classification. 35B32, 35J25, 35J60, 35K57. Key words and phrases. Heterogeneous nonlinear diffusion, fast, slow and linear diffusion. The research of the first named author supported by the Spanish Ministry of Science and Technology under Grants BFM2000-0797 and BFM2003-06466. The research of the second named author was as well supported by REN2003-00707. c 2004 Juliusz Schauder Center for Nonlinear Studies 275 276 J. L´ opez-G´ omez — A. Su´ arez where Ω+and Ω−are two subdomains of Ω of class C2such that (1.2) Ω+⊂Ω,Ω+∩Ω−=∅, and p∈L∞(Ω+)∩C(Ω+), q∈L∞(Ω−)∩C(Ω−)satisfy (1.3) p(x)>0and0<q(y)<1foreach(x, y)∈Ω+×Ω−. Throughout this paper, for any measurable set M⊂Ω, we denote by χMthe characteristic function of M, i.e. χM(x)=1ifx∈M,andχM(x)=0foreach x∈Ω\M.Also,weset Ω1:= Ω \(Ω+∪Ω−), the open set where m= 1, and suppose, by simplicity, that Ω1is connected. Though we allow Ω1,Ω +,orΩ −to be empty, Figure 1.1 shows one of the admissible configurations dealt with in this work. Ω Ω Ω + − 1 Figure 1.1.An admissible configuration Throughout this paper we denote (1.4) m+:= m|Ω+=1+pχΩ+,m −:= m|Ω−=1−qχΩ−. Then, m+(x)>1foreachx∈Ω+and 0 <m −(x)<1foreachx∈Ω−,and hence (1.1) provides us with the steady states of a porous medium equation where diffusion is linear in Ω1and nonlinear in Ω+∪Ω−(slow in Ω+and fast in Ω−). The analysis of these kind of boundary value problems generated a huge industry since the pioneering studies of G. I. Barenblatt ([8]) and A. G. Aronson, L. A. Peletier ([7]), although most of the literature treated the very special case when mis constant. Up to the best of our knowledge, the first work where mhas been allowed to vary is M. Delgado et al. in [12], where the special case when m−=0was treated. The present paper seems to be the first work where the general problem of analyzing the interplay between slow, fast and linear diffusion, simultaneously, Combining Fast, Linear and Slow Diffusion 277 has been addressed. Therefore, most of the results found in this paper are completely new and, undoubtedly, open new research directions that might be of great relevance from the point of view of the applications of the underlying abstract mathematical theory to population dynamics and porous media dynamics. To summarize our main results we need to introduce some basic concepts and notations. As the change of variable u=wmtransforms (1.1) into (1.5) −∆u=λu1/min Ω, u=0 on∂Ω, our efforts will be focused into the problem of analyzing the existence and multiplicity of positive solutions of (1.5). A function u∈H1 0(Ω) ∩L∞(Ω) is said to be a solution of (1.5) if u1/m∈L2N/(N+2)(Ω) and it satisfies the equation in the classical weak sense. By elliptic regularity, any weak non-negative solution u= 0 provides us with an strong solution almost everywhere twice differentiable in Ω and, as a result of the strong maximum principle, u(x)>0foreachx∈Ω and ∂u(x)/∂n < 0foreachx∈∂Ω, where nstands for the outward normal vector-field of Ω. In the remaining of this paper, it should be kept in mind that, as a result of the maximum principle, (1.5) cannot admit a positive solution if λ≤0. Throughout the rest of this paper, for any potential V∈L∞(Ω) we denote by σ[−∆+V; Ω] the principal eigenvalue of −∆+Vin Ω under homogeneous Dirichlet boundary conditions. Note that if Ω+=Ω −=∅, then (1.5) becomes linear and, hence, it possesses a positive solution if, and only if, λ=σ[−∆; Ω]. Therefore, we subsequently assume (1.6) Ω+∪Ω−=∅. Although most of our findings are completely new even in the special case when m−1 does not change of sign, the most interesting results of this paper are those found for the general case when m−1 changes sign, where one must assume m+and m−to be constant to get optimal results. Under these assumptions our main result is Theorem 4.1, which can be rewritten as follows. Theorem 1.1. Suppose Ω+and Ω−are non-empty and m+,m−are constant. Then, there exist λ∗>0and an unbounded component, C,ofthesetof positive solutions (λ, u)of (1.5) such that: (a) (λ, u)=(0,0) ∈C,andΛ:=PλC∈{(0,λ ∗],(0,λ ∗)},forPλ(λ, u):=λ. (b) Problem (1.5) does not admit a positive solution if λ∈(−∞,0]∪(λ∗,∞). 