# Rich dynamics in planar systems with heterogeneous nonnegative weights
††thanks: This paper has been written under the auspices of the Ministry of
Science, Technology and Universities of Spain, under Research Grants
PGC2018-097104-B-I00 and PID2021-123343NB-I00, and of the IMI of Complutense
University.

Julián López-Gómez
Universidad Complutense de Madrid
Instituto de Matemática Interdisciplinar (IMI)
Departamento de Análisis Matemático y Matemática Aplicada
Plaza de las Ciencias 3, 28040 Madrid, Spain
E-mail<EMAIL_ADDRESS>
Eduardo Muñoz-Hernández
Universidad Complutense de Madrid
Instituto de Matemática Interdisciplinar (IMI)
Departamento de Análisis Matemático y Matemática Aplicada
Plaza de las Ciencias 3, 28040 Madrid, Spain
E-mail<EMAIL_ADDRESS>
Fabio Zanolin
Università degli Studi di Udine
Dipartimento di Scienze Matematiche, Informatiche e Fisiche
Via delle Scienze 2016, 33100 Udine, Italy
E-mail<EMAIL_ADDRESS>
###### Abstract

This paper studies the global structure of the set of nodal solutions of a
generalized Sturm–Liouville boundary value problem associated to the
quasilinear equation

$-(\phi(u^{\prime}))^{\prime}=\lambda u+a(t)g(u),\quad\lambda\in{\mathbb{R}},$

where $a(t)$ is non-negative with some positive humps separated away by
intervals of degeneracy where $a\equiv 0$. When $\phi(s)=s$ this equation
includes a generalized prototype of a classical model going back to Moore and
Nehari [35], 1959. This is the first paper where the general case when
$\lambda\in\mathbb{R}$ has been addressed when $a\gneq 0$. The semilinear case
with $a\lneq 0$ has been recently treated by López-Gómez and Rabinowitz [28,
29, 30].

2010 Mathematics Subject Classification: 34B08, 34B24, 35C15.

Keywords and Phrases. Nonlinear differential equations, planar systems,
degenerate weights, Moore–Nehari equation, nodal solutions, global bifurcation
theory, a priori bounds, Poincaré maps.

##  1 Introduction

In this paper we investigate the nodal behavior of the solutions of the
quasilinear differential equation

$-(\phi(u^{\prime}))^{\prime}=\lambda u+a(t)g(u),\quad\lambda\in{\mathbb{R}},$
(1.1)

where:

  * •

$\phi:J_{1}\to J_{2}$ is an increasing homeomorphism defined on the open
interval $J_{1}$, with $-\infty\leq\inf J_{1}<0<\sup J_{1}\leq+\infty$, onto
the open interval $J_{2}$, with $-\infty\leq\inf J_{2}<0<\sup
J_{2}\leq+\infty$, and such that $\phi(0)=0$, as discussed by Bereanu and
Mawhin [2] and Manásevich and Mawhin [33];

  * •

$g:{\mathbb{R}}\to{\mathbb{R}}$ is a continuous function with $g(0)=0$ and
$g(s)s>0$ for $s\not=0$;

  * •

$a:I=[0,L]\to{\mathbb{R}}^{+}:=[0,+\infty)$ satisfies $a\in L^{1}(I)$.

Our main assumption concerning the weight function $a(t)$ is that the interval
$I$ splits into finitely many adjacent subintervals $I^{+}_{i}$, $I^{0}_{i}$,
such that $a\gneqq 0$ on $I^{+}_{i}$ and $a\equiv 0$ on $I^{0}_{i}$. More
precisely, we consider $I^{+}_{i}=[t_{i},s_{i}]$ and
$I^{0}_{i}=(s_{i},t_{i+1})$, where

$0=t_{0}<s_{0}<t_{1}<\dots<t_{i}<s_{i}<\dots<s_{m-1}<t_{m}<s_{m}=L.$ (1.2)

