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

Houston Journal of Mathematics c 2003 University of Houston Volume 29, No. 3, 2003 POSITIVE SOLUTIONS FOR THE DEGENERATE LOGISTIC INDEFINITE SUPERLINEAR PROBLEM: THE SLOW DIFFUSION CASE MANUEL DELGADO AND ANTONIO SU´ AREZ Communicated by Ha¨ım Brezis Abstract. In this work we study the existence, stability and multiplicity of the positive steady-states solutions of the degenerate logistic indefinite superlinear problem. By an adequate change of variable, the problem is transformed into an elliptic equation with concave and indefinite convex nonlinearities. We use singular spectral theory, the Leray-Schauder degree, bifurcation and monotony methods to obtain the existence results, and fixed point index in cones and a Picone identity to show the multiplicity results and the existence of a unique positive solution linearly asymptotically stable. 1. Introduction In this work we analyze the existence, stability and multiplicity of nonnegative and non-trivial solutions of the degenerate logistic indefinite superlinear model (1) ºLwm=λw +a(x)w2in Ω, w= 0 on ∂Ω, where Ω is a bounded and regular domain of IRN,N≥1; m > 1; λ∈IR that it will be considered parameter, a∈Cα(Ω), α∈(0,1), changes sign and Lis a 2000 Mathematics Subject Classification. 35B32; 35J25; 35K57; 92D25. Key words and phrases. Degenerate logistic indefinite equation, singular eigenvalue problems, indefinite superlinear problems, multiplicity results. The second author is grateful to Professors D. Arcoya and J. L´opez G´omez for their helpful comments. The authors thank to CICYT and MCYT of Spain for research support under grants MAR98-0486 and BFM2000-0797 respectively. 801 802 M. DELGADO AND A. SU ´ AREZ second order uniformly elliptic operator of the form (2) Lu:= − N X i,j=1 Di(aijDju) + N X i=1 bi(x)Diu, with aij =aji ∈C1(Ω) and bi∈C1(Ω). We define 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 that A±are open and sufficiently smooth, and that a±are bounded away from zero on compact subsets of A±. Equation (1) can be regarded as a model of a steady-state single species inhabiting in Ω, so w(x) stands for the population density. The parameter λrepresents the growth rate of the species and a(x) describes the limiting effects of crowding in the species in A−and the intraspecific cooperation in A+. Observe that in A0 the population is free from crowding and symbiosis effects. Finally, Lmeasures the diffusivity and the external transport effects of the species. The term m > 1 was introduced in [18] by describing the dynamics of biological population whose mobility depends upon their density. In this context, m > 1 means that the diffusion, the rate of movement of the species from high density regions to low density ones, is slower than in the linear case (m= 1), which seems give more realistic models, see [18]. The change of variable u:= wmtransforms (1) into (3) ºLu=λuq+a(x)upin Ω, u= 0 on ∂Ω, with q= 1/m and p= 2/m. Along this work we suppose (H) 0 < q < 1< p so, we are assuming that 1 <m<2, we call this case the slow diffusion. When q= 1, that is m= 1, (3) has been studied extensively in the last years, see for example [2], [3], [4], [6], [11], [12], [13], [17], [21], [22] and [24]. Roughly speaking, in these works it was proved that from the trivial solution u= 0 at λ=σ1[L] bifurcates an unbounded continuum of positive solutions supercritically (resp. subcritically) if D:= ZΩ aϕp 1ϕ∗ 1<0 (resp. >0,) where σ1[L] = σ1[L∗], ϕ1and ϕ∗ 1stand for the principal eigenvalue and the principal eigenfunction of Land its adjoint L∗in Ω under homogeneous Dirichlet SLOW DIFFUSION AND SUPERLINEAR PROBLEM 803 boundary