HIGH MULTIPLICITY AND COMPLEXITY OF THE BIFURCATION DIAGRAMS OF LARGE SOLUTIONS FOR A CLASS OF SUPERLINEAR INDEFINITE PROBLEMS

This paper 
analyzes the existence and structure of the positive solutions of a 
very simple superlinear indefinite semilinear elliptic prototype 
model under non-homogeneous boundary conditions, measured by $M\leq 
\infty$. Rather strikingly, there are ranges of values of the 
parameters involved in its setting for which the model admits an 
arbitrarily large number of positive solutions, as a result of their 
fast oscillatory behavior, for sufficiently large $M$. Further, 
using the amplitude of the superlinear term as the main bifurcation 
parameter, we can ascertain the global bifurcation diagram of the 
positive solutions. This seems to be the first work where these 
multiplicity results have been documented.

Rather strikingly, although (1) looks so simple, in the general case when b > 0 and 0 < M ≤ ∞ there are very few results concerning the global structure of the solution set of (1). Among them, J. Mawhin, D. Papini and F. Zanolin [21] found some multiplicity results of sign-changing solutions, J. López-Gómez [17] established the existence and global attractive character of the minimal solution of (1) when M = ∞ for sufficiently small b > 0, and, more recently, J. García-Melián established the general shape of the bifurcation diagram of a general multidimensional prototype of (1) for M = ∞ and λ = 0 using b as the main bifurcation parameter. The device of using b as a parameter in the context of indefinite superlinear problems goes back, at least, to J. López-Gómez [15].
The main goal of this paper is ascertaining the global bifurcation diagrams of (1) for −λ > 0 sufficiently large, using b as the main bifurcation parameter, and obtaining quasi-optimal multiplicity results in the appropriate ranges of values of the secondary parameter λ < 0.
As a result of the non-homogeneous boundary conditions, measured by M ∈ (0, ∞], it turns out that there exists m * > 0 such that (1)  Up to the best of our knowledge, this multiplicity result has been observed in this paper for the first time. Based on it, we further perform a global continuation in the parameter b to construct all admissible bifurcation diagrams of (1), using b as the main parameter, at the values of λ < 0 where the multiplicity result holds.
The main tools which allow us to produce very precise bifurcation diagrams for equation (1) are based on a careful and detailed analysis of the time-maps associated to the corresponding phase plane systems. Analogous approaches for different kind of problems have been previously considered by Harris [13] and Dambrosio [7] in connection with non-homogeneous boundary value problems of the form −u = f (u) + h in (0, 1) u(0) = A, u(1) = B. ( In [13] the author studies the case of jumping nonlinearities f (u)/u → C, D as u → ±∞, while in [7] the superlinear case f (u)/u → +∞ as u → ±∞, is discussed.
The main difference of our study with respect to previous ones like [7,13] concerns the fact that in our situation we deal with an indefinite superlinear problem, due to the change of sign in the weight a(t) and, furthermore, we also show the crucial role played by the parameter λ when λ < 0. Such facts produce multiplicity results of positive solutions which are completely different and not comparable to the previous ones. Precisely, for problem (3) in the superlinear case one obtains a large number of sign changing oscillatory solutions (see [7]) and, as proved in [21], such kind of strong oscillatory behavior persists also for blowing-up solutions of −u = a(t)f (u), when a(t) (like in our case) is negative in a neighborhood of t = 0, 1 and positive elsewhere, but these solutions were not, in general, positive, as it occurs in this paper, where we find for problem (1) a new and broad complementary class of multiple large solutions which are positive and oscillate around a positive level. Such multiplicity result relies on the fast oscillation of the solutions of (1) for sufficiently negative λ. The topological structure of the bifurcation curves associated to such positive oscillatory solutions, in dependence of the parameter b, is astonishingly rich and somehow surprising. Moreover, differently from [7,13], we can treat in the same framework both the cases M ∈ R and M = +∞. This gives us the advantage to obtain some new multiplicity results for positive blow-up solutions as well.
As we are describing a new phenomenon in the context of semilinear elliptic problems, which might have crucial implications in a number of fields, in this paper we focus our attention in the simplest prototype model (1), because we are looking for much completeness and depth as possible.
Except for the last section, where we adapt most of the previous results to the special case when M = ∞, this paper focuses into the case M < ∞, and it is distributed as follows.
More precisely, if we denote by Σ 0 the set of solutions of (4), Section 2 shows that is a differentiable decreasing curve. Actually, by the symmetries of (1), Γ 0,M must be the reflection around the u-axis of Γ 1,M . The interest of these curves relies on the fact that the solutions of (1) restricted to the central interval (α, 1 − α) must be solutions of linking Γ 0,M with Γ 1,M in a time 1 − 2α. Conversely, any solution of (5) satisfying this property provides us with a solution of (1). This technical device goes back to J. Mawhin, D. Papini and F. Zanolin [21]. Section 3 gives the main multiplicity result of this paper through a systematic use of phase portrait techniques. Section 4 introduces a series of Poincaré maps which will provide us with the local and global bifurcation diagrams in b of the positive solutions of (1). Section 5 gives a series of global properties of these diagrams which are going to be pivotal in constructing all the global bifurcation diagrams of Section 6. Finally, in Section 7 we adapt most of the previous results to cover the singular case when M = ∞.
2. The problem (1) in the interval [0, α]. In the interval [0, α] the equation (2) reduces to HIGH MULTIPLICITY AND BIFURCATION DIAGRAMS 5 which is autonomous, and, hence, phase portrait techniques can be applied through the equivalent first order system which admits the first integral The main result of this section reveals the structure of the positive solutions of the Cauchy problem in the interval [0, α], for sufficiently large M > 0. Obviously, these solutions include the restrictions to [0, α] of the positive solutions of (1). It can be stated as follows. For any M > m * , there exist v * < v * < 0 satisfying the following properties: i) For every v ∈ (v * , v * ) the unique solution of (8) satisfies u(t) > 0 for all t ∈ [0, α]. ii) Let u * denote the unique solution of (8) with v = v * . Then, u * (t) > 0 for all t ∈ [0, α) and u * (α) = 0. iii) Let u * denote the unique solution of (8) with v = v * . Then, u * (t) > 0 for all t ∈ [0, α) and lim t↑α u * (t) = ∞. iv) For every v > v * , there exists T < α such that the solution of (8) satisfies u(t) > 0 for all t ∈ [0, T ) and lim t↑T u(t) = ∞.
v) For every v < v * , there exists T 0 < α such that the solution of (8) satisfies u(t) > 0 for all t ∈ [0, T 0 ) and u(T 0 ) = 0. Consequently, the candidates to provide us with (positive) solutions of (1) are those of (8) with v * < v < v * .
Proof. The proof will be divided into several steps.
As the generalized potential energy ϕ(u) := λu 2 − 2c p + 1 u p+1 has a quadratic minimum at 0 and a quadratic maximum at ω, (0, 0) must be a nonlinear center, while (ω, 0) is a saddle point of (7). Consequently, the phase portrait of the non-negative solutions of (6) looks like shown in Figure 1. Figure 1. Phase diagram of (6) for λ > 0 By simply looking at Figure 1, it becomes apparent that the solution of the problem (9) cannot blow up at time t = α if m ≤ ω. So, suppose m > ω and denote by t max = t max (m) the global existence time of the solution of (9). Integrating the differential equation we find that, for every t ∈ [0, t max ), t = u(t)/m 1 dθ 2c p+1 m p−1 (θ p+1 − 1) − λ(θ 2 − 1) (10) and, hence, Thus, t max < ∞ and, therefore, u blows up at a finite time. Moreover, letting t ↑ t max in (10), it becomes apparent that According to (11), t max (m) is decreasing with respect to m. Moreover, by continuous dependence, lim Therefore, there exists a unique m * (> ω) such that By construction, This shows the existence and the uniqueness of m * when λ > 0. The previous proof can be easily adapted to cover the general case when λ ≤ 0. In such case, the phase diagram looks like shown in Figure 2 and the proof of the case λ > 0 can be adapted almost mutatis mutandis to cover this case. So, the technical details of the proof in this special case are omitted here in.
u v Figure 2. Phase diagram of (6) in case λ ≤ 0 According to (12), the unique solution of (9) blows up in a time t max (m) ≤ α if m ≥ m * , while it is globally defined in the time interval [0, α] if m < m * .
Step 2: Let M > m * be and denote by v u > 0 the unique value of the derivative v = u for which (M, v u ) lies on the unstable manifold of (ω, 0) (resp. (0, 0)) if λ > 0 (resp. λ ≤ 0). By symmetry, (M, −v u ) is the unique point on the stable manifold of that equilibrium with u = M . Subsequently, for every v ∈ (−v u , 0), we consider the Cauchy problem (8). According to the phase portrait, it is apparent that the solution of (8) needs a time, say t min := t min (v), to reach its minimum, denoted by m := m(v). So, by symmetry, Actually, m > ω if λ > 0.
Step 2 shows that there exists a unique v 0 ∈ (−v u , 0) such that As a byproduct, the boundary value problem possesses a unique solution. Namely, the solution of (8) with v = v 0 .
Indeed, by simply looking at the phase portrait of (6), it becomes apparent that the map m : (−v u , 0) → R + is increasing, since two different trajectories cannot meet. Moreover, On the other hand, by integrating the differential equation, we are easily driven to the identity .