278 J. L´ opez-G´ omez — A. Su´ arez (c) For each λ∈Λ,(1.5) possesses a minimal positive solution, denoted by θλ,andthemapλ→ θλis smooth and increasing. Moreover, σ[−∆−(λ/m)θ1/m−1 λ;Ω]>0if λ∈Λ\{λ∗}, i.e. θλis linearly asymptotically stable with respect to the parabolic counterpart of (1.5),while σ[−∆−(λ∗/m)θ1/m−1 λ∗;Ω]=0 if λ∗∈Λ, i.e. θλ∗is linearly neutrally stable. (d) The component Ccontains the arc of differentiable curve Γ:={(λ, θλ): λ∈Λ\{λ∗}},limλ↓0θλC0(Ω) =0,andlimλ↑λ∗θλ=θλ∗if Λ= (0,λ ∗],whilelimλ↑λ∗θλC0(Ω) =∞if Λ=(0,λ ∗).Actually,C=Γif Λ=(0,λ ∗). (e) If Λ=(0,λ ∗], then there exists λω∈[0,λ ∗)such that (1.5) has two positive solutions, at least, for each λ∈(λω,λ ∗). Actually, if either N∈{1,2},orN≥3and m−>(N−2)/(N+2),thenΛ=(0,λ ∗]and λω=0. (f) For each λ∈Λ,θλprovides us with the unique linearly stable positive solution of (1.5). The distribution of this paper is the following: Section 2 analyzes the case when Ω+=∅, Section 3 analyzes the case when Ω−=∅, and, then, in Section 4, we prove Theorem 1.1. Throughout the manuscript we shortly describe some special perturbation results connecting each of these cases with the remaining one, though we have refrained to include the details of all their proofs to keep the length of the manuscript within a reasonable level. All those results will be deeply discussed and collected elsewhere. 2. The case Ω+=∅ As we are assuming (1.6), we have Ω−=∅and, hence, (1.5) is superlinear within Ω−. The following result holds in the special case when Ω1=∅. Theorem 2.1. Suppose Ω+=Ω 1=∅. Then, the following assertions are true: (a) Under the following condition (2.1) inf Ω− m−>N−2 N+2 if N≥3, problem (1.5) possesses a positive solution for each λ>0. Moreover, if (λn,u n),n≥1, is a sequence of positive solutions of (1.5) such that Combining Fast, Linear and Slow Diffusion 279 limn→∞ λn=0,then (2.2) lim sup n→∞ unC0(Ω) =∞. (b) If m−is constant, then uis a positive solution of (1.5) if, and only if, u=λ−m−/(1−m−)v for some positive solution vof (2.3) −∆v=v1/m−in Ω, u=0 on ∂Ω. In particular, the number of positive solutions of (1.5),foreachλ>0, equals the number of positive solutions of (2.3) and, therefore, the following holds: (b1) Suppose m−>(N−2)/(N+2) if N≥3.Then(2.3) possesses a positive solution, at least, and, actually, each positive solution v of (2.3) provides us with a curve λ→ uλ:= λ−m−/(1−m−)v, λ > 0, of positive solutions of (1.5). Moreover, lim λ↓0uλ=∞and lim λ↑∞ uλ=0 uniformly in compact subsets of Ω. (b2) Suppose N≥3,m−≤(N−2)/(N+2),andΩis star-shaped. Then, (1.5) cannot admit a positive solution. Subsequently, we shall denote by Pρ:R×C 0(Ω) →C 0(Ω) the ρ-projection operator, i.e. Pρ(ρ, u)=ρfor each (ρ, u)∈R×C 0(Ω). Proof of Theorem 2.1. Suppose (2.1) and consider, for each λ>0, the auxiliary problem (2.4) −∆u=µu +λu1/m−in Ω, u=0 on∂Ω, where µ∈Ris regarded as a bifurcation parameter. Thanks to (2.1), the blowing-up argument of B. Gidas and J. Spr¨uck (see [14]) can be easily adapted to show that the positive solutions of (2.4) possess L∞(Ω) a priori bounds uniform in compact intervals of µ∈R. Moreover, thanks to local bifurcation result of M. G. Crandall and P. H. Rabinowitz ([10]), µ:= σ[−∆; Ω] is a bifurcation value to positive solutions of (2.4) from the trivial state (µ, u)=(µ, 0). Actually, by the global unilateral theorem of P. H. Rabinowitz ([22]), the component of positive solutions of (2.4) emanating from (µ, 0) at µ=σ[−∆; Ω], subsequently denoted 280 J. L´ opez-G´ omez — A. Su´ arez by C, must be unbounded in R×C 0(Ω) (cf. E. N. Dancer [11], as well as [19, Chapters 6, 7], for a complete development of the necessary abstract theory, as the original paper of P. H. Rabinowitz [22] contains some serious gaps). Suppose (2.4) possesses a positive solution. Then, (−∆−λu1/m−−1)u=µu and, hence, by the uniqueness of the principal eigenvalue, µ=σ[−∆−λu1/m−−1;Ω]. Thus, since λ>0, it is apparent, from the monotonicity of the principal eigenvalue with respect to the potential, that µ<σ[−∆; Ω], and, hence, PµC⊂(−∞,σ[−∆; Ω]). Actually, thanks to the existence of uniform a priori bounds, PµC=(−∞,σ[−∆; Ω]) and, therefore, 0 ∈P µC. In particular, (1.5) possesses a positive solution. Now, let (λn,u n), n≥1, be a sequence of positive solutions of (1.5) with limn→∞ λn= 0. If there exists a constant M>0 such that unC0(Ω) ≤M, n ≥1, then, by the compactness (−∆)−1(the inverse of the operator −∆ in Ω under homogeneous Dirichlet boundary conditions), along some subsequence of (λn,u n), labeled again by n, lim n→∞ un−u∞C0(Ω) =0, for some strong solution u∞of the problem (2.5) −∆u=0 inΩ, u=0 on∂Ω. Necessarily u∞= 0 and, hence, lim n→∞ unC0(Ω) =0. Now, set vn:= un unC0(Ω) ,n≥1. Then, for each n≥1, we have that vn=(−∆)−1(λnvnu1/m−−1 n), and, hence, along some subsequence, labeled again by n,wehavethat lim n→∞ vn−v∞C0(Ω) =0. Combining Fast, Linear and Slow Diffusion 281 Necessarily, v∞C0(Ω) =1,v∞>0, and v∞solves (2.5). This is impossible, since u= 0 is the unique solution of (2.5). This contradiction shows (2.2) and concludes the proof of (a). (b1) is an easy consequence from (a), and (b2) follows readily from a celebrated identity by S. I. Pohozaev ([21]).  Eveninthecasewhenm−is a constant satisfying (2.1), it is well known that the number of positive solutions of (1.5) is strongly dependent upon the geometry of the domain Ω. Indeed, if Ω consists of n≥2 separated balls joined by n−1 narrow corridors, then (2.3) has 2n−1 positive solutions and, therefore, (1.5) possesses 2n−1 global curves of positive solutions. Eventually, even for the simplest domain geometries, the number of solutions of (1.5) might be strongly dependent upon the local oscillation properties of the function m−(x) (cf. [15], as well as the references there in, for similar closely related discussions). In the general case when N≥3 and the auxiliary function s(x):=m−(x)−N−2 N+2,x∈Ω, changes of sign, the problem of characterizing the existence of positive solutions for (1.5) increases in complexity. The corresponding results will be given elsewhere, as they are still in progress. In the most general case when Ω1=∅the following result is satisfied. Theorem 2.2. Suppose Ω+=∅,Ω1=∅, and consider the function σ(λ):=σ[−∆−λχΩ1;Ω],λ≥0. Then, the exists a unique λ0=λ0(Ω1)>0satisfying σ−1(0) ∩[0,∞)={λ0}. Moreover, (1.5) cannot admit a positive solution if λ≥λ0. Suppose, in addition, that sup Ω− m−<1 and regard to λas a bifurcation parameter. Then λ=λ0is a bifurcation value from (λ, u)=(λ, 0) to an unbounded continuum C⊂(0,λ 0)×C 0(Ω) of positive solutions of (1.5). Moreover, PλC=(0,λ 0)if condition (2.1) is satisfied, though, in general, PλCmight be a proper subinterval of (0,λ 0). Figure 2.1 shows three admissible situations within the setting of Theorem 2.2. Figure 2.1(a) represents Cunder assumption (2.1), while Figure 2.1(b), (c) represent two admissible C’s where (2.1) fails. In case (b), PλC=[λ∗,λ 0), for some λ∗∈(0,λ 0), while, in case (c), PλC=(λ∗,λ 0). In all cases the problem 282 J. L´ opez-G´ omez — A. Su´ arez 0λ λ 0 u 0λ λ 0 u 0λ λ 0 u λλ ∗∗ (a) (b) (c) CCC Figure 2.1.Three admissible bifurcation diagrams might have an arbitrary number of solutions as a result of the geometry of Ω and the local properties of m−. A crucial feature, differentiating the case when Ω1=∅from the case described by Theorem 2.2, is the fact there exists ε>0 such that [λ0−ε, λ0)⊂P λC if Ω1=∅, and, therefore, (1.5) always possesses a positive solutions for each λ<λ 0sufficiently close to λ0, independently of the size of m−; in strong contrast with the situation described by Theorem 2.1, where (1.5) cannot admit a positive solution if Ω is star-shaped, N≥3andm−≤(N−2)/(N+2). If