conditions. Moreover, in [6] and [11], assuming some restrictions on p and on the decay of a+near ∂A+, the authors obtained a priori bounds for λin compact interval of IR. So, if for example D < 0, in [6] was shown the existence of positive solution for λ∈(∞, λ∗] for some λ∗> σ1[L] and the existence of, at least, two positive solutions for λ∈(σ1[L], λ∗). Recently, the existence of a unique solution linearly asymptotically stable in (σ1[L], λ∗) and multiplicity results in such interval have been showed in [17]. The results when q < 1 are completely different. Indeed, in the specific case L=−∆ and a(x) = 1, Ambrosetti, Brezis and Cerami, in the pioneer work [7], proved the existence of, at least, two positive solutions of (3) in the interval (0, λ∗) if p < (N+ 2)/(N−2) and where λ∗is the supremum of the set Λ := {λ > 0 : (3) has a positive solution.} To obtain this result, the authors used the sub-supersolution and variational methods. More recently, in [10] the authors have proved that from the trivial solutions u= 0 emanates an unbounded continuum of positive solutions at λ= 0. Unlike the case q= 1, this continuum emanates supercritically independent of the sign of D, in fact, the bifurcation direction only depends on value of p. This has been proved in [10] even when the operator Lis quasilinear. Then, if a(x)≥a0>0 and p < (N+ 2)/(N−2) they proved the existence of nonnegative solution in (−∞, λ∗) and of, at least, two positive solutions in (0, λ∗). See also similar results obtained in [8] when the operator is the p-Laplacian and [1] when the boundary conditions are Neumann. In all these works, adoes not change sign. When achanges sign, recently in [23] the authors have proved the existence of a weak nonnegative solution if λ≤0 making use of a direct variational approach, see [26] for the case of Neumann boundary conditions. In this work, we improve and generalize the above results. We consider a nonselfadjoint operator L, so it is well-known that the variational methods do not work, and a function achanging sign. As in [10], we prove that an unbounded continuum of nonnegative solutions emanates from the trivial solution u= 0 at λ= 0 supercritically. Moreover, we prove the existence of a minimal solution for λ∈(0, λ∗) and the existence of λ≥λ∗such that for λ > λ, (3) does not admit positive solution. Using the results of Section 4 in [6], and assuming some restrictions on pand a+(see Theorem 5.1), we obtain a priori bounds for the positive solutions of (3) for compact intervals of λ. Finally, under these restrictions, we obtain a unique positive solution linearly asymptotically stable in (0, λ∗), the existence of, at least, two positive solution in (0, λ∗) and other multiplicity result in this interval (see Theorem 6.9). These results were strongly motivated by [17]. 804 M. DELGADO AND A. SU ´ AREZ In order to obtain this result, we have used the Picone identity, so we assume that Lis selfadjoint. Finally, we would like to point out that the stability results are obtained by linearizing at a positive solution. Observe that, since q < 1, the linearized problem is a problem with a potential blowing up near ∂Ω. So, we need some spectral theory with singular potential. For that, we have included some results obtained in [16] and [19]. An outline of this paper is as follows: In Section 2 we have collected some spectral theory with singular potential. In Section 3 we study problem (3) in the case a+= 0. These results come from [15] and will be used in the next sections. In Section 4 we apply the Leray-Schauder degree and bifurcation theory to show the existence of an unbounded continuum of nonnegative solution emanating supercritically at λ= 0 from the trivial solution u= 0. In Section 5 we obtain a priori bounds of the positive solutions of (3) for compact intervals of λ. Finally, in Section 6 we obtain multiplicity and stability results. 