Consequently, as m(v) increases with v, t min (v) decreases. Moreover, t min (0) = 0, by construction, and lim v↓−vu t min (v) = ∞, by continuous dependence. Therefore, (13) holds for a unique v 0 ∈ (−v u , 0). This ends the proof of Step 2. Setting Step 3. This step constructs v * and v * and completes the proof. It should be remembered that, according to Step 2, the solutions of (8) need a time larger than α to reach the u-axis if v ∈ (−v u , v 0 ), while they do it before time α if v ∈ (v 0 , 0).
Next, we suppose M > m * and consider the map , which measures the blow-up time of the solution of (8). According to Steps 1 and 2, T is continuous and decreasing. Moreover, by continuous dependence, and, due to (12), . Moreover, v = v * is the unique shooting speed for which the solution of (8) blows up at time α. Now, we will study the behavior of the solutions of (8) with v ≤ −v u . By simply looking at the phase portrait of (6) (cf. Figures 1 and 2), it becomes apparent that such solutions vanish for some positive time if v < −v u , while they stabilize to ω (resp. 0) if λ > 0 (resp. λ ≤ 0) and v = −v u . Integrating the differential equation, we find that the necessary time to reach the v-axis when v < −v u , denoted by T 0 (v), is given through the formula .
Clearly, T 0 is a continuous and increasing function of v, since v 2 decreases if v < 0 grows. Moreover, by continuous dependence, because −v u is the critical shooting speed of the stable manifold of the equilibrium. Also, letting v ↓ −∞, shows that Necessarily, by the definition of v 0 (cf. (13)), From these features, the proof of the theorem can be easily completed.  (8) for v < v * and v = v * , respectively. By Theorem 2.1, v * is the unique value of v for which the solution u of (8) satisfies u(t) > 0 for all t ∈ [0, α) and u(α) = 0. If v < v * , then u vanishes at some t 0 < α.
The profiles on the second row are two solution plots for v ∈ (v * , v 0 ) and v = v 0 , respectively. According to (13), v 0 is the unique value of v for which the solution u of (8) satisfies u (α) = 0, and, thanks to (14), By (13), t min (v) > α if v < v 0 , and, hence, u (α) < 0 if v < v 0 , as illustrated by the first plot of the second row of Figure 3.
The third row of Figure 3 shows the plots of two (different) solutions of (8) for two values of v ∈ (v 0 , v * ). Finally, in the fourth row we have represented the plots of the solutions of (8) for v = v * and some v > v * , respectively. In the first case, the solution blows up at time t = α, whereas the solution blows up at some Throughout the rest of this paper, for every λ ∈ R and M > m * we denote by Σ 0 the set of solutions of (8) with v ∈ [v * , v * ) and consider the set of points of R 2 reached by the solutions of (8) at time α as the shooting speed v ranges in between v * and v * , Γ 0 := { (u(α), u (α)), u ∈ Σ 0 }. The next result collects the main features of Γ 0 . Theorem 2.2. For every λ ∈ R and M > m * , Γ 0 is a C 1 -curve in R + × R such that: i) π u (Γ 0 ) = R + , where π u stands for the projection of R 2 on the first component.
ii) The value of m 0 defined by (14) satisfies the following properties: Note that, thanks to the proof of Theorem 2.1, is the unique solution of (8), and the Poincaré map Thanks to the proof of Theorem 2.1, P is well defined and According to the theorem of differentiation of G. Peano, P provides us with a diffeomorphism of [v * , v * ) onto Im P. Therefore, P establishes a diffeomorphism between [v * , v * ) and Γ 0 . Consequently, Γ 0 is a curve of class C 1 in R + × R.
As π u is a continuous map, π u (Γ 0 ) must be an interval and, since we can infer that π u (Γ 0 ) = R + , which ends the proof of Part i). Part ii) is an easy consequence of (13) and (14). This ends the proof.
By performing the change of temporal scale one can easily infer the next counterpart of Theorem 2. Moreover: satisfies u(t) > 0 for all t ∈ [1 − α, 1]. ii) Let u * denote the unique solution of (15) with v = −v * . Then, u * (t) > 0 for all t ∈ (1 − α, 1] and u * (1 − α) = 0. iii) Let u * denote the unique solution of (15) with v = −v * . Then, u * (t) > 0 for all t ∈ (1 − α, 1] and lim t↓1−α u * (t) = ∞. iv) For every v < −v * , there exists t max < α such that the solution of (15) satisfies Consequently, the candidates to provide us with positive solutions of (1) are those of (15) Throughout the rest of this paper, for every λ ∈ R and M > m * we denote by Σ 1 the set of solutions of (15) with v ∈ (−v * , −v * ] and consider the set of points of R 2 reached by the solutions of (15) at time 1 − α as v ranges in between −v * and −v * , Obviously, the next counterpart of Theorem 2.2 holds. Corollary 2. For every λ ∈ R and M > m * , Γ 1 is a C 1 -curve of R + × R such that In particular, π u (Γ 1 ) = R + and The next result establishes that Γ 0 is an increasing arc of curve with respect to u.
Setting N := u 1 (θ), it is apparent that u 1 and u 2 solve and, owing to S. Cano-Casanova [6], we find that u 1 = u 2 and, therefore, v 1 = v 2 , which is impossible. This completes the proof.
In Figure 4 we have represented two admissible curves Γ 0 and Γ 1 according to Theorem 2.2, Corollary 2 and Proposition 1. By Corollary 2, Γ 1 is the reflection of Γ 0 around the u-axis.
Our interest in the curves Γ 0 and Γ 1 comes from the fact that the solutions of the nonlinear boundary value problem (1) are given through the solutions of the autonomous equation − u = λu + bu p (17) connecting the curves Γ 0 and Γ 1 in the phase portrait of (u, u ) in time 1 − 2α. Note that 1 − 2α is the length of the subinterval (α, 1 − α) of (0, 1) where (2) becomes provides us with a solution of the superlinear indefinite problem (1). This section concludes with a further fundamental property of Γ 0 which will be used throughout the rest of this paper.
Actually, if (x, y(x)) = P(v), v ∈ [v * , v * ), i.e., x = π u (P(v)), then, where ξ stands for the unique solution of Proof. Subsequently, the notations introduced in the proof of Theorem 2.2 will be maintained. By definition, Thus, according to the theorem of differentiation of G. Peano, we have that As we are imposing λ ≤ 0, and, due to Proposition 1, Therefore, the two components of DP (v) are positive real numbers for all v ∈ [v * , v * ). As DP (v) is the tangent vector to the curve Γ 0 at P (v) for all v ∈ [v * , v * ), the rest of the proof follows easily from well known features on differential geometry of curves. Under this condition, the following result holds. (18) Then, the number of solutions of (1) grows to infinity as λ ↓ −∞. More precisely, if where we are denoting then, (1) possesses, at least, 4n solutions; among them, 2n are symmetric and the remaining 2n are asymmetric.
As ϕ has a quadratic maximum at 0 and a quadratic minimum at Ω, (0, 0) is a saddle point and (Ω, 0) is a center. Moreover, as ϕ(u) is a potential well, there is an homoclinic connection of (0, 0) surrounding (Ω, 0) and any periodic orbit around (Ω, 0). Figure 5 sketches the phase portrait of (17) in case λ < 0, as well as the curves Γ 0 and Γ 1 . Naturally, by the definition of m 0 (cf. (14)), if u (t) stands for the unique solution of −u = λu − cu p , u(0) = M, u(α) = m 0 , then, the function provides us with a symmetric solution of problem (1). Throughout the rest of the paper we will call this function the trivial solution.
