DOUBLE BIFURCATION DIAGRAMS AND FOUR POSITIVE SOLUTIONS OF NONLINEAR BOUNDARY VALUE PROBLEMS VIA TIME MAPS

. In this paper, we consider the existence and exactness of multiple positive solutions for the nonlinear boundary value problem 0 , where λ > 0 is a bifurcation parameter, f ( u ) > 0 for u > 0. We give com- plete descriptions of the structure of bifurcation curves and determine the existence and multiplicity of positive solutions of the above problem for f ( u ) = e u , f ( u ) = a u ( a > 0) , f ( u ) = u p ( p > 0) , f ( u ) = f ( = − 1( a > 1) and f ( u ) = (1 + u ) p ( p > 0). Our methods are based on a detailed analysis of


(Communicated by Rafael Ortega)
Abstract. In this paper, we consider the existence and exactness of multiple positive solutions for the nonlinear boundary value problem where λ > 0 is a bifurcation parameter, f (u) > 0 for u > 0. We give complete descriptions of the structure of bifurcation curves and determine the existence and multiplicity of positive solutions of the above problem for f (u) = e u , f (u) = a u (a > 0), f (u) = u p (p > 0), f (u) = e u − 1, f (u) = a u − 1(a > 1) and f (u) = (1 + u) p (p > 0). Our methods are based on a detailed analysis of time maps.
1. Introduction. In this paper, we study the bifurcation diagrams and multiple positive solutions of the nonlinear boundary value problem where λ > 0 is a bifurcation parameter, f (u) > 0 for u > 0. Our technique is a careful analysis of the so-called time-map G(λ, ρ) (or H(λ, ρ)), a function defined by an elliptic integral, which measures the time an orbit takes to get from one boundary line to another. In fact, the time map method was widely used in Laetsch [14], Crandall and Rabinowitz [5], Smoller and Wasserman [22], Castro and Shivaji [3], Liu and Zhang [18], Wang and Yeh [23], Cheng [4], Addou and Wang [1], Brubaker and Pelesko [2], Pan and Xing [19,20], Zhang and Feng [24] and Huang, Cheng, Wang and Chuang [7].
The motivation for this study comes from Goddard et al. in [6] and Hung et al. in [10]. In [6], J.Goddard. II, E.K. Lee and R. Shivaji have considered the reactiondiffusion model with nonlinear boundary condition given by   where Ω is a bounded domain in R n with n ≥ 1, ∆ is the Laplace operator, λ is a positive parameter, d is the diffusion coefficient, ∂u ∂η is the outward normal derivative, f : [0, ∞) → [0, ∞) is a smooth function, and α(u) ∈ [0, 1] is a nondecreasing smooth function. They discussed the existence of at least two positive radial solutions for λ 1 when Ω is an annulus in R n . Further, they discussed the existence of a double S-shaped bifurcation curve and the existence of six positive solutions for a certain range of positive λ when n = 1, Ω = (0, 1), and f (u) = e au a+u with a 1. However, their results are all based on numerical computations. K. Hung, S. Wang and C. Yu [10] made further investigation on this problem and proved the results of [6] theoretically.
As Goddard et al. in [6] and Hung et al. in [10] pointed out that studying problem (1.1) is equivalent to analyzing the following two boundary problems where Ω is a bounded domain in R n . The equation arises from a model of combustible gas dynamics. Nontrivial solutions of problem (1.4) are steady states for the thermal reaction process. Here λ > 0 is known as the Frank-Kamenetskii parameter. In [15,16], Liang and Wang studied the classification and evolution of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand equation with Dirichlet-Neumann boundary conditions given by where 4 ≤ a < ∞. They proved that the shape of the bifurcation curve depended on the value of a, c. The proofs given in [15,16] are more subtle and complicated. The results substantially improve and generalize those given in [10]. The earlier study of problem (1.2) is in [17,11]. Since then, problem (1.2) has been exuberantly studied in the past decades. There are two main tools to study the exact multiplicity of positive solutions and the shape of the bifurcation curve of problem (1.2). One way is bifurcation theory, see [12,13,21] and the references cited therein. In particular, we would like to mention some excellent results of Shi and Shivaji [21]. Using bifurcation theory, in [21], Shi and Shivaji obtained the exact multiplicity of positive solutions of problem (1.2) with f (0) < 0, which is called semipositone problem, and proved that the bifurcation diagram of (1.2) looks exactly like one of the following two graphs: Another way is time map method. In [8], Hung and Wang studied the bifurcation curve and exact multiplicity of positive solutions of problem (1.2) by using time map analysis, and gave an application to the perturbed Gelfand problem where a > 0 is the activation energy parameter. See Fig.3. The authors proved that, if a ≥ a * ≈ 4.166 for some constant a * defined in [8, Theorem 2.2(i)], the bifurcation curve S of (1.6) is exactly S-shaped on the (λ, u ∞ )-plan. More precisely, there exist two positive numbers λ * < λ * such that (1.6) has exactly three positive solutions for λ * < λ < λ * , exactly two positive solutions for λ = λ * and λ = λ * , and exactly one positive solution for 0 < λ < λ * and λ > λ * . Recently, Huang and Wang [9] proved that there exists a critical bifurcation value a 0 ≈ 4.069 such that, on the (λ, ||u|| ∞ )-plane, the bifurcation curve is exactly S-shaped for a > a 0 and is monotone increasing for 0 < a ≤ a 0 . That is, they proved the long-standing conjecture for the one-dimensional perturbed Gelfand problem. Clearly, C = S S. In this paper, on the (λ, u ∞ )-plane, we will give bifurcation curve S (resp.S) for six cases: see Fig.6, Fig.8, Fig.10, Fig.12, Fig.14 and Fig.16.
For problem (1.3), Goddard, Lee and Shivaji [6] discussed the existence of a Sshaped bifurcation curve by using numerical computations. In [10], Hung, Wang and Yu gave rigorous theoretical proofs of some computational results of [6]. In [15,16], Liang and Wang gave more perfect results on (1.5). Notice that these results are all on exponential nonlinearity. In this paper, we not only give some new results on exponential nonlinearity by establishing a new time map, but also study problem (1.1) with other nonlinearities of f (u) = a u (a > 0), f (u) = u p (p > 0), f (u) = e u − 1, f (u) = a u − 1(a > 1) and f (u) = (1 + u) p (p > 0). In our main results, we obtain the bifurcation curve C = S S of problem (1.1) on the (λ, u ∞ )-plane, and prove that problem (1.1) has multiple positive solutions for a certain range of positive λ. However, the exact shape of bifurcation curve S and the exact number of problem (1.3) need further investigation.
The rest of the paper is organized as follows: In Section 2, we introduce a new time map and analyze the time map which plays a key role in the paper. In Section 3, the bifurcation diagrams and multiple positive solutions of problem (1.1) for the cases of f (u) = e u and f (u) = a u (a > 0) will be stated. In Section 4, we give the bifurcation diagrams and multiple positive solutions of problem (1.1) for the case f (u) = u p (p > 0). In Section 5, we establish the bifurcation diagrams and multiple positive solutions of problem (1.1) for the cases f (u) = e u − 1 and f (u) = a u − 1(a > 1), and we determine the bifurcation diagrams and multiple positive solutions of problem (1.1) for the cases of f (u) = (1 + u) p (p > 0) in Section 6.
2. Time map. In this paper, to prove our main results, we define a new time map, which is different from that of Smoller and Wasserman [22], Goddard, Lee and Shivaji [6] and Huang, Wang and Yu [10].
As is well known, (1.2) is equivalent to We define a new time map as follows: Then positive solutions of (1.2) correspond to u ∞ = ρ and G(λ, ρ) = 1.
Thus, studying of the number of positive solutions of (1.2) is equivalent to studying the shape of the time map G(λ, ρ) on Σ. By calculation, we can get that On the other hand, by [7], problem (1.3) is equivalent to Then positive solutions of (1.3) correspond to u ∞ = ρ and H(λ, ρ) = 1. (2.5) We give some properties of G(λ, ρ) in the following Lemma 2.1. The proof of Lemma 2.1 is similar to Theorem 1.1 in [23], so we omit it.
To study the number of positive solutions of (1.3), we need to analyze the shape of H(λ, ρ). We investigate several important properties of H(λ, ρ) in the following Lemmas.