2. Singular eigenvalue problem In this section we collect some results about the existence of principal eigenvalue for a singular linear eigenvalue problem of the form (4) º(L+M(x))u=σu in Ω, u= 0 on ∂Ω, where M∈C1(Ω) but it can blow-up near ∂Ω at a controlled way. The next result was proved in [19]. Theorem 2.1. Suppose M∈C1(Ω) and there exist two constants K > 0and ε > 0for which (5) |M(x)| ≤ K [dist(x, ∂Ω)]2−εx∈Ω. Then, there exists a unique value of σ, denoted by σΩ 1[L+M]and called principal eigenvalue of (4), for which (4) possesses a weak positive solution (in H1 0(Ω) ∩ L∞(Ω)), unique up to multiplicative constants, denoted by ϕΩ 1and called principal eigenfunction of (4). Moreover, by elliptic regularity, ϕΩ 1∈C1 0(Ω),ϕΩ 1(x)>0for each x∈Ωand ∂ϕΩ 1 ∂n (x)<0for each x∈∂Ω, where nstands for the outward unit normal to Ωat x. SLOW DIFFUSION AND SUPERLINEAR PROBLEM 805 Furthermore, σΩ 1[L+M]is increasing with respect to Mand decreasing with respect to Ω, and if σΩ 1[L+M]>0then u= 0 is the unique weak solution of º(L+M(x))u= 0 in Ω, u= 0 on ∂Ω. The following result was shown in [20] when M∈L∞(Ω), and in [16] when M satisfies (5). Definition 2.2. A function ϕ∈C2(Ω) ∩C1(Ω) is said a supersolution of L+M if (L+M)ϕ≥0in Ωand ϕ≥0on ∂Ω.If in addition, (L+M)ϕ > 0in Ωor ϕ > 0on ∂Ω, then it is said that ϕis a strict supersolution. Theorem 2.3. Assume that Msatisfies (5). Then: (1) σΩ 1[L+M]>0if, and only if, L+Madmits a positive strict supersolution. (2) If there exists ϕ∈C2(Ω) ∩C1(Ω) with ϕ > 0in Ωsuch that ϕ= 0 on ∂Ωand (L+M)ϕ < 0in Ω, then σΩ 1[L+M]<0. We do not write the superindex Ω, when no confusion arises. 3. The sublinear case: a+≡0. In this section we study the sublinear case, that is, when a+≡0. The following result characterizes the existence, uniqueness and linear stability in this case. Theorem 3.1. Assume a+≡0. Then, there exists a unique positive solution of (3) if, and only if, λ > 0. Moreover, if we denote it by θ[λ,a−], then (6) lim λ↓0kθ[λ,a−]k∞= 0. Furthermore, if λ > 0then θ[λ,a−]is linearly asymptotically stable, that is, (7) σ1[L+Mλ(x)] >0, where Mλ:= −λqθq−1 [λ,a−]−pa−θp−1 [λ,a−] Proof. Except (7), the result follows by a similar argument to Theorem 4.2 in [15] where the result was proved when L=−∆. We are going to show (7). Firstly, observe that for λ > 0, (3) satisfies the strong maximum principle, so there exists C > 0 such that Cdist(x, ∂Ω) ≤θ[λ,a−](x),for all x∈Ω, 806 M. DELGADO AND A. SU ´ AREZ and so, Mλsatisfies (5). On the other hand, by (H) and Theorem 2.1, it follows 0 = σ1[L − λθq−1 [λ,a−]−a−θp−1 [λ,a−]]< σ1[L+Mλ]. This completes the proof. £ The following result will be used in the next sections. Lemma 3.2. Assume a+≡0. Then, (8) lim λ↓0σ1[L+Mλ] = σ1[L − qzq−1]>0, where zis the unique positive solution of (9) ºLz=zqin Ω, z= 0 on ∂Ω. Proof. The existence of a unique positive solution for (9) follows by the subsupersolution method, see [15] for details. Moreover, again by the strong maximum principle, zq−1satisfies (5) and so it is well-defined σ1[L − zq−1]. By (H) and Theorem 2.1, we have (10) 0 = σ1[L − zq−1]< σ1[L − qzq−1]. In order to prove (8), by (6) it is sufficient to show that (11) ξλ:= λ1/(q−1)θ[λ,a−]→zas λ↓0. It is not hard to prove that ξλsatisfies Lξλ=ξq λ+a−λ(p−1)/(1−q)ξp λin Ω, ξλ= 0 on ∂Ω. By (H), it follows (11), and thanks to (10) we obtain the result. £ 4. Bifurcation from the trivial solution In this section we will show that a bifurcation from the trivial solution of (3) occurs at λ= 