To construct more solutions, one should note that the limiting period of the small amplitude periodic oscillations around (Ω, 0), as the amplitude goes down to zero, equals the period of the solutions of the linearized equation which is given through Having a glance at Figure 5 it becomes apparent from Proposition 1 that Γ 0 meets the homoclinic of (0, 0) twice. Let (x − 0 , y − 0 ) denote the second crossing point, as time passes by, between them. By Theorem 2.3, y − 0 = y(x − 0 ). Note that x − 0 < Ω. For every (x, y) ∈ Γ 0 with x ∈ (x − 0 , Ω) (necessarily y = y(x)), consider the initial value problem −u = λu + bu p , Let τ 1 (x) denote the time taken by the trajectory of the solution of (20) to reach Γ 1 for the first time, and τ (x) the period of the solution of (20). By definition, τ 1 (x) < τ (x) for all x ∈ (x − 0 , Ω). Thus, by continuous dependence and taking into account that lim it is easily seen that Thus, if we assume that then, τ 1 (x) < 1 − 2α for some x < Ω, x ∼ Ω, and, hence, by the continuity of τ 1 in (x − 0 , x) along Γ 0 , there exists (x 1 , y 1 ) ∈ Γ 0 , y 1 = y(x 1 ), with x 1 ∈ (x − 0 , x) such that τ 1 (x 1 ) = 1 − 2α; τ 1 is continuous by the transversality of the trajectories with the curves Γ 0 and Γ 1 . Consequently, reasoning as at the end of Section 2, the unique solution of (17) such that (u(α), u (α)) = (x 1 , y 1 ) provides us with another symmetric solution of (1) having a unique critical point (a local minimum) in the interval (α, 1 − α). Now, for every (x, y) ∈ Γ 0 with x ∈ (x − 0 , Ω), let τ 2 (x) denote the necessary time to reach Γ 1 exactly twice. As for (21), (22) implies that and, therefore, there exists (x 2 , y 2 ) ∈ Γ 0 (y 2 = y(x 2 )), with x 2 ∈ (x 0 , Ω), such that Consequently, the unique solution of (17) such that (u(α), u (α)) = (x 2 , y 2 ) provides us with an asymmetric solution of (1) under condition (22). This solution has two critical points in (α, 1 − α): a local minimum and a local maximum. Naturally, if we denote it by u(t), then, the reflected functioñ provides us with another asymmetric solution of (1). The associated orbit (ũ,ũ ) leaves Γ 0 atũ(α) > m 0 and meets Γ 1 twice ending on it. This solution also has a local minimum and a local maximum. The four solutions that we have just constructed have been represented in the first row of Figure 6. It should be noted that they do exist provided (22) holds, which has been emphasized in the first column of Figure 6, where the requested conditions for the existence of the solutions of the corresponding row are given. Except for the solutions of the first column, the remaining solutions of Figure 6 are constructed by adding n ≥ 1 laps around (Ω, 0) to the solution at the top of the corresponding column. Figure 6 consists of 12 pictures and three conditions. Each of the pictures exhibits two crossing curves. As we move from the left to the right, one of them increases and the other decreases. The increasing one represents Γ 0 , whereas the decreasing one stands for Γ 1 . The arcs of curve connecting them stand for trajectories of solutions of (17). The number of arrows counts how many times the corresponding piece of trajectory between Γ 0 and Γ 1 is run. One should always start at Γ 0 and end at Γ 1 .
Next, we will construct the first solution of the second column. Let (x + 0 , y + 0 ) be the first crossing point, as time passes by, between Γ 0 and the homoclinic of (0, 0). Obviously, Ω < x + 0 and, thanks to Theorem 2.3, y + 0 = y(x + 0 ). For every (x, y) ∈ Γ 0 with x ∈ (Ω, x + 0 ), let τ 3 (x) denote the time needed by the solution of (20) to reach Γ 1 for the first time after a complete lap. By definition, τ 3 < 2τ Ω and, arguing as above, it becomes apparent that Consequently, under condition 2 2π

Condition
Symmetric solutions Asymmetric solutions provides us with a symmetric solution of (1) which has three critical points in (α, 1 − α): two local maxima and one local minimum. The remaining solutions of the first column of Figure 6 are constructed from this one by adding an additional lap every time we pass from one row to the next one, as it happens with the remaining three columns. Consequently, under condition (1) possesses, at least, 4n solutions, 2n among them being symmetric (those on the first two columns), and the remaining 2n solutions (those on the last two columns) asymmetric. The proof is complete.
By Theorem 3.1, it is natural to introduce the following concept.
Definition 3.2. A solution u of (1) is said to be of type n ≥ 0, which can be shortly expressed by writing u ∈ T n , if it has n ≥ 0 strict critical points in the central interval (α, 1 − α). According to this terminology, the trivial solution u 0 is of type 0, i.e., u 0 ∈ T 0 .
The proof of Theorem 3.1 reveals that, under conditions (18) and (19), the problem (1) has, at least, one solution of type 1 and two solutions of type j, for every j ∈ {1, . . . , n}, besides the trivial solution (cf. Figure 6).

3.1.
A sharp pivotal property of the time-map τ 1 . The time map τ 1 defined in the proof of Theorem 3.1 can be extended for every x > Ω in a natural way. Indeed, for each x > Ω, τ 1 (x) is the minimal necessary time to reach Γ 1 by the solution of the problem (20), where y(x) is the function introduced by Theorem 2.3.
The next result shows that the singularity Ω can be overcome.
for which the trajectory of the solution of (20) reaches (x L , 0) at a (minimal) time τ 1 (x)/2. As the nonlinearity of (20) is analytic at Ω, the differential equation of (20) can be equivalently written as follows for some coefficients h j , j ≥ 2, whose knowledge is not relevant in this proof. The series is absolutely convergent in some interval around Ω. Multiplying (24) by (u − Ω) = u and rearranging terms, we are driven to Therefore, the orbit of the periodic solution of (20) is given through Consequently, shortening the notations by setting it is easily seen that Equivalently, by performing the change of variable θ = u − Ω, u ∼ Ω, On the other hand, since we find the estimate Consequently, shortening the notations by naming and it follows from (25) and (26) that and, therefore, to prove the first identity of (23) it suffices to show that It should be noted that, in order to do this, we must make sure that the functions are non-negative, at least for x in a neighborhood of Ω. Through some elementary manipulations, it is easily seen that (28) implies Now, we need to ascertain the asymptotic expansion of x L (x) in terms of x − Ω, as x ↑ Ω. As the solution of (20) satisfies necessarily for all x < Ω, x ∼ Ω. By Theorem 2.3, we already know that Moreover, a direct calculation shows that because Ω p−1 = −λ/b. Thus, substituting these expansions into (31) and rearranging terms, it follows that as x ↑ Ω. By expanding the left hand side of this identity in powers of x − Ω, or, alternatively, differentiating twice with respect to x and particularizing at x = Ω, it becomes apparent that as x ↑ Ω. Actually, x L is a function of class C ∞ in a neighborhood of Ω by the theorem of differentiation of G. Peano, because y(x), and so Γ 0 , is of class C ∞ outside the origin. Consequently, substituting (32) and (33) into (27), we are led to Using these asymptotic expansions, it is straightforward to check that the function g(θ) defined above is indeed non-negative and that As a byproduct, by (33), we also have that Consequently, letting x ↑ Ω, it becomes apparent that This proves (29) and ends the proof of the first identity of (23). The previous argument can be easily adapted, almost "mutatis mutandis", to show the validity of the second identity of (23). So, we omit the technical details here. The fact that  (20) satisfies x p+1 M and, hence, for every x > x + 0 , .
As lim x↑∞ x M (x) = ∞, letting x ↑ ∞ in the previous estimate shows (34) and ends the proof.
As expected, the value of τ 1 (Ω) for the nonlinear problem (20) given through Theorem 3.3 coincides with the value of τ 1 (Ω) for its linearized problem at the steady-state Ω Indeed, the general solution of the differential equation of (35) can be expressed as where A, B ∈ R are arbitrary integration constants, and, hence, and, therefore, the unique solution of (35) is given through As τ 1 (x)/2 is the first time where u vanishes, it becomes apparent that and, consequently, Letting x → Ω in (36), indeed gives the value of τ 1 (Ω) calculated through Theorem 3.3.

3.2.
The existence of solutions of type T 1 . The following result establishes an almost optimal condition for the existence of solutions of type T 1 , in the sense that it would be a characterization theorem if τ 1 (x) would be decreasing. But, we have not been able to prove this monotonicity property yet.
Theorem 3.4. Suppose λ < 0 and (18). Then, the following assertions are true: Note that, besides these solutions of type T 1 , the problem admits the trivial solution u 0 .
Proof. The notations introduced in the proof of Theorem 3.1 will be kept through this proof. Under condition (37), there exists Now, suppose (38), instead of (37). Then, there exists x > Ω such that τ 1 (x) > 1 − 2α. Therefore, owing to (34), it becomes apparent that there existsx > x such that τ 1 (x) = 1 − 2α. This completes the proof.  As far as concerns to solutions of type T 1 , it should be noted that Theorem 3.4 is substantially sharper than Theorem 3.1. Not only the solutions given by Part (b) were left aside of Theorem 3.1, but the condition τ Ω < 1 − 2α is much stronger than (37), since, actually, we do have The next result complements Theorem 3.4 ensuring that there are open ranges of values of the parameters involved in the setting of (1) for which any of the conditions (37) and (38) can be satisfied.
By the theorem of differentiation of G. Peano, it follows from Theorem 2.3 that the positive real number y (Ω) can be regarded as a continuous function of λ ≤ 0, say y (Ω) = y (Ω, λ). Therefore, and, consequently, Theorem 3.3 implies that which completes the proof of Part (b).

3.3.
The time-maps τ j , j ≥ 2. Sharpening Theorem 3.1. Throughout this section, we suppose λ < 0 and (18). As for τ 1 , the time map τ 2 constructed in the proof of Theorem 3.1 can be extended, in a rather natural way, to be defined for all , where, following the notations introduced in the proof of Theorem 3.1, (x + 0 , y(x + 0 )) and (x − 0 , y(x − 0 )) are the first and the second crossing points, respectively, as time increases, between Γ 0 and the homoclinic orbit through (0, 0); x − 0 < Ω = m 0 < x + 0 . Similarly, the map τ 3 can be extended to be defined in J * too. Also, by construction (see Figure 5, if necessary), we have that where τ (x) stands for the period of the solution of (20). More generally, throughout the rest of this section, for every integer number n ≥ 2, we consider the time-maps τ 2n and τ 2n+1 defined in J * through By (40), (41) does actually make sense for every n ≥ 1. According to Theorem 3.3, one can easily infer, by the symmetry of the problem, that Consequently, for all n ≥ 1, and, therefore, all these time-maps can be extended to J so that τ j ∈ C(J), j ≥ 1, by simply setting By continuous dependence, all these time-maps satisfy Moreover, by construction, Actually, τ 1 is globally defined in (x − 0 , ∞) and, due to Theorem 3.3, lim x↑∞ τ 1 (x) = 0. Figure 8 represents the plots of τ n , 1 ≤ n ≤ 7, that we have computed using Mathematica for an appropriate choice of the several parameters involved in the formulation of (1). The numerics gave the nice monotonicity properties shown in Figure 8, though we were not able to prove them analytically.