Thus, we assume to have $m+1$ intervals where the weight function $a(t)$ is
nontrivial, separated by $m$ intervals where $a\equiv 0$. Our main aim is to
prove the existence of multiple solutions to equation (1.1) with a prescribed,
arbitrarily large, number of oscillations in the intervals $I^{+}_{i}$ and a
“linear” behavior in the intervals $I^{0}_{i}$. The equation (1.1) is
complemented with either Dirichlet, Neumann, or mixed boundary conditions in
$[0,L]$. In the past decades a great deal of research has been devoted to the
so-called nonlinear problems with sign indefinite weight, in which the domain
of the function $a(t)$ splits into some intervals where $a>0$ and $a<0,$
respectively (see, for instance, [32] and the references therein for an
investigation on the multiplicity result in the case of the boundary condition
$u(0)=u(L)=M$). The equations studied in the present paper exhibit some
peculiar dynamical features which look analogous to the sign indefinite case
when $\lambda<0$ and are completely different for $\lambda\geq 0$; in
particular, our aim is to enlighten the role that the intervals of degeneracy
of the weight function have in producing new multiplicity results.

This analysis is motivated by some recent researches in semilinear boundary
value problems with nonnegative weights that are “degenerate” in some parts of
their domains. The interest in these problems goes back to the pioneering work
of Moore and Nehari [35], 1959, concerning the Dirichlet boundary value
problem

$\left\\{\begin{array}[]{l}-u^{\prime\prime}=a(t)u^{2n+1}\quad\hbox{in}\;\;[0,L],\\\
u(0)=u(L)=0,\end{array}\right.$ (1.3)

where $n\geq 1$ is an integer, $L=\alpha+1$ for some $\alpha>1$, $a(t)\equiv
1$ for $t\in[0,1]\cup[\alpha,\alpha+1]$ and $a(t)\equiv 0$ in $(1,\alpha)$.
Under these assumptions, Moore and Nehari [35] proved the existence of three
positive solutions for some values of $\alpha>1$. It turns out that non-
negative weight functions possessing zones of degeneracy in their domains are
interesting also in wide classes of reaction-diffusion equations arising in
Population Dynamics where the coefficients governing the interaction between
the species may vanish in some regions of their domains. The readers are sent
to López-Gómez and Sabina [31], the authors’s papers [21, 22, 24] and the
monograph [17], as well as to the lists of references therein, for some
paradigmatic examples where the weight functions vanish either in the spatial
or the temporal variables.

More recently, the existence of nodal solutions for the symmetric variant of
the Moore–Nehari problem

$\left\\{\begin{array}[]{l}-u^{\prime\prime}=a(t)|u|^{p-1}u\quad\hbox{in}\;\;[0,L],\\\
u(0)=u(L)=0,\end{array}\right.$ (1.4)

has been accomplished in Gritsans and Sadyrbaev [12] for $p>1$, referred to as
the superlinear case here, as well as in Kajikiya [13] for $0<p<1$, usually
refereed to as the sublinear case. In these works, the weight function $a(t)$
is defined on the symmetric interval $[-1,1]$ and it is a positive constant in
a neighborhood of $-1$ and $1$, while $a=0$ in the central interval. Solutions
with an arbitrarily large number of zeroes in the intervals where $a>0$ are
shown to exist according to the length of the interval $a^{-1}(0)$.

A different line of research had been already introduced by Rabinowitz [39,
42] in the early seventies, where the nonlinear eigenvalue problem

$\left\\{\begin{array}[]{l}-u^{\prime\prime}=\lambda
u-a(t)|u|^{p-1}u\quad\hbox{in}\;\;[0,L],\\\ u(0)=u(L)=0,\end{array}\right.$
(1.5)

was analyzed adopting the point of view of global bifurcation theory, settled
in the foundational papers of Rabinowitz [40, 41]. According to [39, 42], in
(1.5), $a(t)>0$ for all $t\in[0,L]$, and $p>1$. The degenerate problem where
the interior of $a^{-1}(0)$ is nonempty in (1.5) has been analyzed (for the
case $p>1$), much more recently, by López-Gómez and Rabinowitz [28, 29, 30],
where it has been established that the structure of the set of nodal solutions
can be far more intricate than in the classical case when $a(t)$ is everywhere
positive (see the numerical experiments of Molina-Meyer in [19]).