0. For that, we consider the Banach space X:= C0(Ω), denote Bρ:= {u∈X:kuk∞< ρ}and take K > 0 sufficiently large. We extend the function f(λ, x, s) := λsq+a(x)sp+Ks by taking f(λ, x, s) := 0 if s < 0. Note that fcan take negative values. Finally, we define the map Kλ:X7→ X;Kλ(u) := u−(L+K)−1(f(λ, x, u)) where (L+K)−1is the inverse of the operator L+Kunder homogeneous Dirichlet boundary conditions, which is well-defined since σ1[L+K]>0. Indeed, since positive constants are supersolutions of L, then σ1[L]>0, SLOW DIFFUSION AND SUPERLINEAR PROBLEM 807 whence it follows that σ1[L+K]>0. Now, we can prove that uis a nonnegative solution of (3) if, and only if, uis a zero of the map Kλ. It is clear that every nonnegative solution is a zero of Kλ; conversely, if uis a zero of Kλthen, multiplying (3) by u−, we obtain (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 uniformly elliptic operator, it follows that u−≡0. Observe that a nonnegative solution u∈Xof (3), it belongs to C1+ν(Ω) ∩C1 0(Ω) for ν:= min{α, q}. The main result of this section is: Theorem 4.1. The value λ= 0 is the only bifurcation point from the trivial solutions for (3). Moreover, there exists a continuum C0of nonnegative solutions of (3) unbounded in IR ×Xemanating from (0,0). In addition, C0bifurcates to the right of λ= 0, i.e., it is supercritical. In order to prove this result we use the Leray-Schauder degree of Kλon Bρ with respect to zero, denoted by deg(Kλ, Bρ), and the index of the isolated zero uof Kλ, denoted by i(Kλ, u). In the following results, we use homotopies which were used in [10], see also [9]. Lemma 4.2. If λ < 0, then i(Kλ,0) = 1. Proof. Fix λ < 0. Define the map H1: [0,1] ×X7→ X;H1(t, u) := (L+K)−1(tf(λ, x, u)). We claim that there exists δ > 0 such that u6=H1(t, u) for u∈Bδ,u6= 0 and t∈[0,1]. Indeed, suppose that there exist sequences un∈X\{0}with kunk∞→0 and tn∈[0,1] such that un=H1(tn, un). We know that un≥0. Since kunk∞→0 and λ < 0, there exists n0∈IN such that for n≥n0, it holds Lun≤0 in Ω, which is impossible. 808 M. DELGADO AND A. SU ´ AREZ Taking now ε∈(0, δ], the homotopy defined 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. If λ > 0, then i(Kλ,0) = 0. Proof. Fix λ > 0 and φ∈X, φ > 0. We define the map H2: [0,1] ×X7→ X;H2(t, u) := (L+K)−1(f(λ, x, u) + tφ). We will show that there exists δ > 0 such that u6=H2(t, u) for all u∈Bδ,u6= 0 and t∈[0,1]. Indeed, suppose the contrary: there exist sequences un∈X\ {0} with kunk∞→0 and tn∈[0,1] such that un=H2(tn, un). Since tnφ≥0, multiplying by u−, and by a similar argument to the used in (12), we obtain that un≥0. Moreover since λ > 0, by the strong maximum principle un>0. We fix M≥σ1[L]. Since kunk∞→0 and λ > 0, there exists n0∈IN such that for n≥n0we get Lun=λuq n+a(x)up n+tnφ > Mun+tnφ, and so, (L − M)un>0. So, unis a positive strict supersolution of L − M, and by Theorem 2.3, we get σ1[L − M]>0, and so M < σ1[L]. This is impossible. This proves that the homotopy defined by H2is admissible. Then, if we take ε∈(0, δ] we have i(Kλ,0) = deg(Kλ, Bε) = deg(I− H2(0,·), Bε) = deg(I− H2(1,·), Bε)=0. This last equality is true because the problem Lu=λuq+a(x)up+φhas no solution in Bεbecause we have shown that u6=H2(1, u) for all u∈Bδ,u6= 0. £ Proof of Theorem 4.1: The fact that λ= 0 is a bifurcation point follows by Lemma 4.2 and Lemma 4.3. Moreover, from Lemma 4.2, (3) does not have bifurcation points in (−∞,0). Assume that there exists a sequence of solutions (λn, un) such that λn→λ0>0 and kunk∞→0. We take M≥σ1[L], so there exists n0∈IN such that λnuq n+a(x)up n> Munfor all n≥n0. As in the proof of Lemma 4.3, we obtain that σ1[L − M]>0, a contradiction. SLOW DIFFUSION AND SUPERLINEAR PROBLEM 809 