The graphs of the curves τ n , 1 ≤ n ≤ 7.
By Corollary 2 and the symmetry of the equation (17), the solutions of type T 2n , n ≥ 2, must appear by pairs (u,ũ) with similarly to the case n = 1 already analyzed in the proof of Theorem 3.1.
It should be noted that, according to (42), By simply counting the number of roots of τ n = 1 − 2α, for every n ≥ 1, one can get the next substantial improvement of Theorem 3.1.
Theorem 3.5. Suppose λ < 0, (18) and for some n ≥ 2. Then, (1) admits, at least, one solution of type T 1 and two solutions of type T j for all 2 ≤ j ≤ n, besides the trivial solution u 0 ∈ T 0 . When n = 1, the problem possesses at least one solution of type T 1 plus the trivial solution, as already established by Theorem 3.4(a).

Local perturbation from
The main goal of this section is to analyze the behavior of the solutions already constructed in Section 3 as the parameter b perturbs from b * . In Section 5 we will ascertain the global behavior of these solution as b separates away from b * , obtaining in this way the corresponding global bifurcation diagrams. One of the main differences between the cases b = b * and b = b * is that in case b = b * all solutions of odd type must be symmetric, whereas in case b = b * the problem (1) can admit asymmetric solutions of odd type, which play the same role as solutions of even type. (17) looks like shows Figure 9, where we also have superimposed Γ 0 and Γ 1 .
Besides the homoclinic through the origin, we have represented three trajectories. One exterior to the homoclinic and two interior orbits. The interior ones are rather special, as the inner one is the unique orbit which is tangent to Γ 0 at (x t , y(x t )), and the exterior one is the unique orbit passing through the crossing point between Γ 0 and Γ 1 , (m 0 , 0). The points x − 0 and x + 0 are defined as in the proof of Theorem 3.1. We have denoted by x m0 the unique value of x = m 0 for which (x, y(x)) ∈ Γ 0 lies on the orbit through (m 0 , 0). All these points are going to play a very important role in the subsequent analysis. Naturally, except m 0 , all of them are regular functions of b, Figure 9 illustrates the special case when in general, this condition does not need to be satisfied.
Besides (0, 0), the homoclinic connection meets the u-axis at (u M , 0), where Figure 9 represents the phase portrait of (17) . This is the range of values of b where we are going to focus our attention in this section. Figure 10 describes the most relevant general features of the phase portrait of (17) as b increases from b * . Precisely, Figure  10A shows the phase portrait for b * < b < b m0 . As b reaches the critical value b m0 ( Figure 10B) and crosses it, m 0 moves outside the homoclinic until b attains a further critical value, say b t > b m0 , where the homoclinic is tangent to Γ 0 and Γ 1 ( Figure  10C). As b > b t , the homoclinic connection cannot meet Γ 0 ∪ Γ 1 , and, actually, it shrinks to (0, 0) as b ↑ ∞. These features are straightforward consequences of the fact that the family of homoclinic connections shrinks monotonically to (0, 0) as b increases. As in Section 3, the solutions of (17) connecting Γ 0 with Γ 1 in a time 1 − 2α provide us with solutions of (1), as it was described in Section 2. But, since the phase portrait changes when b increases from b * , the time-maps τ n , n ≥ 1, also change. Most of our efforts to understand what is going on when b perturbs from b * rely on the appropriate definitions of these time-maps, which will be subsequently denoted by τ n (x, b), in order to emphasize their dependence on b.
For every b ∈ (b * , b m0 ), we denote by the Poincaré map defined, for every x ∈ D 1 , as the minimal time needed by the solution of (20) to reach Γ 1 . Figure 11 shows the corresponding orbits of two of these solutions for some x < x t (A) and x > m 0 (B). By continuous dependence, because m 0 is not an equilibrium of (17). Also, the proof of (34) (for b = b * ) adapts mutatis mutandis to show that is the minimal time needed by the solution of (20), with x = x t , to connect (x t , y(x t )) ∈ Γ 0 with Γ 1 .
Subsequently, for every b ∈ (b * , b m0 ), we denote by the Poincaré-map defined, for every x ∈ (x t , m 0 ), as the minimal time needed by the solution of (20) to reach Γ 1 exactly twice, while where, for every x ∈ (x − 0 , x + 0 ), τ (x, b) stands for the period of the orbit through (x, y(x)). Figure 12 shows the corresponding orbits of two of these solutions for some x ∼ x t (A) and x ∼ m 0 (B).
Similarly, we may introduce the Poincaré map , as the minimal time needed by the solution of (20) to reach Γ 1 twice (see Figure 13A), and, for every x ∈ [x t , m 0 ], as the minimal time needed to reach Γ 1 for the first time (see Figure 13B).
and, hence, by (44), we obtain that though, by definition, Incidentally, the solutions of τ 1 = 1 − 2α and τ 1,s = 1 − 2α provide us with symmetric solutions of (1), whereas the solutions constructed from τ 1,a = 1 − 2α are asymmetric, except at x = x t , and, by definition of the time maps, all these solutions provide us with solutions of (1) with a single critical point. Conversely, any solution of problem (1) with a single critical point in (α, 1 − α) must be of some of these forms. Actually, this is why we have introduced all these time-maps. Naturally, in order to look for solutions of type T 2 of (1), we must introduce the Poincaré map defined, for every x ∈ D 2 \ {x m0 , m 0 }, as the minimal time needed by the solution of (20) to reach Γ 1 exactly twice, and By continuous dependence and the symmetry properties of our problem, these identities are consistent. Figure 14 shows the corresponding orbits of two of these solutions for some cannot admit a solution of type T 2 passing through (x, y(x)) if x m0 < x < m 0 . Incidentally, by symmetry, each of the solutions on Figure 14(A) must be a reflection around t = 1/2 of some solution in Figure 14(B). More generally, we may introduce the time maps for all n ≥ 1. According to (43), it follows from (44) and (48) that Thus, the graph of τ 1,s in the interval [x t , m 0 ] does connect the graph of , as illustrated by Figure 15, where those graphs have been plotted together. Consequently, by (48), the graph of τ 2n+1,s in [x t , m 0 ] must connect the graph of τ 2n+1 in the interval (x − 0 , x t ] with the graph of τ 2(n+1)+1 in [m 0 , x + 0 ), for all n ≥ 1. Similarly, by (47), the graph of τ 1,a in [x m0 , m 0 ] does connect the graph of τ 2 in (x − 0 , x m0 ] with the graph of τ 2 in [m 0 , x + 0 ). As, according to (45), the graphs of τ 1 , τ 1,s and τ 1,a cross at x t ; owing to (46) and (48), it becomes apparent that the graphs of all these functions look like shown by Figure 15, where we have collected the plots of the first seven time maps for an appropriate special case with b = b * + 0.01. Note that Figure 15. The graphs of the curves (48) and (49) for b * < b < b m0 More precisely, Figure 15 represents the graphs of the functions θ j , j ≥ 1, defined by for all n ≥ 0, and for all n ≥ 1. According to (48) and (49), it is apparent that θ 2n+1 := θ 1 + nτ, θ 2n := θ 2 + (n − 1)τ, n ≥ 1.
The next result collects some global monotonicity properties of these auxiliary functions.
Lemma 4.1. For every integer n ≥ 0 and b ∈ (b * , b m0 ), we have that Moreover, for every Note that Figure 15 is in complete agreement with Lemma 4.1 .
the graphs of all these θ j 's in the intervals [x m0 (b), m 0 ] shrink either to a single point, or a certain segment, as b ↓ b * . Moreover, the crossing points between the homoclinic and Γ 0 satisfy where x ± 0 = x ± 0 (b * ) are those introduced in the proof of Theorem 3.1. Naturally, by well known results on continuous dependence, the next result holds.
Proposition 3. For every n ≥ 1 and ε > 0 sufficiently small, . Proposition 3 guarantees that, for any n ≥ 1, the graphs of the time map τ n (·, b) constructed in this section converges, as b ↓ b * , to the graph of the corresponding time map τ n constructed in Section 3; the convergence being uniform in the complement in (x − 0 (b * ), x + 0 (b * )) of any neighborhood of m 0 . This has been illustrated by Figure 15, where the graphs of the perturbed Poincaré maps, τ n (·, b), for b > b * , plotted with continuous lines, are very close to the graphs of the corresponding unperturbed time maps τ n , plotted with dashed lines. Not surprisingly, in between x m0 and m 0 (very close if b ∼ b * ), the nature of these graphs changes drastically. Indeed, in the interval [x m0 (b), m 0 ] each τ 2n+1 breaks down and its perturbed left branch can be connected with the perturbed right branch of τ 2(n+1)+1 through the graph of τ 2n+1,s , while the graph of τ 2n breaks down and its perturbed left branch can be connected with its perturbed right branch through the graph of τ 2n−1,a . In other words, the unperturbed τ n 's of Section 3 perturb into the θ n 's constructed through (50) and (51). The next theorem provides us with the precise behavior of these time-maps as b ↓ b * .