Ωδ 1,δ∈[0,1], stands for an increasing family of smooth domains such that Ω1 1=Ω 1and limδ↓0Ωδ 1=∅, then, limδ↓0λ0(Ωδ 1)=∞(cf. the details of the proof of [13, Theorem 12]). Actually, the corresponding bifurcation diagrams approximate, as δ↓0, to the bifurcation diagram of the problem in case Ω1=∅, though, being outside the general scope of this work, this sharper analysis will appear elsewhere. Proof of Theorem 2.2. By the monotonicity of the principal eigenvalue with respect to the potential, the function σ(λ) is decreasing with λ.Moreover, σ(0) = σ[−∆; Ω] >0, and, for any ball B⊂Ω1and λ>0, we have that σ(λ)<σ[−∆−λ;B]=σ[−∆; B]−λ, and, hence, limλ↑∞ σ(λ)=−∞. This shows the existence and the uniqueness of λ0. Suppose (1.5) possesses a positive solution u. Then, (−∆−λχΩ1)u=λχΩ−u1/m−>0 and, hence, uis a strict positive supersolution of −∆−λχΩ1in Ω under homogeneous Dirichlet boundary conditions. Thus, thanks to [17, Theorem 2.5], σ[−∆−λχΩ1;Ω]>0 and, therefore, λ<λ 0. Now, we regard to λas the main bifurcation parameter and consider the nonlinear operator F:R×C 0(Ω) →C 0(Ω) defined by (2.6) F(λ, u):=u−(−∆)−1(λχΩ1u+λχΩ−|u|1/m−), Combining Fast, Linear and Slow Diffusion 283 whose positive fixed points provide us with the positive solutions of (1.5). For each λ∈R,F(λ, 0) = 0. Moreover, Fis continuous and admits the decomposition F(λ, u)=L(λ)u−λ(−∆)−1(χΩ−|u|1/m−), where L(λ)u:= u−λ(−∆)−1(χΩ1u),u∈C 0(Ω). Therefore, it adjusts to the abstract setting of [19, Chapter 6]. It should be noted that condition supΩ−m−<1 cannot be relaxed, because otherwise the nonlinearity would not be o(uC0(Ω)). Let ϕ0>0 denote a principal eigenfunction associated to σ[−∆−λ0χΩ1;Ω]. Then, (2.7) N[L(λ0)] = span[ϕ0]and d dλL(λ0)ϕ0∈ R[L(λ0)], where, given any linear continuous operator L,N[L]andR[L] stand for the null space and the range of L, respectively. Indeed, the first identity of (2.7) is true by construction. For the second, suppose (2.8) −(−∆)−1(χΩ1ϕ0)=u−λ0(−∆)−1(χΩ1u) for some u∈C 0(Ω). Then, (−∆−λ0χΩ1)u=−χΩ1ϕ0 and multiplying this identity by ϕ0and integrating by parts in Ω gives Ω1 ϕ2 0=0, which is impossible, since ϕ0(x)>0foreachx∈Ω. Therefore, since L(λ)is a Fredholm operator of index zero, λ0is a 1-transversal eigenvalue of the family L(λ) and, hence, the generalized algebraic multiplicity χ[L;λ0] introduced in [19, Chapter 4] equals 1. Therefore, thanks to [19, Theorem 4.2.4], λ0is a nonlinear eigenvalue of L(λ). Actually, this fact is a direct consequence from the main local bifurcation theorem of M. G. Crandall and P. H. Rabinowitz ([10]). It should be noted that the main theorem of [10] does not apply in order to get the existence of a curve of positive solutions of (1.5) emanating from u=0at λ=λ0, because our nonlinearity does not have the required regularity. But this is far from being a trouble, since, due to [19, Theorem 5.6.2], the index – local topological degree – of L(λ) at zero, Ind(L(λ),0), λ∼λ0,λ=λ0, must change as λcrosses λ0, because χ[L;λ0] = 1. Therefore, thanks to [19, Theorem 6.2.1] there is a component, C, of the set of nontrivial solutions of (1.5) such that (λ0,0) ∈C. Finally, the proof of [19, Theorem 6.5.5] carries over mutatis mutandis to show the existence of an unbounded subcomponent of C, C, entirely consisting of positive solutions of (1.5) and such that (λ0,0) ∈C.It 290 J. L´ opez-G´ omez — A. Su´ arez Thus, there exists a continuum of positive solutions of (1.5) emanating from u=0at λ=0. The maximal continuum, for the inclusion, provides us with the component C. Proof. The proof of (4.9) and (4.10) is based upon some homotopies coming from A. Ambrosetti and P. Hess ([5]), and D. Arcoya et al. ([6]). Fix λ<0 and consider the map H1:[0,1] ×C 0(Ω) →C 0(Ω) defined by H1(t, u):=u−(−∆)−1(tf(λ, ·,u)). Since the nontrivial zeroes of H1(t, ·) are the positive solutions of (4.11) −∆u=tλu1/min Ω, u=0 on∂Ω, and tλ ≤0, we obtain that H1(t, u)=0ift∈[0,1] and