The first goal of this paper is adapting the perspective of global bifurcation
theory to analyze the structure of the set of non-trivial solutions, positive,
negative and nodal, of the semilinear boundary value problem

$\left\\{\begin{array}[]{l}-u^{\prime\prime}=\lambda
u+a(t)|u|^{p-1}u\quad\hbox{in}\;\;[0,L],\\\ u(0)=u(L)=0,\end{array}\right.$
(1.6)

in the general case when $a\in\mathcal{C}[t_{i},s_{i}]$ satisfies $a(t)>0$ for
all $t\in[t_{i},s_{i}]$ and $i\in\\{0,...,m\\}$, while $a=0$ in $I^{0}_{j}$
for all $j\in\\{0,...,m\\}$. Thus, there are $m+1$ intervals where the weight
function $a(t)$ is positive, separated away by $m$ intervals where $a\equiv
0$. In particular, $a(t)$ is far from being constant on the intervals
$I_{i}^{+}=[t_{i},s_{i}]$, $i\in\\{0,...,m\\}$. The existence of a priori
bounds for the problem (1.6) when $p>0$, $p\neq 1$, can be derived with the
blowing-up techniques of Amann and López-Gómez [1], going back to Gidas and
Spruck [11]. Our main findings can be packaged into the next result, where we
are denoting $\sigma_{n}:=\left(\frac{n\pi}{L}\right)^{2}$ for all integer
$n\geq 1$. Note that, by the oddity of the nonlinearity of (1.6) its solutions
arise by pairs: $(\lambda,u)$ and $(\lambda,-u)$. Moreover, any non-zero
solution, $(\lambda,u)$ with $u\neq 0$, has finitely many interior zeroes, or
nodes, in $(0,L)$, and any zero, $\tau\in[0,L]$, must be simple, i.e.,
$u^{\prime}(\tau)\neq 0$. In particular, either $u^{\prime}(0)>0$, or
$u^{\prime}(0)<0$, and, similarly, $u^{\prime}(L)\neq 0$.

######  Theorem 1.1.

Suppose $p>0$, $p\neq 1$. Then, for every integer $n\geq 1$, the problem (1.6)
has a solution with $n-1$ interior nodes if, and only if,
$\lambda<\sigma_{n}$. In such case, (1.6) has, at least, two solutions of this
type: One with $u^{\prime}(0)>0$ plus the opposite. Consequently:

  * •

For every $\lambda<\sigma_{1}$ and any integer $\kappa\geq 1$, (1.6)
possesses, at least, two solutions with $\kappa-1$ zeroes in $(0,L)$.

  * •

For every pair of integers $\kappa\geq n\geq 1$ and
$\lambda\in[\sigma_{n},\sigma_{n+1})$, (1.6) has, at least, two solutions with
$\kappa$ zeroes in $(0,L)$.

Figure 4 shows the minimal global bifurcation diagram of (1.6) in both cases:
$p>1$ and $p\in(0,1)$. When $p>1$, the nodal solutions with $n-1$ interior
nodes bifurcate from $u=0$ at $\lambda=\sigma_{n}$. When $p\in(0,1)$, they do
bifurcate from infinity at $\lambda=\sigma_{n}$. In both cases, these
bifurcations are subcritical. It turns out that, in the special case when
$a\equiv\mu>0$, by using some standard phase-portrait techniques, it is easily
seen that (1.6) admits, at most, two nodal solutions with $n-1$ interior nodes
for each integer $n\geq 1$. Naturally, as illustrated by the example of Moore
and Nehari [35] and the more recent works [12, 13], the number of nodal
solutions with a fixed number of nodes for a particular value of $\lambda$ can
vary with the value of $m$ and the length of the intervals $I^{+}_{i}$ and
$I^{0}_{i}$ of $a(t)$. Thus, besides the components plotted in Figure 4, (1.6)
might have one, or several, additional components that have not been plotted
in these (minimalist) global bifurcation diagrams.