Theorem 4.2. The following assertions are true: , as well as for the linearized problem.
Thus, owing to Parts (a), (b) and (45), the function τ 1,s must jump from a value very close to τ 2 (Ω) to a value very close to τ (Ω) in the interval [x t (b), m 0 ] for sufficiently close b > b * , whereas, due to Parts (b), (c) and Proposition 3, the graph of τ 3 (·, b) approximates the graph of Naturally, thanks to (48) and (49), we also have that for all n ≥ 1.
Proof. Part (b) is an immediate consequence of (44), as . Part (c) follows directly from (48) and (43), as . Therefore, we only have to prove Part (a).
We begin with the study of the linearized problem −u = λu + pbΩ p−1 u = λ(1 − p)u, whose integral curve through a generic point (x, y) in the phase plane (u, v) is the ellipse with equation Figure 16 represents three of these integral curves. Precisely, Figure 16 shows the tangent straight lines to the curves Γ 0 and Γ 1 at (m 0 , 0) (subsequently referred to asΓ 0 andΓ 1 , respectively), the integral curves through (m 0 , 0), (x t , y t ), and a generic point (x, y) with where (x m0 , y m0 ) stands for the other crossing point between the integral curve through (m 0 , 0) and the lineΓ 0 , and (x t , y t ) denotes the crossing point withΓ 0 of the unique integral curve tangent to it.
For every x ∈ [x m0 , x t ), we defineτ 1,a (x) as the time needed to crossΓ 1 twice, starting at the point (x, y) ∈Γ 0 and moving along the orbit up to reach (x 2 , y 2 ), whileτ 1,a (x t ) stands for the necessary time to reachΓ 1 for the first time starting at (x t , y t ). By continuous dependence,  We want to prove that To this end, we will show thatτ 1,a (x) is actually constant as a function of x, and thatτ 1,a (x t ) = τ 2 (Ω). Substituting (55) in (54) and solving yields Similarly, substituting (55) in imposing that x t is a double solution, shows that Consequently, and, therefore, since m 0 > Ω, x t > Ω. As it is apparent from Figure 16, Subsequently, we denote by x L = x L (x) and x M = x M (x) the minimum and maximum values of the u-coordinate along the orbit E x passing through (x, y), and, for every pair of points, (x a , y a ), (x b , y b ) ∈ E x ,t xa,x b stands for the necessary time to reach (x b , y b ) from (x a , y a ) for the first time along E x . By definition, Figure 16).
To calculatet x L ,x M , we consider the general solution of the linearized differential equation, which is given by subjected to the initial conditions (u(0), u (0)) = (x L , 0). Then,t x L ,x M is the first (positive) time where u vanishes. As and, hence, the first (positive) time where u = 0 is Similarly, the unique solution of (57) such that (u(0), u (0)) = (x, y) satisfies and, therefore, the first (positive) time where u = 0 is where it should be noted that Analogously, the unique solution such that (u(0), u (0)) = (x 2 , y 2 ) is given by (57) with where x 2 is given by (56). Thus, sincet x2,x M is the first positive time where u = 0, we obtain that Consequently, setting differentiating with respect to x, and using (56), we find that, for all x = Ω, because (x, y), (x 2 , y 2 ) ∈ E x . As, thanks to the theorem of differentiation of G. Peano, the functionτ 1,a (x) is differentiable in x, necessarily it must be constant for all x. Therefore,τ Adapting our previous calculations, it is easily seen that which equals by Theorem 3.3, since arctan θ + arctan 1/θ = π/2 for all positive θ. This completes the proof of Part (a) for the linearization. Now, we will prove Part (a) for the nonlinear problem. The basic idea will be adapting the technical device introduced in the proof of Theorem 3.3 by expanding x m0 (b) and x t (b) as power series centered at b = b * and then letting b → b * in the formulas for the time maps. By construction, and, hence, x t (b) may be characterized as Note that, thanks to the implicit function theorem and the theorem of differentiation of G. Peano, as a consequence of (y (m 0 )) 2 + λ(1 − p) > 0 it follows the existence and the uniqueness of a real analytic function Similarly, x m0 (b) is given through To show its analyticity in b one can argue as follows. Setting we consider the map for all b ∼ b * , uniformly in s ∈ [0, 1]. Let x L (s, b) denote the minimum u-coordinate of the orbit through the point (x(s, b), y(x(s, b)). It satisfies The fact that it is real analytic can be shown by brute force by computing all the Taylor series and checking that the coefficients can be locally estimated by those of a convergent series. Differentiating twice with respect to b in (63) we find that For every s ∈ [0, 1] and b ∼ b * , let (x 2 (s, b), y 2 (s, b)) be the second crossing point of the orbit through (x(s, b), y(x(s, b)) with the curve Γ 1 , starting at (x(s, b), y(x(s, b)). It is given by Being the existence already known, to prove its analyticity in b, we introduce the mappings Consequently, the analyticity of x 2 (s, b) is an easy consequence of the implicit function theorem. Moreover, by implicit differentiation, we obtain that By developing f around (x, b, u) = (m 0 , b * , m 0 ), we have that for some constants c ijk whose explicit knowledge is not important in this proof. According to (62), (64) and (65), and taking into account that u ∈ [x L (s, b), x(s, b)] in the first integral, while u ∈ [x L (s, b), x 2 (s, b)] in the second one, we find that for some positive c h (s), h ≥ 3. Subsequently, we set we find that for all s ∈ [0, 1] and b > b * sufficiently close to b * . Now, performing the change of variable θ = u − m 0 and completing squares in the radicand of all the integrals, one can proceed as in the proof of Theorem 3.3 to obtain that .

Now
. Thus, and, consequently, thanks to (68), we find that  (65), and differentiating with respect to s, it is easily seen, after some tedious, but straightforward, manipulations, that for all s ∈ [0, 1], and, therefore, for all s ∈ [0, 1]. As and all these times equal the corresponding times when x runs over the interval , the proof is complete.
The results established by Theorem 4.2 are extremely sharp. Indeed, since x t (b) ∈ (x m0 (b), m 0 ), Part (a), in particular, entails that lim b↓b * τ 1 (x t (b), b) = τ 2 (Ω), which is far from being obvious, as using directly some (not valid) continuous dependence argument, one might have been tempted to argue that τ 1 (x t (b), b) approximates τ 1 (Ω) as b ↓ b * , which is not true. Moreover, by Proposition 3, which, according to Theorem 4.2(c), shows that one cannot commute the double limit. This is utterly attributable to the lack of uniformity in the limit as b ↓ b * .
It should be noted that if the equation has some solution x ∈ (x m0 (b), m 0 ) for a b sufficiently close to b * , then the graph of the orbit through (x, y(x)) ∈ Γ 0 must approach (m 0 , 0) as b ↓ b * . Therefore, these solutions must perturb from the trivial solution u 0 . Similarly, if for some of these b's τ 2n−1,a (x, b) = 1 − 2α, then the graph of the orbit through (x, y(x)) must approximate (m 0 , 0) as b ↓ b * , and, therefore, these asymmetric solutions must perturb from u 0 too. Throughout the rest of this paper, we will use the following terminology, extending Definition 3.2. Definition 4.3. Let u be a solution of (1) and an integer n ≥ 1. Then: • u is said to be of type T n if τ n (u(α), b) = 1 − 2α.
It should be noted that the solutions of type T n possess n local extrema in (α, 1 − α) and that, similarly, the solutions of type T 2n−1,j , j ∈ {s, a}, exhibit 2n − 1 local extrema there in. Also, note that the solutions of type T 2n and T 2n−1,a are asymmetric, whereas the solutions of type T 2n−1 and T 2n−1,s are symmetric.
Proof. For Part (b), one should take into account that, thanks to Theorem 4.2, m 0 ], and, consequently, u 0 must have a perturbed solution of type T 1 . Actually, the proof of Theorem 4.2 can be adapted to show that lim b↓b * τ 1, the end-points reached at x t (b) and m 0 , respectively. This information is needed in the proof of Part (d).
4.2. The case b < b * . Now, Ω > m 0 and the phase portrait of (17) looks like shows Figure 17 for all 0 < b < b * . Naturally, Figure 17 is reminiscent of Figure 9 and, consequently, we will not paraphrase its construction again. The main difference we can observe between the cases b > b * and b < b * is that the relative positions of the points x m0 , x t and m 0 have been interchanged, so that, in case b < b * , We already know that the solutions of (17) connecting Γ 0 with Γ 1 in a time 1−2α provide us with the solutions of (1), as described in Section 2. But, as the phase portrait changes when b separates away from b * , the time-maps τ n = τ n (·, b), n ≥ 1, also change with b. In order to understand what it is going on when 0 < b < b * we should first introduce an exhaustive list of Poincaré maps, much like in the previous section, which will provide us with all the solutions of (1). Figure 17. The phase portrait of (17) for 0 < b < b * For every b ∈ (0, b * ), we denote by the Poincaré map defined, for every x ∈ D 1 , as the minimal time needed by the solution of (20) to reach Γ 1 . Figure 18 shows the corresponding orbits of two of these solutions for some x < m 0 (A) and x > x t (B). Note that τ 1 (x t , b) is the minimal time needed by the orbit through (x t , y(x t )) to reach Γ 1 .