u=0. Thus,foreach R>0, the homotopy invariance of the topological degree gives 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 the proof of (4.9). Now, fix λ>0, φ∈C 0(Ω), φ>0, and consider the map H2:[0,1] ×C 0(Ω) → C0(Ω) defined by H2(t, u):=u−(−∆)−1(f(λ, ·,u)+tφ). We claim that there exists δ>0 such that H2(t, u)=0foreacht∈[0,1] and u∈Bδ\{0}. Note that, in particular, this shows that u= 0 is an isolated solution of (1.5). We shall proceed by contradiction. First, note that if H2(t, u)=0for some t∈[0,1] and u=0,then −∆u=λf(λ, ·,u)+tφ and, hence, Ω|∇u−|2=0,sincetφ ≥0. Consequently, u>0. Now, suppose there is a sequence (tn,u n)∈[0,1] ×(C0(Ω) \{0}),n≥1, such that limn→∞ un=0andH2(tn,u n)=0foreachn≥1. Then, for each n≥1, we have that un>0and −∆un=λu1/m+ n+tnφ≥λun1/m+−1 C(Ω+)un+tnφin Ω+. Moreover, un>0on∂Ω+.Thus,un|Ω+provides us with a strict positive supersolution of −∆−λun1/m+−1 C(Ω+) Combining Fast, Linear and Slow Diffusion 291 in Ω+, under homogeneous Dirichlet boundary conditions. Thus, thanks to [17, Theorem 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 contradiction shows the claim above. Now, thanks to the homotopy invariance of the topological degree, we obtain that 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)=0foreachu∈Bδ.This concludes the proof of (4.10). Now, fix λ1<0<λ 2,pickε>0 such that K(λj,u)=0foreachj∈{1,2} and u∈Bε\{0}, and consider the cylinders Qη:= [λ1,λ 2]×Bη⊂R×C 0(Ω),η∈(0,ε]. Fix η∈(0,ε]. We claim that there exist λη∈[λ1,λ 2]anduη∈∂Bηsuch that K(λη,u η)=0. Note that, necessarily, λη>0. Indeed, thanks to (4.9) and (4.10), if this were not true, then, by the homotopy invariance of the degree, we would get 1=Deg(K(λ1,·),B η)=Deg(K(λ2,·),B η)=0, which is a contradiction. By the compactness of K, it follows that there exists a sequence ηn∈(0,ε), n≥1, such that lim n→∞ ηn= 0 and lim n→∞(ληn,u ηn)=(0,0). Actually, thanks to a celebrated result by G. T. Whyburn ([24]), there is a continuum of non-trivial zeroes of Kconnecting (0,0) with uC0(Ω) =η.Asthe technical details of the proof have been already given in the proof of [19, Theorem 6.2.1], we will omit them here in (cf. [1, Theorem 3.1] and [6, Theorem 4.4] as well). This concludes the proof.  4.3. The existence and linear stability of the minimal solution. The main result of this section is the following. 292 J. L´ opez-G´ omez — A. Su´ arez Proposition 4.4. Suppose (1.5) possesses a positive solution. Then, it possesses a minimal positive solution, denoted by θλ. By minimal it is meant that θλ<ufor any other positive solution uof (1.5). Moreover, θλis linearly stable, i.e. (4.12) σ−∆−λ mθ1/m−1 λ≥0. Proof. Suppose (1.5) has a positive solution, say u. Necessarily, λ>0. Let Bbe any ball such that B⊂Ω+,denotebyψthe unique positive eigenfunction associated to σ[−∆; B], normalized so that ψC0(B)=1,andset Ψ:=ψin B, 0inΩ\B. Then, for sufficiently small ε>0, the function εΨ provides us with a subsolution of (1.5) such that εΨ<u. As a consequence, (1.5) possesses a minimal positive solution in the order interval [εΨ,u]ofC0(Ω). Thus, it possesses a minimal positive solution in the order interval [0,u], since λcannot be a bifurcation value to positive solutions from u= 0, because of Proposition 4.2. Let θu λdenote the minimal positive solution in [0,u]andletu(x, t;εΨ) be the unique solution of the parabolic counterpart of (1.5) starting at εΨ<θ u λ≤u. Thanks to the theory of D. Sattinger [23], u(·,t;εΨ) is increasing in time and it approaches θu λas t↑∞. Suppose vis another positive solution of (1.5) and shorten ε, if necessary, so that εΨ<v. Then, by the uniqueness of the limit limt↑∞ u(·,t;εΨ), we find that θu λ=θv λand, therefore, θu λis independent of the positive solution u.Thus, it provides us with the minimal positive solution θλof (1.5). Relation (4.12) follows from [2, Proposition 20.4] (cf. [4, Lemma 3.5] as well).  