Some indefinite nonlinear eigenvalue problems

May 13, 2004 10:52 WSPC/Trim Size: 9in x 6in for Proceedings mawhinsuarez SOME INDEFINITE NONLINEAR EIGENVALUE PROBLEMS 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: suar[email protected] Dedicated to Prof. Jean Mawhin for his first 60 years of Nonlinear Analysis In this work we study the structure of the set of positive solutions of a nonlinear eigenvalue problem with a weight changing sign. Specifically, the reaction term arises from a population dynamic model. We use mainly bifurcation methods to obtain our results. 1. Introduction The aim of this work is to study some nonlinear indefinite eigenvalue problems of the form ½−∆u=λm(x)f(u) in Ω, u= 0 on ∂Ω, (1) where Ω ⊂IRNis a bounded domain with a regular boundary ∂Ω, m∈C(Ω) changes sign, fis a regular function and λplays the role of real parameter. We focus our attention on the case f(0) = 0 and λ > 0; similar results can be obtained for negative values of λ. Depending of the shape of f, Eq. (1) models different situations: population dynamics, population genetics, combustion theory,... see [10]. In the linear case, i.e., f(u) = u, (1) is the eigenvalue problem ½−∆u=λm(x)uin Ω, u= 0 on ∂Ω. (2) ∗Supported by the Spanish Ministry of Science and Technology under grants BFM2000- 0797 and BFM2003-06446. 1 May 13, 2004 10:52 WSPC/Trim Size: 9in x 6in for Proceedings mawhinsuarez 2 It is well known (see for instance [19] and [23]) that there exist two values of λ,λ−(m)<0< λ+(m), called principal eigenvalues because they have associated positive eigenfunctions. In the present work, given q∈L∞(Ω) we denote by σΩ 1[−∆ + q] (we delete the superscript Ω when no confusion arises) the principal eigenvalue of the problem −∆u+q(x)u=λu in Ω, u= 0 on ∂Ω. When in (1) the weight does not appear, i.e., m≡1, the nonlinear problem ½−∆u=λf(u) in Ω, u= 0 on ∂Ω, (3) has been extensively studied. Classical references are [2] and [21], but many others can be given where, as well as existence results, uniqueness ones are shown: [4], [14], [26], [20], [22] and references therein. Much less is known for problem (1). In [19], assuming for example that f0(0) >0, the authors showed that there exists an unbounded continuum of positive solutions bifurcating from the trivial solution at λ=λ+(m)/f0(0). In [8] the authors assumed that f:I7→ IR+,I⊂IR, and f00 <0 and showed that every positive solution of (1) is stable. If, moreover, I= [0,1], f(1) = 0 and f0(0) >0 they proved that there exists a positive solution if, and only if, λ>λ+(m)/f0(0), and in this case the solution is unique. Similar result was shown in [13], although the authors’ motivation was to study the problem in the whole space. Very recently, in [9] the authors analyze the particular cases f(u) = gi(u), i= 1,2 with g1(u) = u−u2, g2(u) = u+u2.