By continuous dependence, it is apparent that, for every b ∈ (0, b * ), since m 0 is not an equilibrium. In particular, τ 1 (·, b) admits a continuous extension to m 0 by setting τ 1 (m 0 , b) = 0. Next, we denote by x t x 0 Figure 19. The time-map (x, y(x)) → τ 1,s (x, b) for 0 < b < b * Figure 19 shows the orbit of one of these solutions with x ∈ (m 0 , x t ). Now, for every 0 < b < b * , we consider the map defined, for every x ∈ [m 0 , x t ], as the minimal time needed by the solution of (20) to reach Γ 1 for the first time (see Figure 20A), and, for every x ∈ (x t , x m0 ], as the minimal time needed by the solution of (20) to cross Γ 1 twice (see Figure 20B). Since lim it follows from (72) that Moreover, By construction, the solutions of τ 1,s = 1 − 2α and τ 1 = 1 − 2α provide us with symmetric solutions of (1), whereas the solutions given by τ 1,a = 1 − 2α are asymmetric, except at x = x t . All these solutions provide us with solutions of (1) with a single critical point in (α, 1 − α). Conversely, any solution of (1) with a single critical point in (α, 1 − α) must be of one of these forms. As in the previous sections, we also introduce the Poincaré map for every x ∈ D 2 (b) \ {m 0 }, as the minimal time needed by the solution of (20) to reach Γ 1 exactly twice, while τ 2 (m 0 , b) is the minimal time needed by the solution of (20) to reach Γ 1 . Figure 21 shows the corresponding orbits of two of these solutions for some x ∈ (x − 0 , m 0 ) (A) and x ∈ (x m0 , x + 0 ) (B). It should be x t x 0 Figure 21. The time-map (x, y(x)) → τ 2 (x, b) for 0 < b < b * observed that (1) cannot admit a solution of type T 2 passing through (x, y(x)) if By continuous dependence and symmetry, it is easily seen that a (m 0 , b), and, by symmetry, each of the solutions on Figure 21(A) must be a reflection of some solution in Figure 21(B). Once defined the time maps τ 1 (·, b), τ 1,a (·, b), τ 1,s (·, b) and τ 2 (·, b) in the whole interval 0 < b < b * , one can introduce mutatis mutandis, as in Section 4.1, the Poincaré maps (48) and (49). Similarly, one can extend the θ n 's introduced in (50) and (51) to the case 0 < b < b * by setting , for all n ≥ 1. These functions satisfy the following counterpart of Lemma 4.1, whose repetitive proof will be omitted here.
Theorem 4.6. The following assertions are true: . Naturally, thanks to (48) and (49), we also have that for all n ≥ 1.
According to these properties, the graphs of these functions, for 0 < b < b * should be of the type illustrated in Figure 22. Through the remaining of this paper we will use the concepts introduced in Definition 4.3 extended to 0 < b < b * . By simply looking at Figure 22 one can easily infer the next counterpart of Theorem 4.4 for b ∈ (0, b * ). Figure 22 collects the plots of the first seven Poincaré maps for an appropriate choice of the parameters with b = b * − 0.01. (a) Suppose τ 1 (Ω) > 1 − 2α. Then, there exists ε > 0 such that (1) has, at least, two solutions of type T 1 for each b ∈ (b * − ε, b * ). Moreover, the trivial solution u 0 perturbs into a solution of type T 1 as b separates away from b * . (b) Suppose τ 1 (Ω) < 1 − 2α < τ 2 (Ω). Then, there exists ε > 0 such that (1) has, at least, two solutions of type T 1 for every b ∈ (b * − ε, b * ). Moreover, the trivial solution u 0 perturbs, at least, into one solution of type T 1 as b separates away from b * . (c) Suppose (69) for some integer n ≥ 1. Then, there exists ε > 0 such that (1) possesses, at least, 2(2n + 1) solutions for each b ∈ (b * − ε, b * ). More precisely, (1) possesses, at least, two solutions of type T j , for each 2 ≤ j ≤ 2n + 1, one solution of type T 1 and an additional solution of type T 2n+1 . Moreover, the trivial solution u 0 perturbs, at least, into a solution of type T 2n+1 . (d) Suppose (70) for some integer n ≥ 1. Then, there exists ε > 0 such that (1) possesses, at least, 4n solutions for every b ∈ (b * − ε, b * ). More precisely, (1) possesses, at least, two solutions of type T j for every j ∈ {2, . . . , 2n}, one solution of type T 1 , and a solution of type T 2n−1,s . Moreover, the trivial solution u 0 perturbs into one solution of type T 2n−1,s . (e) Suppose (71) for some integer n ≥ 1. Then, there exists ε > 0 such that (1) possesses, at least, 4n solutions for every b ∈ (b * − ε, b * ). More precisely, (1) possesses, at least, two solutions of type T j for every j ∈ {2, . . . , 2n}, one solution of type T 1 , and a solution of type T 2n+1 . Moreover, the trivial solution u 0 perturbs into one solution of type T 2n+1 .

5.
Some general existence and non-existence results.
Lemma 5.1. Suppose that b ∈ (0, b m0 ) and (1) possesses a solution of type T n for some n ≥ 2. Then, (a) (1) has, at least, two solutions of type T j for all 2 ≤ j ≤ n, if n is even.
(b) (1) has, at least, two solutions of type T j for all 2 ≤ j ≤ n − 1, if n is odd. In both cases, (1) exhibits, at least, a solution of type T 1 .
Proof. Under the assumptions of the lemma, there exist (73) Consequently, thanks to Lemmas 4.1 and 4.5, the equation has, at least, two solutions for all 2 ≤ j ≤ n − 1 (see Figures 8, 15 and 22) and one solution for j = 1. If, in addition, n is even, then, by reflection and symmetry, the solutions of (73) must appear by pairs. This ends the proof. Proof. Remember that b m0 is the unique value of b > b * for which (m 0 , 0) belongs to the homoclinic trajectory through (0, 0) (cf. Figure 10B if necessary), and that b t > b m0 is the unique value of b for which this homoclinic is tangent to Γ 0 , and, hence, to Γ 1 (see Figure 10C). Suppose By simply looking at the phase portraits of (17) it is easily seen that (1) cannot admit any solution of type T 2 . Indeed, the exterior trajectories of the solutions of (20) look like shown in Figure  23A and, consequently, they might provide us, at most, with solutions of type T 1 with a single local maximum in (α, 1 − α). The interior trajectories are orbits of periodic solutions around Ω. Let u be denote the solution of (20) with P = (x, y(x)) ∈ Γ 0 . Its first critical point occurs at some time, say t 1 such that u(t 1 ) = x L . The second one at some further time t 2 > t 1 such that u(t 2 ) = x M . But (u(t), u (t)) cannot reach the curve Γ 1 without crossing again (x L , 0). Therefore, (1) cannot admit a solution of type T 2 . Consequently, it cannot admit any solution of even order, however it might admit solutions of type T 2n+1 , T 2n+1,s , and T 2n+1,a , for some integer n ≥ 0. The proof of the lemma in this case can be adapted mutatis mutandis to cover the case b = b m0 . Now, suppose b ≥ b t . Then, the proof of the lemma is straightforward, as a glance to the phase portrait reveals that (1) might only admit solutions of type  (1) admits a solution of type T n , n ≥ 1, then the problem must have solutions of type T j for all 1 ≤ j ≤ n provided 0 < b < b m0 . On the other, Lemma 5.2 establishes that, in case b m0 ≤ b < b t , (1) might have a solution of type T 2n+1 but not solutions of type T 2n . As a byproduct, the structure of the diagram of Poincaré maps given by Figure 22, if b  *  < b < b m0 , should change as b crosses b m0 . Actually, the graphs of the τ 2n 's in Figure 22 should disappear when b crosses b m0 . In order to realize what is going on, we will study all the possible solutions of (1) in case As in Section 4, we introduce the points x ± 0 = x ± 0 (b) and x t = x t (b) similarly, and define the time map , as the minimal time needed by the solution of (20) to reach Γ 1 . Figure 25 shows two trajectories corresponding to x ∈ (x − 0 , x t ) and x > m 0 .
Similarly, we introduce the map , defined, for every x ∈ D 1,s (b), as the time needed by the solution of (20) to reach Γ 1 exactly twice (see Figure 26). By continuous dependence, we have that Thus, τ 1,s may be regarded as a sort of continuous prolongation of τ 1 from (x − 0 , x t ] to the wider interval (x − 0 , x + 0 ). In addition, we introduce the map   Note that, by construction, x ∈ (x t , x + 0 ). Consequently, the relative positions of the graphs of these curves are in complete agreement with Figure 28.