4.4. Solution curves through linearly stable solutions. The main result of this section reads as follows. Note that, thanks to Proposition 4.4, it reveals some crucial properties satisfied by all minimal solutions θλof (1.5). Theorem 4.5. Suppose (λ0,u 0)is a positive solution of (1.5). (a) If (4.13) σ−∆−λ0 mu1/m−1 0;Ω >0, then, there exist ε>0and a real analytic map U:(λ0−ε, λ0+ε)→ C1+α 0(Ω),0<α<1, such that U(λ0)=u0and (λ, U(λ)) is a positive solution of (1.5) for each λ∈(λ0−ε, λ0+ε). Moreover, the map λ→ U(λ)is point-wise increasing and there exists a neighbourhood N of (λ0,u 0)in (0,∞)×C 0(Ω) such that if (λ, u)∈N solves (1.5),then u=U(λ). Combining Fast, Linear and Slow Diffusion 293 (b) If (4.14) σ−∆−λ0 mu1/m−1 0;Ω =0, then, there exist ε>0and a real analytic map (Λ,U): (−ε, ε)→(0,∞)× C1+α 0(Ω),0<α<1, such that (Λ(0),U(0)) = (λ0,u 0)and for each s∈(−ε, ε),(Λ(s),U(s)) is a positive solution of (1.5). Moreover, there exists a neighbourhood Nof (λ0,u 0)in (0,∞)×C 0(Ω) such that if (λ, u)∈N solves (1.5),then(λ, u)=(Λ(s),U(s)) for some s∈(−ε, ε). Furthermore, if Φ>0denotes a principal eigenfunction associated with the principal eigenvalue (4.14), then the function U(s)can be chosen so that the auxiliary map s→ V(s)defined by (4.15) V(s):=U(s)−u0−sΦ,|s|<ε, satisfy ΩV(s)Ψ = 0 and V(s)=O(s2),ass→0. Also, for this 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, for each s∈(−ε, ε), (4.17) sign dΛ ds (s)=signσ−∆−Λ(s) mU(s)1/m−1;Ω . Summarizing, around any linearly asymptotically stable positive solution the set of solutions of (1.5) consists of a smooth curve of linearly asymptotically stable solutions, while around any linearly neutrally stable positive solution the set of solutions consists of a second order sub-critical turning point whose upper curve is filled in by linearly unstable positive solutions, whereas its lower curve is filled in by linearly asymptotically stable positive solutions. For a more detailed discussion we send to the interested reader to [15] and [16], where the linear diffusion case was treated. Proof of Theorem 4.5. Part (a) is an easy consequence from the implicit function theorem applied to the operator Kdefined in Section 4.2. As any nontrivial solution pair (λ, u) must have the second component, u, in the interior of the cone of positive functions of C0(Ω) and we are assuming that Ω+⊂Ω, the map u→K(λ, u)isanalyticforeachλ>0. Thus, the implicit function theorem provides us with an analytic solution curve. The existence and the uniqueness of the curve (Λ(s),U(s)) in Part (b), as well as (4.17), have been already shown in [2, Proposition 20.8]. Actually, they can be obtained by applying the implicit function theorem to a certain operator related to Kthrough a Lyapunov–Schmidt decomposition parallel to span[Φ]. It should 294 J. L´ opez-G´ omez — A. Su´ arez be noted that, thanks to (4.14), Λ(0) = 0, where stands for differentiation with respect to the pseudo-length of arc of curve s. Consequently, the proof will be completed if we show that λ2=Λ (0)/2 satisfies (4.16). Indeed, for each s∈(−ε, ε)wehavethat (4.18) −∆[u0+sΦ+V(s)] = [λ0+s2λ2+O(s3)][u0+sΦ+V(s)]1/m, and, hence, differentiating (4.18) twice with respect s, particularizing the resulting expression at s= 0 and rearranging terms gives (4.19) −∆−λ0 mu1/m−1 0V(0) = 2λ2u1/m 0+λ0 m1 m−1u1/m−2 0Φ2. It should be noted that the second term in the right hand side of (4.19) makes sense since u−2 0Φ2∈C(Ω). Now, multiplying (4.19) by Φ, integrating in Ω and applying the formula of integration by parts gives λ2=λ0 2ΩΦ3u1/m−2 0 1 m1−1 mΩ u1/m 0Φ. Thus, to conclude the proof, it remains to show that (4.20) ΩΦ3u1/m−2 0 1 m1−1 m<0. As in [15] and [16], this inequality will be obtained from a celebrated variational identity attributed to M. Picone [20] (cf. e.g. [9, Section 4] and [18, Lemma 4.1]). For any u,v∈C 1 0(Ω) twice differentiable a.e. in Ω and such that v/u ∈ C(Ω)∩C1(Ω), and every Υ ∈C 1([0,∞); R), the following identity, usually referred to as Picone’s identity, holds (4.21) Ω Υv u(−v∆u+u∆v)=−Ω Υv uu2∇v u 2 . Choosing Υ(t)=t2,v=Φ,u=u0, identity (4.21) gives (4.22) ΩΦ3u1/m−2 01−1 m=ΩΦ u02 (−Φ∆u0+u0∆Φ)<0, since Φ cannot be a multiple of u0. Clearly, (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−xfor each x∈R. This shows (4.20) and concludes the proof of the theorem.  