1. Introduction
In this paper we analyze the nodal solutions of the onedimensional nonlinear weighted boundary value problem
(1.1) 
where are functions that change sign in and are regarded as bifurcation parameters. More precisely, is the primary parameter, and the secondary one. All the numerical experiments carried out in this paper have been implemented in the special case when
(1.2) 
because this is the weight function considered by LópezGómez and MolinaMeyer in [28] to compute the global bifurcation diagrams of positive solutions there in. In this paper we pay a very special attention to the particular, but very interesting, case when
for some integer .
Up to the best of our knowledge, this is the first paper where the problem of the existence and the structure of the nodal solutions of a weighted superlinear indefinite problem is addressed when changes of sign. The existence results of large solutions of Mawhin, Papini and Zanolin [39] required , as well as the results of LópezGómez, Tellini and Zanolin [35], where the attention was focused on the problem of ascertaining the structure of the set of positive solutions. Most of the available results on nodal solutions dealt with the special cases when , and is a positive function with (see Rabinowitz [40, 41, 42]), or with the degenerate case when is a continuous positive function such that (see LópezGómez and Rabinowitz [36, 37, 38], and LópezGómez, MolinaMeyer and Rabinowitz [33]). In strong contrast with the classical cases when , in the degenerate case when with the set of nodal solutions might consist of two, or even more, components, depending on the nature of the weight function (see [33] and [38] for any further required details). Nevertheless, as for the special choice given by (1.2), is negative in the intervals and , while it is positive in the central interval , this is the first time that the problem of analyzing the structure of the nodal solutions in this type of superlinear indefinite problems is addressed.
A natural strategy for constructing the nodal solutions of (1.1) with interior zeroes, or nodes, consists in linearizing (1.1) at the trivial solution,
, and then searching for the eigenvalues of the linearization having an associated eigenfunction with exactly
interior nodes in , for as these values of the parameters will provide us, through the local bifurcation theorem of Crandall and Rabinowitz [12], with all the small nodal solutions of (1.1) bifurcating from . This strategy provides us in a rather natural way with the linear weighted eigenvalue problem(1.3) 
By the Sturm–Liouville theory, the problem (1.3) has a sequence of eigenvalues
which are algebraically simple. Moreover, associated with each of them there is an eigenfunction, , with , unique up to a multiplicative constant, with exactly interior nodes, necessarily simple, in . By uniqueness,
(1.4) 
It turns out that the set of all the possible bifurcation points from to solutions of (1.1) with interior zeroes are provided by the values of and for which
So, the huge interest in analyzing them. Throughout this paper, we will denote
(1.5) 
Then,
and for all . Based on a classical result of Kato [21] on perturbation from simple eigenvalues, for every , is analytic in . A proof of this can be easily accomplished from [26, Ch. 9] and Section 5 of Antón and LópezGómez [2], where the result was established when . An extremely important property of is its strict concavity with respect to the parameter (see Berestycki, Nirenberg and Varadhan [5], CanoCasanova and LópezGómez [10] and Chapter 9 of [26]). According to it, for all , , for all , and
(1.6) 
Since , this property entails that, for every , consists of two values of ,
which are the unique bifurcation values to positive solutions from of (1.1) (see LópezGómez and MolinaMeyer [28]). Even dealing with general second order elliptic operators under general mixed boundary conditions of nonclassical type, the strict concavity of relies on the strong ellipticity of the elliptic operator (see, e.g., Chapter 8 of [26]).
For analytic semigroups the spectral mapping theorem holds (see, e.g., [3, 4]), i.e,
Thus, the spectral radius of the associated semigroup is given through the formula
Hence, is logarithmically convex, which is a classical property going back to Kato [20], because is convex. Rather astonishingly, there are examples of weight functions for which none of the remaining eigenvalues , , is concave with respect to . Figure 1 shows one of these examples for the special choice .
In this case, is the unique eigencurve which is concave, for as the remaining ones, , , are far from concave. Indeed, all of them are symmetric functions of , with a quadratic local minimum at , as illustrated by Figure 1. This fact has dramatic implications from the point of view of the structure of the set of nodal solutions of the problem (1.1). Indeed, setting
(1.7) 
it becomes apparent that for all and, hence, for every and any , consists of two negative eigenvalues, , and two positive eigenvalues such that
Therefore, for this range of ’s we expect that the solutions with interior nodes of (1.1) will bifurcate from the trivial solution at each of the four values
By simply having a look at Figure 1, it is easily realized that
Moreover,
at least for .
As illustrated by Figure 2, the number of eigencurves, , , which are concave in might vary with the weight function . Indeed, when , it turns out that not only but also is strictly concave, while the remaining eigencurves, , with , are not concave. Similarly, when , then are concave for , while they are not concave for .