Finally, for every integer n ≥ 1, we introduce the maps is the period of the solution of (20). According to Lemma 5.2, if the solution of (20) provides us with a solution of (1), necessarily Conversely, if x ∈ Σ, then, the solution of (20) provides us with a solution of (1). It should be noted that, by construction, we have that x ∈ (x t , x + 0 ), for all n ≥ 0. Consequently, the set of graphs of all these Poincaré maps looks like shown in Figure 28, which shows the plots of some of these time maps. To explain the transition from the set of τ n 's described in Figure 15 to the scheme shown in Figure 28, as b crosses the critical value b m0 , one should be aware that Actually, i.e., the domains of definition of the τ 2n 's shrink towards the end points of the intervals as b ↑ b m0 , which explains why such time maps cannot be defined for b ≥ b m0 . Actually, by continuous dependence, it is easily seen that for any sequences ε n > 0, n ≥ 1, such that lim n→∞ ε n = 0, and x n ∈ D 2 (b m0 − ε n ), n ≥ 1, one has that Therefore, the graphs of the τ 2n 's in Figure 15 grow up to infinity as b ↑ b m0 while their domains of definition shrink to the limiting points, x ± 0 (b m0 ). Consequently, letting b ↑ b m0 in Figure 15, we exclusively keep the local structures bifurcated when b separated away from b = b * , i.e., the left branch of τ 2n+1 , and the whole branches of τ 2n+1,s and τ 2n+1,a , for all n ≥ 0. As in Figure 15, τ 2n+1,s linked the left branch of τ 2n+1 with the right branch of τ 2(n+1)+1 , and τ 2n+1,a linked the left and the right branches of the τ 2n 's, it becomes apparent that indeed there is a continuous transition between all these time maps as b crosses b m0 , though, at first glance, the natures of Figures 15 and 28 might look so different. It should be remarked that such a transition still respects Lemma 4.1 as b crosses b m0 , because the τ 2n 's gradually shorten their supports around their edges. This allows us to overcome the apparent paradox between the results established by Lemmas 5.1 and 5.2.
In the light of this discussion, it becomes apparent how emphasizing the superlinear character of (1), by increasing the value of b, has the effect of magnifying dramatically the local effects emerged from Ω = m 0 as b separated away from b * .
Later, as b increases and crosses b t , the point x − 0 (b) increases, while x + 0 (b) decreases, until they meet at b = b t . Therefore, all the graphs shown in Figure 28 collapse as b crosses b t , except the right branch of τ 1 emerging from zero at m 0 . A careful analysis of the phase portrait of (17) as b crosses b t reveals that the graphs of the Poincaré maps τ 2n+1 , τ 2n+1,s and τ 2n+1,a , n ≥ 0, do actually blow up as The next result sharpens Lemma 5.2(b) by establishing that (1) cannot admit any solution for sufficiently large b.
Proof. Let x > m 0 , b ≥ b t , and denote by u the unique solution of (20). Then, arguing as in the last part of the proof of Theorem 3.3 and setting 1−α] u > m 0 , it becomes apparent that .
Thus, letting b ↑ ∞ in this estimate, we are led to Similarly, the following result holds.
Lemma 5.4. For sufficiently small b > 0, the problem (1) cannot admit a solution of type T n , nor of type T 2n−1,s , nor T 2n−1,a , with n ≥ 2.
Proof. On the contrary, suppose that (1) admits one of such solutions. Then, there are t 1 , t 2 ∈ [α, 1 − α], with t 1 < t 2 , such that with u (t) > 0 for all t ∈ (t 1 , t 2 ). Thus, setting v = u , we have that By looking at the phase portrait for the range 0 < b < b * , it should be noted that u(t) must be periodic and that Thus, , and, hence, Moreover, u 2 u 1 > Ω(b) m 0 and, consequently, we find that .
As lim b↓0 the estimate (74) cannot be satisfied for small b > 0. This concludes the proof.
The proof of Lemma 5.4 does actually entail that and that the period of any periodic solution around Ω blows up to ∞ as b ↓ 0. On the other hand, we already know that Consequently, it seems a reasonable conjecture that It should be noted that, since if condition (76) holds, then Consequently, the solutions U [b] bifurcate from infinity at b = 0.
Proof. Throughout this proof it is appropriate to denote to emphasize the dependence of the weight function a on the parameter b ≥ 0. According to Theorem 4.4 of [18], the problem (1)  Subsequently, we consider the nonlinear differential operator where ω > −π 2 is fixed. Naturally, the change of variable establishes a bijection between the solutions of (1) and the zeros of F. By construction, we have that Moreover, differentiating with respect to v leads to 6. Global bifurcation diagrams. The main goal of this section is to ascertain all the possible global bifurcation diagrams of the set of solutions of (1) regarding b ≥ 0 as the main bifurcation parameter. It turns out that the structure of these global diagrams becomes more and more complex the more negative the secondary parameter λ is, in complete agreement with the multiplicity results found in Section 3.
In the light of the analysis already done in the previous sections, it becomes apparent that, as a general tendency, the τ n (·, b)'s decrease with b for b < b * , whereas they are increasing functions of b when b > b * ; independently on whether the conjecture (76) holds or not. Indeed, by construction, (75) implies that τ n (·, b), n ≥ 3, must grow as b ↓ 0 and adjust to the profiles of the τ n 's constructed in Section 3 at the critical value of the parameter b = b * , which plays the role of a sort of organizing center in our mathematical analysis. Similarly, thanks to Lemmas 5.3 and 5.4, the τ n 's must grow as b > b * separates away from b * , except for the right branch of τ 1 which approximates zero uniformly as b ↑ ∞. Incidentally, this does not mean that all these functions will be pointwise monotonic in b. As a matter of fact, our numerical computations show that they are not.
At a first instance, in order to construct the global bifurcation diagrams of (1), it might be of great help to assume that all the graphs of the Poincaré maps τ n , τ 2n−1,a , τ 2n−1,s , n ≥ 1, are well-shaped, as those already represented in Figures 15  and 22. As those shown in these figures, and some others that we have computed but not included here, adjust to this property, there is no serious reason to doubt that, in general, they will have a similar shape. Anyway, even if in the general case this is not true, the lack of such assumption would not change substantially the topological nature of the global bifurcation diagrams, and the readers should be able to implement very easily by themselves all the necessary changes in the forthcoming discussion.
Throughout this section we assume conjecture (76). Consequently, according to Lemmas 5.4 and 5.5, for small b > 0 all the global bifurcation diagrams consist of two curves consisting of solutions of type T 1 : one emanating from the unique solution of (1) for b = 0, and the other one bifurcating from infinity at b = 0.
6.1. The case τ 1 (Ω) < 1 − 2α < τ 2 (Ω). Throughout this section we assume that According to Theorems 3.4, 4.4(b) and 4.7(b), the local bifurcation diagram around b * looks like shown in Figure 29. The trivial solution, u 0 , exhibits a bilateral bifurcation to one solution of type T 1 . In Figure 29, as in the remaining global bifurcation diagrams of this section, we are representing the value of u(α) versus the parameter b. It should be remembered that (u(α), y(u(α))) ∈ Γ 0 . Although in Figure 29, and in all subsequent bifurcation diagrams, we are representing a bifurcation from infinity at b = 0, it should be noted that (77) does not necessarily entail lim Actually, because of the superlinear character of the boundary value problem (1), the solution u(t) := u(t; b) might exhibit a spike-like behavior around 1/2 with u(α, b), b ∼ 0, bounded. In such cases, we are representing such a bifurcation from infinity as an abuse of notation. Figure 29 provides us with the same bifurcation diagram recently found by J. García-Melián [9] for a general multidimensional prototype of (1) with λ = 0 and M = ∞, which was already suggested by the mathematical analysis of J. López-Gómez [17].
6.2. The case τ 2 (Ω) < 1 − 2α < τ (Ω). Throughout this section we assume that A significant difference with respect to the previous case is that now the trivial solution, u 0 , exhibits a bilateral bifurcation to a solution of type T 1,s . By simply having a look at Figures 15 and 22 it is easily realized that these solutions will be defined until we reach the first values of b where min τ 1,s (·, b) = 1 − 2α, which has been marked with a thick dot on the bifurcation diagram. In the case represented, there are two such values of b. Beyond, the solutions become of type one. Another significant difference is that, under condition (79), the problem (1) with b = b * possesses two additional solutions of type T 2 , which can be globally pathfollowed in the parameter b until At these two values of b, the solutions of type T 2 become solutions of type T 1,a until they further meet the branch of solutions of type T 1,s . Figure 30 shows an admissible global bifurcation diagram under condition (79). In each of these values two different situations might occur. Either, or not. When condition (80) holds, the three solution curves meet at the solution u determined by the unique x such that which has been the situation illustrated by Figure 30 at b + . If not, the branch of solutions of type T 1,a must exhibit a turning point besides its crossing point with the branch of solutions of type T 1,s , which has been the situation illustrated by Figure 30 at b = b − . The remaining features of the diagram can be easily inferred in the light of the results of Section 5.
6.3. The case τ (Ω) < 1 − 2α < τ 3 (Ω). Figure 31 shows an admissible global bifurcation diagram in case The unique significant difference with respect to the diagram of Figure 30 is that, in the considered situation, according to Theorems 4.4(e) and 4.7(e), the trivial solution u 0 perturbs, as b separates away from b * , into a solution of type T 3 , instead of type T 1,s as it happened under condition (79). Further, these solutions of type T 3 become solutions of type T 1,s as |b − b * | increases, and beyond the situation evolves as in the previous case.