As an immediate consequence from Theorem 4.5, the following result holds. Combining Fast, Linear and Slow Diffusion 295 Corollary 4.6. Let (λ0,u 0)be a positive solution of (1.5) satisfying (4.14). Then, there exists ε>0such that for each λ∈[λ0−ε, λ0),(1.5) has, at least, two positive solutions; one of them linearly asymptotically stable and the other linearly unstable. Moreover, there exists a neighbourhood Nof (λ0,u 0)in R× C0(Ω) such that (1.5) cannot admit a positive solution in Nif λ>λ 0. 4.5. Local structure of Cat (λ, u)=(0,0).The main result of this section reads as follows. Proposition 4.7. There exist ε>0and β>0such that, for each λ∈(0,ε], the minimal positive solution θλis the unique positive solution of (1.5) in Bβ. In particular, C∩[(0,ε]×Bβ]={(λ, θλ):0<λ≤ε}. Actually, thanks to Corollary 4.6,foreachλ∈(0,ε], the following holds σ−∆−λ mθ1/m−1 λ;Ω >0 and, therefore, thanks to Theorem 4.5(a),C∩[(0,ε]×Bβ]is a compact arc of analytic curve. Proof. Thanks to Proposition 4.2 and Theorem 4.3, there exists R>0 such that (1.5) has a positive solution, at least, for each λ∈(0,R], because PλCis a connected interval of (0,∞). Actually, due to Proposition 4.4, (1.5) possesses a minimal solution, θλ,foreachλ∈(0,R]. Thus, θλis well defined for any sufficiently small λ>0. Suppose (1.5) possesses, for some λ∈(0,R], a further solution 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, therefore, thanks to the strong maximum principle, (4.23) σ−∆−λχΩ1−λ m+ χΩ+θ1/m+−1 λ−λ m− χΩ−u1/m−−1 λ;Ω ≤0. The proof of the proposition will follow from (4.23), arguing by contradiction. Suppose there exists a sequence (λn,u λn), n≥1, of positive solutions of (1.5) such that lim n→∞(λn,u λn)=(0,0),u λn>θ λn>0,n≥1. 296 J. L´ opez-G´ omez — A. Su´ arez Then, thanks to (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. Moreover, thanks to the estimate (4.7), we have that θλn≥λm+/(m+−1) nv1,n≥1, and, hence, −λn m+ χΩ+θ1/m+−1 λn≥− 1 m+ χΩ+v1/m+−1 1,n≥1. Thus, thanks to (4.24) and (4.25), passing to the limit as n→∞gives σ−∆−1 m+ χΩ+v1/m+−1 1;Ω ≤0, which is impossible, since v1is a non-degenerate solution of (4.6) with λ=1. This contradiction concludes the proof of the proposition.  4.6. The component Cis unbounded. The main result of this section is the following. Proposition 4.8. The component Cis unbounded in R×C 0(Ω). Proof. We will argue by contradiction. Suppose Cis bounded. Then, the extended component C0:= C∪{(0,0)} is bounded in X:= R×C0(Ω), and, hence, it is compact, since it consists of fixed points of the compact operator Kdefined in Section 4.2. Thus, since K−1(0) ∩({0}×C 0(Ω)) = {(0,0)}, it is apparent, from Proposition 4.7, that there exists η∈(0,ε] such that (4.26) C0∩([0,η]×C 0(Ω)) = {(λ, θλ):0≤λ≤η}. Subsequently, we use the notations introduced in the statement of Proposition 4.7. Set δ:= β/2 and consider the open neighborhood of C0defined by U:= C0+[(−η/2,η/2) ×Bδ], as well as the set of non-trivial zeroes of K S:= {(λ, u)∈X:K(λ, u)=0,u=0}∪{(0,0)}. Combining Fast, Linear and Slow Diffusion 297 Subsequently, a bounded open set O⊂Xis said to be an open isolating neighborhood of C0in Xif C0⊂Oand (4.27) ∂O∩S=∅. If ∂U ∩S=∅,thenUprovides us with an open isolating neighborhood of the component C0, but, in general, ∂U ∩S=∅. When this is the case, Whyburn’s Lemma [24] uses the fact that C0is a maximal compact and connected subset of Sto show the existence of an open isolating neighbourhood Oof C0such that C0⊂O⊂U (cf. e.g. the proof of [19, Theorem 6.3.1]). Now, for each λ>0weset Oλ:= {u∈X:(λ, u)∈O}. By construction, Oη/3∩S={θη/3}. Thus, combining Leray–Schauder’s formula with Proposition 4.7 gives Deg(K(η/3,·),Oη/3) = Ind(K(η/3,·),θ η/3)=1 and, hence, by homotopy invariance, Deg(K(λ, ·),Oλ) = 1 for all λ>0. On the other hand, for sufficiently large λwe have that Oλ=∅and, hence, Deg(K(λ, ·),Oλ) = 0. This contradiction concludes the proof.  