Quite astonishingly, as suggested by our numerical computations, the more wiggled is the higher number of modes is concave. This astonishing feature might have some important implications in quantum mechanics.
The distribution of this paper is as follows. Section 2 studies some global properties of the eigencurves for all and analyzes their concavities in the special case when, for some ,
(1.8) 
Section 3 provides us some global bifurcation diagramas of nodal solutions of (1.1) with one and two interior nodes, which are superimposed to the global bifurcation diagrams of positive solutions of LópezGómez and MolinaMeyer [28]. Finally, in Section 4 we describe, very shortly, the numericical schemes used to get the global bifurcation diagrams of Section 3.
2. Some global properties of the nodal eigencurves
Throughout this paper, for any given with and every continuous function , we denote by , , the th eigenvalue of the eigenvalue problem
(2.1) 
The next properties are well known (see, e.g., [9]):

Monotonicity of with respect to : If satisfy , then

Monotonicity of with respect to the interval: If , then
Based on these properties, as suggested by Figures 1 and 2, the next result holds.
Proposition 2.1.
Suppose that there exist such that , i.e., changes the sign in . Then, for every ,
(2.2) 
Proof.
Consider a sufficiently small such that
Then, by the monotonicity properties of , for every and , we have that
Thus, letting , the second relation of (2.2) holds. The first one follows by applying this result to the weight function . This ends the proof. ∎
The fact that all the eigencurves plotted in Figures 1 and 2 are symmetric about the ordinate axis is a direct consequence of the next general result, because
for all integer and .
Proposition 2.2.
Suppose that is a continuous function in such that
(2.3) 
this holds under condition . Then, for all and any integer . In particular,
(2.4) 
where we are denoting .
Proof.
Since , either there exists such that , or for some . Suppose the first alternative occurs. Then, by (2.3), we also have that
and hence, changes the sign in . In particular, (2.2) holds.
Pick an integer , a real number , and let be an eigenfunction associated to . Then, possesses zeros in , , and
for all . Thus, setting
it is easily seen that
and hence, for every ,
Consequently, is an eigenfunction associated to with interior zeros. Therefore, by the uniqueness of , it becomes apparent that
Since is an analytic function of , necessarily . This ends the proof. ∎
By having a glance at Figure 3, it is easily realized that the function might not be an even function of if condition (2.3) fails.
The next result establishes that, as already suggested by Figures 1 and 2, the nodal eigencurves, , cannot be concave for the choice (1.8) if . We conjecture that, in general, for that particular choice, is concave if . Therefore, should be concave if, and only if, . But the analysis of the concavity when for the choice (1.8) remains outside the general scope of this paper.
Theorem 2.1.
Proof.
Since is algebraically simple for all , we already know that is analytic, by some well known perturbation results of Kato [20]. Moreover, the eigenfunction associated to , denoted by , can be chosen to be analytic in by normalizing it so that
(2.6) 
By definition, and
(2.7) 
Thus, since , particularizing (2.7) at and taking into account (2.6), it becomes apparent that actually is an analytic perturbation of the eigenfunction
Moreover, differentiating (2.7) with respect to yields
(2.8) 
Thus, since and , particularizing (2.8) at shows that solves the problem
(2.9) 
In order to find out , we first determine the general solution of the linear inhomogeneous equation
(2.10) 
To get it, we will set in order to vary coefficients in the first order system associated to (2.10),
(2.11) 
Since
is a fundamental matrix of solutions for the homogeneous linear system associated to (2.11), the change of variable
transforms (2.11) into the equivalent system
whose solution, according to Cramer rule, is given through
Thus,
for some constants . Therefore, the general solution of (2.10) is given by
where
(2.12) 
is a particular solution of (2.10). It is the solution obtained by making the choice . Obviously, . Moreover, by (1.8),
because the integrand,
satisfies for all
and hence, it is odd about
. As we are interested in solving (2.9), we should make the choiceThus,
for some constant . To determine , we can proceed as follows. Differentiating (2.6) with respect to and particularizing the resulting identity at yields
Consequently,
and therefore,
(2.13) 
To find out , we can differentiate with respect to the identity (2.8). After rearranging terms, this provides us with the identity
Thus, particularizing at yields
(2.14) 
and hence, multiplying (2.14) by and integrating in it is apparent that
(2.15) 
Therefore, substituting (2.13) into (2.15) and using (1.8) yields
Finally, we need the trigonometric formulas
(2.16)  
(2.17) 
to simplify the integrands arising in integrals of . First, we will ascertain the function . For this, we use the formula (2.16) on and then the formula (2.17) to simplify the integrand in . Then, integrating yields
(2.18) 
After substituting (2.18) into the formula for , we can again use the formulas (2.16) and (2.17) to simplify the underlying integrands, which can then be directly integrated. The result can be simplified to get the final formula
Obviously, if , and therefore . Hence, the eigencurves for are convex in a neighborhood of and thus they cannot be globally concave. ∎
3. Global bifurcation of nodal solutions
Since , for every the set consists of two points,
such that
Moreover, owing to Theorem 9.4 of [26],
Thus, by the main theorem of Crandall and Rabinowitz [12] (one can see also Chapter 2 of [25]), are the unique bifurcation values of to positive solutions of (1.1) from . The first plot of Figure 1 of LópezGómez and MolinaMeyer [28] shows one of those bifurcation diagrams for the special choice (1.2) of with
(3.1) 
Trying to complement the numerical experiments of [28] with our new findings here, all the numerical experiments of this section has been carried out for this special choice of . As grows up to reach the critical value , the set of positive solutions of (1.1) bifurcating from consists of one single closed loop bifurcating from at the single point . These loops, separated away from , are persistent for a large range of values of , until they shrink to a single point before disappearing at some critical value of the parameter (see [28, Fig. 1]).