XUEMEI ZHANG AND MEIQIANG FENG
Proof. We need only to prove (iii). By (2.4) we have , Proof. dy.
By the third formular of (2.6), we can easily obtain that   Thus we can get that Lemma 3.3. There existsλ > 0 such that: This and (2.4) show that

Remark 2.
We can extend the above results to the case f (u) = a u .    Fig.8). Let f (u) = a u (a > 1) in (1.1). Then we have same results as Theorem 3.1.
Proof. By computation we have Then we easily obtain the results of Lemma 4.1. .
Proof. By Lemma 2.1-Lemma 2.3 and Lemma 4.1-Lemma 4.4, it is easy to obtain the results of Theorem 4.1, so we omit it. Fig.11. Graph of G(λ, ρ) for fixed λ in the case f (u) = e u − 1.  Fig.11). If f (u) = e u − 1, then for fixed λ > 0, G(λ, ρ) is decreasing on ρ, and Proof. It is easy obtained by Lemma 2.1 and (2.3). So we omit it.
Lemma 5.2 (See Fig.12). If f (u) = e u − 1, then there exists λ * = π 2 such that:  We can extend the above results to the case f (u) = a u − 1(a > 1).   Proof. If p > 1, by Theorem 1.1 of [23] we obtain the result of (i). If 0 < p ≤ 1, then by (2.3) we can easily obtain that θ (u) > 0. It follows that ∂ ∂ρ G(λ, ρ) > 0, i.e. G(λ, ρ) is increasing on ρ. Several limits can be obtained by calculation. So we omit it.  Proof. It is similar to that of Lemma 4.3. So we omit it.