Now, even though our map Kλdoes not satisfy exactly the hypotheses of Theorem 1.3 in [25], the proof can be modified to obtain the result, see Theorem 3.1 in [1] and Theorem 4.4 in [10], and we can conclude the existence of a continuum of solutions of (3) such that meets (0,0) either infinity or (λ0,0) with λ06= 0. We can discard the last possibility by the above reasoning, and so the existence of an unbounded continuum of solutions of (3) follows. We are going to prove that the bifurcation is supercritical, for which plays an essential role that p > 1. Indeed, assume that there exists a sequence (λn, un) of solutions of (3) such that λn≤0 and un≥0, un6= 0 with λn→0 and kunk∞→0. Since σ1[L]>0, there exists a sufficiently small ε > 0 such that (13) σ1[L − ε]>0. For such ε > 0, there exists n0(ε)∈IN such that for n≥n0, we get Lun=λnuq n+a(x)up n≤a(x)up n< εun whence by (13) we obtain a contradiction. £ The next result shows that for λlarge, (3) has no solution. Proposition 4.4. There exists λ > 0such that for λ > λ,(3) has no solution. Proof. We fix δ > 0 sufficiently small and define the set Dδ:= {x∈A+:dist(x, ∂A+)> δ} 6=∅. Then, there exists a positive constant c+(δ)>0 such that a+(x)≥c+(δ)>0 in Dδ. Observe that Dδhas only finitely many connected components, say Dδ i,i= 1,...,r. Since λ > 0, the strong maximun principle assures that any nonnegative and nontrivial solution of (3) is in fact strictly positive. So, by Theorem 2.1 is welldefined σ1[L − λuq−1−a(x)up−1], and we have (14) 0 = σ1[L − λuq−1−a(x)up−1]< σDδ 1 1[L − λuq−1−a(x)up−1]. Let ϕDδ 1 1the principal eigenfunction associated with Lin Dδ 1. We claim that there exists λ > 0 such that for λ > λ,ϕDδ 1 1is a strict subsolution of L1:= L − λuq−1−a(x)up−1in Dδ 1, and so by Theorem 2.3 σDδ 1 1[L1]<0 816 M. DELGADO AND A. SU ´ AREZ Proposition 6.7. Assume Lselfadjoint (bi= 0 in (2)) and let (λ0, u0)be a positive solution of (3) with λ=λ0>0, such that σ1[L+Rλ0]=0.Then, λ2<0, where λ2is defined (18). Proof. By Lemma 6.6, for s∈J, we have 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, differentiating twice with respect s, taking account that (L+Rλ0)Φ0= 0, we obtain (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 the Fredholm alternative λ2=1 2 ZΩ Φ3 0uq−2 0(q(1 −q)λ0+p(1 −p)a(x)up−q 0) ZΩ uq 0Φ0 . Observe that, since u0and Φ0are strictly positive, there exist Ci>0, i= 1,2, such that Φ3 0uq−2 0≤C1dist(x, ∂Ω)3−2+q≤C1, and so λ2is well-defined. To prove that λ2<0, the basic tool is a Picone identity (see Section 4 in [12] and Lemma 4.1 in [21], for instance). Let u, v ∈C2(Ω) ∩C1 0(Ω) be such that v/u ∈C(Ω) ∩C1(Ω) and Υ : [0,∞)7→ IR of class C1. Then (20) ZΩ Υ(v u)(vLu−uLv) = −ZΩ Υ0(v u)u2 N X i,j=1 aijDi(v u)Dj(v u). We take Υ(t) = t2,v= Φ0and u=u0. Observe that v/u ∈C(Ω) ∩C1(Ω) by the strong maximum principle. Hence, by (20) and since ucannot be a multiple of v, we obtain 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 therefore λ2<0. £ As an easy consequence of Lemma 6.6, relation (19) and Proposition 6.7, we obtain: Corollary 6.8. Assume Lselfadjoint and let (λ0, u0)be a positive solution of (3) with λ=λ0>0, such that σ1[L+Rλ0]=0.Then, there exists ε > 0such that for each λ∈(λ0−ε, λ0),(3) has two positive solutions, one of them linearly asymptotically stable and the other one linearly unstable. Moreover, there exist a neighborhood Nof (λ0, u0)in IR ×Psuch that (3) does not have a positive solution in Nfor λ>λ0. We are ready to prove the main result of this section. Theorem 6.9. Assume Lselfadjoint and that the hypotheses of Theorem 5.1 are satisfied. Then, (1) Λ=(−∞, λ∗], (2) There exist, at least, two positive solution in (0, λ∗), (3) There exists a unique