According to Theorem 2.1, is not concave if (3.1) holds, which is clearly illustrated by simply looking at the plot of superimposed in Figure 1. This feature has important implications concerning the structure of the set of 1node solutions of (1.1). Indeed, according to the plot of , for every , the set consists of two single values with and . Thus, according to [26, Th. 9.4], the transversality condition of Crandall and Rabinowitz [12] holds at . Thus, an analytic curve of 1node solutions of (1.1) emanates from at each of these values of , . Figure 3(a) shows the plots of these two curves for the value of the parameter . Our numerical experiments suggest that they are separated away from each other. In this bifurcation diagram, as well as in all the remaining ones, we are representing the values of the parameter , in abscisas, versus the norm of the computed solutions, in ordinates. So, each point on the curves of the bifurcation diagrams, , represent a value of and a nodal solution of (1.1) for that particular value of .
When grows up to reach the critical value , the two previous components become closer and closer until they meet at at , where the set of bifurcation points to 1node solutions from consists of the points plus . This is the situation sketched by Figure 3(b), where we have plotted the global bifurcation diagram computed for
When , where is given by (1.7), the set consists of four values: two negative, , plus two positive, . Moreover, by Proposition 2.2, it is apparent that
Furthermore, as suggested by our numerical experiments,
Thus, again the transversality condition of [12] holds at each of these critical values of the parameter . Therefore, (1.1) should possess four analytic curves filled in by 1node solutions bifurcating from at each of these critical values of the parameter . Figure 3(c) shows the global bifurcation diagram of 1node solutions bifurcating from these four bifurcation points that we have computed for the choice . Once again, the set of 1node solutions consists of two components.
Actually, as soon as the transversality condition of Crandall and Rabinowitz [12] holds, the generalized algebraic multiplicity of Esquinas and LópezGómez [15, 25], , equals 1 and hence, thanks to Theorem 5.6.2 of LópezGómez [25], the Leray–Schauder index of , as a solution of (1.1), changes as crosses each of these values. Therefore, each of the components of the set of nontrivial solutions of (1.1) emanating from at each of these critical values of the primary parameter satisfies the global alternative of Rabinowitz [40], i.e., either it is unbounded in
, or it meets the trivial solution in, at least, two of these singular values.
Each of the two components plotted in Figure 3(c) bifurcates from two different points of and, according to our numerical experiments, both seem to be unbounded. The problem of ascertaining their precise global behavior remains open in this paper. As increases and crosses the critical value , these two components abandone the trivial curve and stay separated away from the trivial solution. So, they became isolas. Figure 3(d) shows the plots of these components for the choice . In Figure 5 we have plotted some distinguished solutions with 1node along some of the pieces of the global bifurcation diagrams already plotted in Figure 4. Precisely, Figure 4(b) shows a series of solutions with one node along the bifurcation diagram plotted on Figure 4(a), which is a magnification of a piece of the left component of Figure 3(a), and Figure 4(d) shows a series of solutions with one node along the bifurcation diagram plotted in Figure 4(c), which is a magnification of a piece of the left component plotted in Figure 3(d). The colors of each of these 1node solutions corresponds with the color of the piece of the bifurcation diagram on the left where they are coming from.