(4) Observe that the result of [8] can only be applied to g1. In [9], without the assumption that ftakes only values in [0,1], the main result of [8] was improved showing (by variational method) that, assuming some restriction in the space dimension, there exists positive solution if λ∈(0, λ+(m)). For the case, f=g2, they also proved the existence of positive solution for λ∈(0, λ+(m)) and that there does not exist positive solution at λ= λ+(m). In [16] these results have been again completed. We prove for f=g1that there exist at least two positive solutions in λ∈(λ+(m),∞), one of them linearly asymptotically stable and that for f=g2there exists positive solution if, and only if, λ∈(0, λ+(m)). In this work, we are going to analyze the following nonlinearities f1(u) = u−u2−Ku 1 + u, f2(u) = u+u2−Ku 1 + u,(5) May 13, 2004 10:52 WSPC/Trim Size: 9in x 6in for Proceedings mawhinsuarez 3 where K∈IR. Observe that the functions in (4) are included in (5). These last nonlinearities arise in population dynamics. Indeed, when K= 0, f1 is the classical logistic reaction term and for K6= 0 the predation one Ku/(1 + u) is called the Holling-Tanner term, see for example [7] for an ecological interpretation. In order to state our main results we need some notations. Specifically, assume that M±:= {x∈Ω : m±>0} are open and regular sets, where m±represent the positive and negative part of mrespectively; and suppose that m±(x)≈[dist(x, ∂M±)]γ±for x close to ∂M±and some γ±≥0. The following condition will provide us with a priori bounds of the solutions 2<min ½N+1+γ± N−1,N+ 2 N−2¾.(6) Finally, we define for K6= 1 the values λ+:= λ+(m) 1−Kλ−:= λ−(m) 1−K, and Π : IR ×C(Ω) 7→ IR the projection map onto IR, i.e. Π(µ, u) = µ. The main results are: Theorem 1.1. Assume that K6= 1 and (6). (1) There exists an unbounded continuum Cof positive solutions of (1) bifurcating from the trivial solution at λ=λ+if K < 1and λ=λ− if K > 1. (2) The bifurcation is supercritical for f=f1and for f=f2and K < −1or K > 1and subcritical for f=f2and K∈[−1,1). (3) If f=f1and K < 1(resp. f=f2and K > 1), then Π(C) = (λ+,∞)(resp. (λ−,∞)). Moreover, if (λ, uλ)∈ C, then uλis linearly asymptotically and such that uλ≤√1−K(resp. √K−1). Furthermore, there exists another positive solution vλfor all λ > 0. (4) If f=f1and K > 1(resp. f=f2and K < −1) then Π(C) = (0, λ∗]for λ∗> λ−(resp. λ+). Moreover, there exist λ0and λ∗ with λ0< λ∗such that for λ≥λ∗the problem (1) does not admit positive solutions and it possesses at least two positive solutions for λ∈(λ−, λ0)(resp. (λ+, λ0)). (5) If f=f2and K∈[−1,1) there exists positive solution for λ∈ (0, λ+)and (1) does not admit positive solutions for λ≥λ∗. May 13, 2004 10:52 WSPC/Trim Size: 9in x 6in for Proceedings mawhinsuarez 4 (6) In any case, if there exists a solution vλfor λ > 0, then limλ→0kvλk∞= +∞. Theorem 1.2. Assume K= 1 and (6). Then there exists at least a solution uλfor λ > 0and limλ→0kuλk∞= +∞. Remark 1.1. (1) The existence of Cis true without assuming (6). In the cases (4) and (5) of Theorem 1.1, Ccould “go to infinity” in a value λ0. (2) In the particular case f=f2and K= 0, in [16] it was proved using a Picone inequality that (1) possesses a positive solution if, and only if, λ∈(0, λ+). In Figs. 1 and 2 we have summarized these results (the case f=f2and K= 1 is similar to f=f1and K= 1). λ || . || || . || || . || a) b) c) +λ−λ λ λ Figure 1. Bifurcation diagrams for f=f1: a) K < 1; b) K= 1; c) K > 1. The rest of the paper is organized as follows: Secs. 2 and 3 are devoted to prove Theorems 1.1 and 1.2, respectively. 