6.4. The case τ 3 (Ω) < 1 − 2α < τ 4 (Ω). Now, we assume that As in the previous cases, a careful analysis based on Figures 15 and 22 and the general properties of Section 5 reveals that an admissible global bifurcation diagram is the one sketched in Figure 32. As already predicted by Theorem 3.5, the problem (1) has, at least, six solutions for b = b * . Among them, two of type T 3 , two of type T 2 , and one of type T 1 , besides Basically, much like in Figure 31, the diagram consists of two main curves. One filled in by solutions of odd type, joining to infinity the solution of the sublinear problem associated to (1) by switching to zero the parameter b, plus a sort of close loop filled in by asymmetric solutions, which bifurcates from the odd curve at some points on the arcs of solutions of type T 1,s , and behaves globally like the loop shown in Figure 31. Along the odd curve, the trivial solution perturbs into solutions of type T 3 , which further switch to solutions of type T 1,s before reaching, eventually, the solution arcs consisting of solutions of type T 1 . Note that, in any circumstances, m 0 = u 0 (α) is the value of u(α) where solutions of type T 3 become of type T 1,s . So, these transition points are at the same level in Figure 32.
A careful comparison between the structure of the global bifurcation diagrams of Figures 31 and 32 reveals that they are really very similar, in the sense that we are going to explain now. Indeed, under condition (81), as τ 3 (Ω) decreases crossing the value 1 − 2α, there is some critical value of the parameters where the trivial solution u 0 must perturb into three solutions of type T 3 . This entails that the structure of the bifurcation diagram in a neighborhood of u 0 is S-shaped for slightly perturbed values of the parameters. Initially, the S is small, as it perturbs from u 0 , but, as τ 3 (Ω) separates away from 1 − 2α, the size of the S gradually grows until it reaches a significant size, which has been the situation illustrated in Figure 32.
The structure of the solutions on the diagram fits the theoretical analytical results of Section 5, of course. In particular, for every b ∈ (0, b * ) the problem has a solution of type T 2 whenever it has a solution of type T 3 , however in Figure 32 there are some values of b > b * where the model has some solution of type T 3 and no solution of type T 2 . According to Lemma 5.2, necessarily b ≥ b m0 . 6.5. The case τ 4 (Ω) < 1 − 2α < 2τ (Ω). When a further loop of higher order asymmetric solutions emerges from the solutions of type T 4 of (1) at b = b * . These solutions must bifurcate from the solutions of type T 3,s perturbed from the trivial solution. An admissible bifurcation diagram following these patterns has been represented in Figure 33. 6.6. The case 2τ (Ω) < 1 − 2α < τ 5 (Ω). When the only difference with respect to the previous case is the fact that the trivial solution perturbs into solutions of type T 5 , instead of solutions of type T 3,s , as b moves away from b * . These solutions of type T 5 further become of type T 3,s as b reaches some critical values. Figure 34 shows an admissible bifurcation diagram. 6.7. The case τ 5 (Ω) < 1 − 2α < τ 6 (Ω). When the main difference with respect to the previous case is that the odd curve exhibits an additional wind emerged from the trivial solution as τ 5 (Ω) crossed 1−2α. The other features of the bifurcation diagram can be explained reasoning as in the previous cases.
At this step, it should be rather apparent how to get all the admissible global bifurcation diagrams using b as the main parameter as the value of the secondary parameter λ becomes more and more negative.
do actually possess similar properties to those of Γ 0,M and Γ 1,M , and play an analogous role in the construction of the solutions of (1) carried out in Sections 3, 4, 5 and 6. According to Theorem 2.3, Γ 0,M is the graph of a smooth function denoted by y(x). In this section, it is appropriate to rename this function by y M and set Although most of the proofs in this section follow the general patterns of the proofs of Section 2, by the sake of completeness, we will detail their main parts. We begin our analysis with the following pivotal proposition which will provide us with Γ 0,∞ in a compact way, without passing to the limit as M ↑ ∞.
Proposition 5. Suppose λ ≤ 0. Then, the following properties hold: (a) For every x ≥ 0, the singular boundary value problem has a unique positive solution.
Throughout the rest of this section, we will consider the curve Although (86) holds, the proof is postponed.
Proof. First, we will prove the existence and the uniqueness of m 0,∞ . Let t ∞ min (x) denote the time needed by a solution of (6) with u(0) = ∞ to reach the u-axis in the phase plane (see Figure 2) at the point x, i.e., the necessary time to attain its minimum value x. Then, x p−1 (θ p+1 − 1) and, hence, t ∞ min is decreasing and it satisfies lim Therefore, there exists a unique value of x > 0 for which t ∞ min (x) = α. Let us denote it by m 0,∞ . Due to the monotonicity of t ∞ min , we have that Note that t ∞ min (0) = ∞. This proves the first sentence of Part (b). To show that (88) admits a unique solution, we proceed by contradiction. Suppose it admits two solutions, u 1 = u 2 . Then, by the uniqueness of the Cauchy problem at t = α, we infer that u 1 (α) = u 2 (α), but this contradicts the uniqueness of m 0,∞ and completes the proof of Part (b).
Obviously, T ∞ (v) is increasing in (−∞, 0) and, due to (89), it satisfies Therefore, there exists a unique v < 0 such that T ∞ (v) = α, which completes the proof of Part (a) in this case. Suppose x > m 0,∞ and let v u (x) > 0 denote the intersection between the unstable manifold passing through (0, 0) in the phase plane of (6) and the straight line u = x. For every v ∈ (0, v u (x)), let (m(v), 0) be the crossing point between the orbit through (x, v) and the u-axis. With these notations, it is easy to realize that the backward blow-up time of the solution of (90) is given by ) is the necessary time to reach x = u(α) from the minimum m(v). According to (89), we find that lim v↓0 T ∞ (v) = t ∞ min (x) < α.
Moreover, by continuous dependence, which implies the existence of a v(x) ∈ (0, v u (x)) such that T ∞ (v(x)) = α. To show the uniqueness and, hence, complete the proof of Part (a), we will prove the monotonicity of T ∞ (v). As m(v) is decreasing in (0, v u (x)), t ∞ min (m(v)) is increasing. Moreover, , and, consequently, t x (m(v)) is increasing. Therefore, we conclude that T ∞ (v) is also increasing when x > m 0,∞ . This completes the proof of Part (a). Part (c) is a straightforward consequence of the construction that we have just carried out in this proof.
The next two results are corollaries from Proposition 5 and its proof. Then, u (α) > y ∞ (x).
The next result establishes the monotonicity in M of the functions y M (x) constructed in Theorem 2.3 to parameterize the curves Γ 0,M . We want to prove that, under these conditions, y M1 (x) = u 1 (α) > u 2 (α) = y M2 (x).
As an immediate consequence of these results, we find the next one. In particular, the first relation of (86) holds. Moreover, y ∞ ∈ C 1 [0, m 0,∞ ) and it satisfies the implicit relation .
As G ∞ (x, ·) is (strictly) increasing for y ≤ 0, necessarilỹ This ends the proof of the first assertion. The regularity of y ∞ in [0, m 0,∞ ) is a straightforward consequence of the implicit function theorem applied to the equation G ∞ (x, y ∞ (x)) = α, 0 ≤ x < m 0,∞ , based on the fact that because y ∞ (x) < 0 for all x ∈ [0, m 0,∞ ). The proof is complete.
We need a last preliminary result about the positive blow-up solutions of (6). Proof. The unique delicate point is the monotonicity of T . To show it we will proceed by contradiction. Suppose that there are 0 ≤ x 1 < x 2 such that T (x 1 ) > T (x 2 ) and let u i , i = 1, 2, denote the unique solution of −u = λu − cu p u(0) = x i , u (0) = v.
The last result we need in order to cover the general case when M ≤ ∞ is the next counterpart of Theorem 2.3 for M = ∞.
The differential of the Poincaré map P α−ε is given by (DS(α), d dt DS(α)), where DS(t) is the solution of and u is the solution of (93). Obviously, the same device used in the proof of Theorem 2.3 shows that both components of DP α−ε are strictly positive. Consequently, Γ 0,∞ is the graph of an increasing C 1 -function. Therefore, y ∞ must be increasing and of class C 1 . The proof is complete.
Naturally, the change of variablest = 1 − t provides us with the counterparts of the previous results in the interval [1 − α, 1]. As in Section 2, Γ 1,∞ must be the reflection around the u−axis of Γ 0,∞ . Remark 1. (a) Combining the results of this section with the techniques of Sections 3-6, it is easily realized that all the results for M < ∞ found in this paper hold for M = ∞ too, except for the local perturbation result established by Lemma 5.5, which can be proven following the general patterns of M. Bertsch and R. Rostamian [5]. J. López-Gómez [17] and J. García-Melián [9], though the proof, being rather technical and outside the scope of this work, will appear elsewhere. (b) Throughout all this section we have used that some integrals, like the one defining t ∞ min (x) in the proof of Proposition 5, are finite and tend to 0 as x ↑ ∞. This is a very special case of the so-called Keller-Osserman condition, which allows us to generalize, substantially, our results by dealing with more general classes of nonlinearities, but this is far from being our goal in this work. So, we stop our analysis here.