4.7. Proof of Theorem 4.1. Suppose there exist  λ>0andu  λ=θ  λsuch that ( λ, u  λ) is linearly stable (either neutrally stable, or asymptotically stable). Then, thanks to Proposition 4.4 and Theorem 4.5 (cf. Corollary 4.6), by global continuation to the left of  λ, (1.5) must admit two linearly asymptotically stable solutions for each λ∈(0, λ). As the solutions in each of the corresponding curves are increasing with λ, thanks to Proposition 4.7, (1.5) must admit a positive solution for λ= 0. This is impossible. Therefore, for each λ>0, θλis the unique linearly stable positive solution of (1.5) if it admits a solution. This shows (f). It should be noted that, thanks to Proposition 4.2, λ= 0 is the unique bifurcation value to positive solutions from u=0. Let λ∗be the maximal λ>0 satisfying the following condition (4.28) σ−∆−λ mθ1/m−1 λ;Ω >0,λ∈(0,λ ∗). Thanks to Propositions 4.2, Proposition 4.7, λ∗is well defined. Moreover, since Cis the maximal connected set such that (0,0) ∈C, (4.29) γ:= {(λ, θλ):λ∈(0,λ ∗)}⊂C, because γis connected. 298 J. L´ opez-G´ omez — A. Su´ arez Either γis bounded in R×C0(Ω), or it is unbounded. Suppose γis bounded. Then, uλ∗:= lim λ↑λ∗θλ provides us with a solution of (1.5) for λ=λ∗. Moreover, by the continuous dependence of the principal eigenvalue with respect to the potential, (4.28) implies σ−∆−λ∗ mu1/m−1 λ∗;Ω =0, because of the maximality of λ∗.Asθλ∗is the unique linearly stable solution, necessarily uλ∗=θλ∗. Actually, thanks to Corrollary 4.6, around (λ∗,θ λ∗), Cconsists of a second order sub-critical turning point. In particular, there exists λω∈[0,λ ∗) such that C possesses two solutions, at least, for each λ∈(λω,λ ∗); this shows the first claim of Part (e). Note that there exists an open set Osuch that: (1) {(λ, θλ):λ∈(0,λ ∗]}⊂O. (2) Any solution of (1.5) in Olies in C. (3) Any positive solution of (1.5) in ∂Ois linearly unstable. Clearly, (0,λ ∗]⊂Λ:=PλC. We claim that Λ = (0,λ ∗]. Indeed, suppose there exists  λ>λ ∗such that  λ∈Λ. Then, by global continuation from ( λ, θ  λ)totheleftof λone can construct a linearly stable positive solution of (1.5), outside O,e.g.forλ=λ∗.This contradicts the uniqueness of the stable solution, and, therefore, Λ=(0,λ ∗]. To complete the proof of the theorem when γis bounded it remains to show that Cpossesses two positive solutions for each λ∈(0,λ ∗)ifeitherN∈{1,2}, or N≥3andm−>(N−2)/(N+ 2). It suffices to show that, under these conditions, the component Cis bounded in [ε, λ∗]×C 0(Ω) for any ε∈(0,λ ∗). Pick one of those ε’s. Then, the blowing-up argument of B. Gidas and J. Spr¨uck ([14]) carries over mutatis mutandis to show the existence of a positive constant M>0 such that uλC(Ω−)≤M for any positive solution (λ, uλ) of (1.5) with λ∈[ε, λ∗]. Thus, uλ|Ω1∪Ω+is a subsolution of (4.30)      −∆u=λu1/min Ω1∪Ω+, u=0 on∂Ω, u=Mon ∂Ω−. Combining Fast, Linear and Slow Diffusion 299 Now, we have to distinguish two different cases. Assume Ω1=∅. Then (4.30) possesses a unique positive solution for each λ>0, say vλ, and, as an easy consequence from the strong maximum principle, uλ|Ω+≤vλin Ω+, for each λ∈[ε, λ∗], which provides us with the desired a priori bounds. If Ω1=∅, then, thanks to Proposition 4.2, λ∗<λ + 0, and, similarly, uλ|Ω1∪Ω+is bounded above by the unique positive solution of (4.30). The existence and the uniqueness of the positive solution of (4.30) follows with the same argument used in [13] to treat the case of homogeneous Dirichlet boundary conditions. This concludes the proof of the theorem when γis bounded. Now, suppose γis unbounded (cf. 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