2. Proof of Theorem 1.1 2.1. Local bifurcation In this subsection we show the direction of bifurcation from the trivial solution for both cases f1and f2. For that, we write the nonlinearity of the following manner f(u) = u∓u2−Ku 1 + u=u(1 −K) + u2(K 1 + u∓1). May 13, 2004 10:52 WSPC/Trim Size: 9in x 6in for Proceedings mawhinsuarez 5 λ λ λ λλλ a) b) c) + + − || . || || . || || . || Figure 2. Bifurcation diagrams for f=f2: a) K < −1; b) K∈[−1,1); c) K > 1. It is clear that to study (1) is equivalent to find zeros of L(λ)u−N(λ, u) = 0, where L(λ)u:= u−λ(−∆)−1m(x)(1 −K)u, N(λ, u) := λ(−∆)−1m(x)u2(K 1 + u∓1). We can prove that N(L(λ+)) = Span < ϕ+>and d dλL(λ+)ϕ+/∈R(L(λ+)) (7) where, given any linear continuous operator L,N[L] and R[L] stand for the null space and the range of L, respectively, and −∆ϕ+=λ+(m)m(x)ϕ+in Ω, ϕ+= 0 on ∂Ω.(8) The first equality of (7) is trivial, for the second expression we need the following result. Lemma 2.1. For any p≥2we have that ZΩ m(x)(ϕ+)p>0. Proof: Multiplying (8) by (ϕ+)p−1we get λ+(m)ZΩ m(x)(ϕ+)p=ZΩ (−∆ϕ+)(ϕ+)p−1= (p−1) ZΩ|∇ϕ+|2(ϕ+)p−2>0. ¦ Now, we show (7). Assume that there exists usuch that d dλL(λ+)ϕ+=−(−∆)−1m(x)(1 −K)ϕ+=u−(−∆)−1m(x)λ+(1 −K)u, May 13, 2004 10:52 WSPC/Trim Size: 9in x 6in for Proceedings mawhinsuarez 6 then (−∆−λ+(m)m(x))u=−(1 −K)m(x)ϕ+, and so, multiplying by ϕ+we get a contradiction using Lemma 2.1. Now, we can apply the Crandall-Rabinowitz Theorem [15] and conclude that there exists δ > 0 such that in a neighborhood of (λ+,0) the nontrivial solutions of (1) are of the form u(s) = sϕ++s2ϕ2+s3ϕ3+o(s3), λ(s) = λ++sλ1+s2λ2+o(s2). Introducing these terms in (1), using (8) and a Taylor expression of the function 1/(1 + u(s)), we get (−∆−λ+(m)m(x))ϕ2=λ+m(x)(ϕ+)2(K∓1) + λ1m(x)(1 −K)ϕ+, and so, λ1=−λ+(K∓1) 1−K ZΩ m(x)(ϕ+)3 ZΩ m(x)(ϕ+)2 .(9) Observe that in the particular case f=f2and K=−1, λ1= 0, and so we have to calculate λ2. It can be proved that λ2=−λ+ 2 ZΩ m(x)(ϕ+)4 ZΩ m(x)(ϕ+)2 .(10) From (9) and (10), we conclude the paragraph (2) of Theorem 1.1. Analogously it can be treated the case λ−. 2.2. Non-existence results Lemma 2.2. Assume f=f1and K > 1or f=f2and K < 1. Then, there exists λ∗>0such that for λ≥λ∗(1) does not have positive solutions. Proof: Assume f=f1and K > 1. Firstly observe that h(x) := x(K 1 + x−1) ≤(√K−1)2,∀x≥0.(11) May 13, 2004 10:52 WSPC/Trim Size: 9in x 6in for Proceedings mawhinsuarez 7 Let ube a positive solution of (1). Then, using the monotony of the principal eigenvalue with respect to the domain and (11) we get 0 = σ1[−∆−λm(x)(1 −K)−λm(x)u(K 1 + u−1)] < < σM− 1[−∆−λm(x)((1 −K)+(√K−1)2)] = =σM− 1[−∆−λm(x)2(1 −√K)], which is an absurdum for λlarge. Now, assume f=f2and K < 1. In this case, x(K 1 + x+ 1) ≥0,if K≥ −1, ∀x≥0, x(K 1 + x+ 1) ≥ −(√−K−1)2,if K < −1, ∀x≥0. So, if −1≤K < 1 we have 0 = σ1[−∆−λm(x)(1−K)−λm(x)u(K 1 + u+1)] < σM+ 1[−∆−λm(x)(1−K)]; on the other hand, for K < −1, 0 = σ1[−∆−λm(x)(1−K)−λm(x)u(K 1 + u+1)] < σM+ 1[−∆−λm(x)2√−K], in both cases a contradiction for large λ.