Similarly, according to Theorem 2.1, for the special choice (3.1), the third eigencurve, , is far from concave if (3.1) holds. This becomes apparent by simply having a look at the plot of superimposed in Figure 1. According to it, for every , the set consists of two negative eigenvalues, , plus two positive eigenvalues, . Moreover, by Proposition 2.2,
and, according to our numerical experiments,
Thus, the transversality condition of [12] holds at each of these critical values. Therefore, owing to the local bifurcation theorem of [12], an analytic curve of 2node solutions emanates from at each of these four singular values of . The first three plots of Figure 6 show these curves for three different values of the secondary parameter . Namely: , and , respectively. All these values of are bellow . The last plot of Figure 6 has been computed for and shows three components of 2node solutions separated away from . For this value of no solution with 2 interior nodes can bifurcate from .
More precisely, at the problem (1.1) possesses three components of solutions with two interior nodes. Two of them bifurcating from at and , respectively, and the third one linking with . According to our numerical experiments these components are unbounded in , and are persistent for all further value of bellow some critical value, , where the three components meet. Thus, for there is a component of the set of nontrivial solutions of (1.1) bifurcating from at four different values of : and . The plot in Figure 5(b) shows the corresponding global bifurcation diagram for , a value of slightly greater than , where the three components of set of nontrivial solutions are very close. By comparison with the global bifurcation diagram for , it becomes apparent that a global imperfect bifurcation phenomenon has happened at the critical value . As a consequence of this imperfect bifurcation one of the components bifurcating from links with , another links with , while the third one remains separated away from . Actually, the latest one is separated away from zero for any further value of . Therefore, there have occurred a sort of reorganization in components of the set of 2node solutions of (1.1) as the parameter crossed the critical value . The pictures in Figures 5(c), 5(d) show the plots of the corresponding components for and , where the previous bifurcations from of these components are lost. For larger values of the solutions along these three components become larger and larger and it remains an open problem to ascertain whether, or not, (1.1) can admit some 2node solution for sufficiently large . Figure 7 shows the plots of some distinguished 2node solutions of (1.1) along some of the curves of the bifurcation diagrams plotted in Figure 6.
Finally, Figure 8 superimposes the global bifurcation diagrams of positive solutions found in [28] (in blue) with the global bifurcation diagrams of nodal solutions with one node (in red) and two nodes (in black) computed in this paper for four different values of : , , and . Although all the components of nodal solutions persist for these values of , the component of positive solutions shrinks to a single point and disappear at a value of above but very close to it. In Figure 7(b) one can still see an small piece of blue trace component shortly before disappearing for an slightly grater value of .
4. Numeric of bifurcation problems
To discretize (1.1) we have used two methods. To compute the small solutions bifurcating from we implemented a pseudospectral method combining a trigonometric spectral method with collocation at equidistant points, as in most of our previous numerical experiments (see, e.g., [17, 18, 27, 29, 30, 31, 32]). This gives high accuracy (see, e.g., Canuto, Hussaini, Quarteroni and Zang [11]). However, to compute the large solutions we have used a centered finite differences scheme, which gives high accuracy at a lower computational cost, for as it provides us with a much faster code to compute large pieces of curves of the global bifurcation diagrams.
The pseudospectral method is easier to use and more efficient for choosing the shot direction from the trivial solution in order to compute the small nodal solutions of (1.1), as well as to detect bifurcation points along the bifurcation diagrams. Its main advantage for accomplishing this task relies on the fact that it provides us with the true bifurcation values from the trivial solution, while the differences scheme only provides with an approximation to these bifurcation values.
For general Galerkin approximations, the local convergence of the solution paths at regular, turning and simple bifurcation points was proven by Brezzi, Rappaz and Raviart in [6, 7, 8] and by LópezGómez et al. in [27, 34] at codimension two singularities in the context of systems. In these situations, the local structure of the solution sets for the continuous and the discrete models are known to be equivalent. The global continuation solvers used to compute the solution curves of this papers, as well as the dimensions of the unstable manifolds of all the solutions along them, have been built from the theory on continuation methods of Allgower and Georg [1], Crouzeix and Rappaz [13], Eilbeck [16], Keller [22], LópezGómez [24] and LópezGómez, Eilbeck, Duncan and MolinaMeyer [27].
The complexity of the bifurcation diagrams, as well as their quantitative features, required an extremely careful control of all the steps in subroutines. This explains why the available commercial bifurcation packages, such as AUTO07P are unuseful to deal with differential equations with heterogeneous coefficients. As a matter of fact, Doedel and Oldeman admitted in [14, p.18] that
“Note that, given the nonadaptive spatial discretization, the computational procedure here is not appropriate for PDEs with solutions that rapidly vary in space, and care must be taken to recognize spurious solutions and bifurcations.”
This is just one of the main problems that we found in our numerical experiments, as the number of critical points of the solutions increases according to the dimensions of unstable manifolds, and the turning and bifurcation points might be very close.
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