positive solution in (0, λ∗)linearly asymptotically stable, (4) If we assume that (3) has a finite number of non-degenerate positive solutions, say u1,...,ur, then r= 2kfor some k≥1, and exactly kamong them have index −1, and the other khave index 1. Proof. To show the first paragraph it remains to prove that there exists solution for λ=λ∗. Let (λn, un) a sequence of solutions with 0 < λn< λ∗and λn→λ∗. By Theorem 5.1 and a standard compactness argument, we obtain that un→u∗, with u∗solution of (3) for λ=λ∗. Moreover, u∗6= 0 because of λ= 0 is the unique bifurcation value from the trivial solution, hence u∗>0. We will show that the minimal solution uλis the unique linearly asymptotically stable. Indeed, we take λ1>0 sufficiently small such that σ1[L+Rλ1]>0, for (λ1, uλ1). This is possible by Lemma 6.3, Proposition 6.4 and Corollary 6.8. By continuation to the left of λ1, and thanks of Proposition 6.4 and Proposition 6.7, we obtain that uλis asymptotically stable for 0 < λ ≤λ1. 818 M. DELGADO AND A. SU ´ AREZ Now, we prolongate to the right of λ1to reach a value λ2≤λ∗where σ1[L+Rλ]> 0 for 0 <λ<λ2and σ1[L+Rλ2]=0. If λ2=λ∗, we have just proved the existence of a linearly asymptotically stable positive solution for λ∈(0, λ∗). So, assume λ2< λ∗and take λ3∈(λ2, λ∗) and consider (λ3, uλ3). In any case, σ1[L+Rλ3] = 0 or σ1[L+Rλ3]>0, by Corollary 6.8 we can take λ4∈(λ2, λ3] such that σ1[L+Rλ4]>0. We can prolongate to the left of λ4by a branch, say uλ, of linearly asymptotically stable positive solution, see Proposition 6.7. This branch can not degenerate in the branch uλdue to the uniqueness of positive solution around the minimal solution uλ. Neither, it can degenerate to 0 in λ= 0, because of Proposition 6.4. Hence, there exists a positive linearly asymptotically stable for λ= 0, say u0, which is impossible by Lemma 6.5. A similar argument can be used to show the uniqueness of positive linearly asymptotically stable solution. Moreover, Theorem 5.1 and Theorem 4.1 show the second paragraph. Now, we take Γ := [0, b] with b>λ∗. By Theorem 5.1, there exists a positive constant C(independent from λ) such that kuk∞≤Cfor all λ∈Γ. We take R:= C+ 1 and then for all λ∈Γ (21) iP(K, PR)=0. Indeed, we can consider the homotopy H: [0,1] ×P7→ P, H(t, u) := (L+K)−1((λ(1 −t) + tb)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 the only solution for λ>λ∗and by Lemma 4.3. Moreover, by paragraph 3 and the Leray-Schauder formula we have iP(K, uλ)=1. Without lost of generality we can suppose that u1=uλ. Now, for each nondegenerate solutions u2,...,urwe have iP(K, ui)=(−1)ni SLOW DIFFUSION AND SUPERLINEAR PROBLEM 819 where niis the sum of the algebraic multiplicities of all the eigenvalues greater than one of the linearized of Kat ui. Since u= 0 has index zero by Lemma 4.3 and (21), we obtain 0 = 0 + 1 + r X i=2 (−1)ni from where the result follows. £ Remark 6.10. Observe that we only have used that Lis selfadjoint in Proposition 6.7 in order to apply the Picone identity. So, paragraphs (1) and (2) of Theorem 6.9 are true if Lis a general operator as (2). References [1] S. 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Tarfulea, Positive solution of some nonlinear elliptic equation with Neumann boundary conditions, Proc. Japan Acad. Ser. A, 71 (1995), 161-163. Received April 10, 2001 Revised version received December 14, 2001 (M. Delgado) Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Fac. Matem´ aticas, C/ Tarfia s/n, C.P. 41012, Univ. Sevilla, Spain. E-mail address:[email protected] (A. Su´arez) Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Fac. Matem´ aticas, C/ Tarfia s/n, C.P. 41012, Univ. Sevilla, Spain. E-mail address:[email protected]