¦ 2.3. Multiplicity results To obtain multiplicity results, we include (1) in the more general equation ½−∆u=µm(x)(1 −K)u+λm(x)g(u) in Ω, u= 0 on ∂Ω, (12) where gsatisfies (Hg)g(0) = g0(0) = 0, g00(u)<0,lim s→+∞ g(s) s2=β < 0. Problem (12) has attracted a great deal of attention during last years (see for example [1], [3], [5], [6], [18] and [24]) when m≡1 in the first term on the right-hand side of (12) and in [11], [12] and [13] with the right-hand side of the form µh(x)u+g(x)upand restrictive conditions on hand gwhich are not satisfied in our case. In [16] was proved (see Fig. 3): Proposition 2.1. Assume that gsatisfies (Hg),(6),K6= 1 and fix λ > 0. Denote by Λ+:= λ+(m(x)(1 −K)),Λ−:= λ−(m(x)(1 −K)). May 13, 2004 10:52 WSPC/Trim Size: 9in x 6in for Proceedings mawhinsuarez 8 Then, (12) possesses a positive solution if µ > Λ−. Moreover, from the trivial solution u= 0 emanate two unbounded in IR×C(Ω) continua of positive solutions C+:= {(µ, uµ)}and C−:= {(µ, wµ)}at µ= Λ+and µ= Λ−, respectively. Both continua bifurcate to the right and Π(C−)⊃(Λ−,+∞), Π(C+) = (Λ+,+∞). Finally, for µ > Λ+,uµis linearly asymptotically stable and uµ6=wµ. Remark 2.1. Observe that for K < 1, Λ+=λ+and Λ−=λ−, and for K > 1, Λ+=λ−and Λ−=λ+. Indeed, for example for K > 1, it follows that Λ+=λ+(m(x)(1 −K)) = λ+(−m(x)) K−1=−λ−(m(x)) K−1=λ−(m(x)) 1−K=λ−. µλ λ C C || . || + + − − Figure 3. Bifurcation diagram for (12) and K < 1. 2.4. Proof of Theorem 1.1: Before proving the result, we generalize a well-known result for m≡1. The proof is coming from [8]. Lemma 2.3. Assume that fis a regular function and f(0) = 0. Let u0be a positive solution of (1) such that f(u0)>0, it holds: May 13, 2004 10:52 WSPC/Trim Size: 9in x 6in for Proceedings mawhinsuarez 9 (1) If f00(u0)<0, then u0is linearly asymptotically stable. (2) If f00(u0)>0, then u0is unstable. Proof: We have to calculate the sign of the eigenvalue σ1[−∆− λm(x)f0(u0)]. Take ψ:= f(u0)>0, then (−∆−λm(x)f0(u0))ψ=−f00(u0)|∇u0|2. So, if fis concave (resp. convex) the function ψis a supersolution (resp. subsolution) of −∆−λm(x)f0(u0), and then (see [23]) σ1[−∆− λm(x)f0(u0)] >0 (resp. <0). ¦ The following result is proved in Theorem 3.4 of [3] and provides us with a priori bounds for the positive solutions of (1). Lemma 2.4. Assume (6). If (λ, u)is a positive solution of (1) and λ∈J, where Jis a compact subset such that J⊂(0,∞), then there exists a positive constant C(independent from λ) such that kuk∞≤C. Finally, the following result is proved in [17]. Lemma 2.5. Assume that Σ⊂I×C2 0(Ω),I⊂IR an interval, is a connected set of positive solutions of (1). Consider u:I7→ C2 0(Ω) a continuous map of supersolution for each λ∈I, but not a solution. If u0< u(λ0)for some (λ0, u0)∈Σ, then u < u(λ)for all (λ, u)∈Σ. We are ready to prove the result. By subsec. 2.1 we know that there exists bifurcation from the trivial solution at λ=λ+or λ=λ−when K < 1 or K > 1, respectively. Moreover, we can apply Theorem 6.4.3 of [25], and conclude that from λ=λ+or λ=λ−bifurcates an unbounded continuum Cof positive solutions of (1). We would like to remark that the a detailed proof that Cis unbounded and it does not satisfy the other alternatives of the above mentioned result will be presented elsewhere. Now assume f=f1and K < 1. It is clear that u:= √1−K is a supersolution of (1). So, we can apply Lemma 2.5 (taking λ0=λ+) and conclude that for all (λ, uλ)∈ C, we have that uλ<√1−K. (13) Moreover, f1(uλ)>0 and f00 1(uλ)<0, and so by Lemma 2.3 we get that uλis linearly asymptotically stable.