Although some available global bifurcation theorems for
$\mathcal{C}^{1}$-Fredholm operators can be invoked to get some global results
concerning the existence of non-trivial solutions with a prescribed nodal
behavior for the quasilinear boundary value problem

$\left\\{\begin{array}[]{l}-(\phi(u^{\prime}))^{\prime}=\lambda
u+a(t)|u|^{p-1}u\quad\hbox{in}\;\;[0,L],\\\ u(0)=u(L)=0,\end{array}\right.$
(1.7)

it turns out that these results inherit, essentially, a local character, as in
many circumstances the classical solutions of (1.7) can develop singularities
as the solutions separate away from $u=0$ becoming somewhere larger. Indeed,
by making the special, but important, choice

$\phi(s):=\frac{s}{\sqrt{1+s^{2}}},\qquad s\in(-\infty,+\infty),$ (1.8)

with $p>1$, one can apply the global bifurcation theorem of López-Gómez and
Mora-Corral [20] and López-Gómez [18] to show the existence of a component of
solutions with $n-1$ interior zeroes bifurcating from $u=0$ at
$\lambda=\sigma_{n}$ for every integer $n\geq 1$. Alternatively, one can adapt
the global bifurcation theorem of López-Gómez and Omari [25], which provides
us with a global component of bounded variation positive solutions. However,
it is folklore that the classical solutions of (1.7), with the choice (1.8),
can develop singularities when they separate away from zero (see, eg., Cano-
Casanova et al. [5] and López-Gómez and Omari [27]). Thus, this methodology
does not seem the most appropriate to get multiplicity results of classical
nodal solutions of (1.7) without some additional, very serious, effort to show
the regularity of its non-zero solutions. Therefore, to accomplish our second
goal in this paper, which consists in studying the generalized quasilinear
problem (1.7), we have preferred here to adopt a different approach based on
planar shooting type methods. This allows us to study in a rather compact way
problems involving the $p$-Laplacian, the mean-curvature operator, or the
Minkowski operator (see, e.g., López-Gómez and Omari [26], Bereanu and Mawhin
[2, 3], and the references therein).

Precisely, to study the quasilinear equation (1.1) we will analyze the
dynamical system associated with the underlying system

$\begin{cases}x^{\prime}=h(y),\\\ y^{\prime}=-\lambda x-a(t)g(x),\end{cases}$
(1.9)

where

$h(y):=\phi^{-1}(y).$

By the definition of $\phi$, $h$ is a continuous and monotone increasing
function, defined on the open interval $J_{2}:=(\varrho_{-},\varrho_{+})$,
with $-\infty\leq\varrho_{-}<0<\varrho_{+}\leq+\infty$, and such that
$h(0)=0$. Solutions of (1.9) are intended in the Carathéodory sense and are
interpreted as solutions of (1.1) as well. Namely, $u(\cdot)$ is a solution
(1.1) if it is a $C^{1}$-function with $\phi(u^{\prime})$ absolutely
continuous and (1.1) is satisfied for almost every $t\in[0,L]$. In our
applications the weight function $a(t)$ will be chosen as a piecewise constant
function. Thus, the solutions $(x,y)$ of system (1.9) have $x$ of class
$C^{1}$, while $y$ is continuously differentiable except at the points where
$a(t)$ is discontinuous.

The analysis of (1.9) is of huge interest on its own, regardless whether or
not it is related to an equation like (1.1). Indeed, examples of (1.9) (with
$\lambda=0$) arise, rather naturally, in dealing with some important classes
of Lotka–Volterra predator–prey systems where seasonal interaction effects
play an important role (see the recent paper of the authors [23]). Actually,
analyzing the dynamics of (1.9) allows us to deal with a broad class of
boundary value problems of generalized Sturm–Liouville type related to the
quasilinear equation (1.1). These problems extend considerably the Dirichlet
problem (1.7). Our results can be applied, similarly, to deal with the
periodic problem, by assuming $a:{\mathbb{R}}\to{\mathbb{R}}$ a $T$-periodic
function with a certain number of intervals where $a(t)$ is positive separated
by intervals of degeneracy in $[0,T].$

The plan of this paper is the following. In Section 2 we analyze the
semilinear problem (1.6) in the general case when $a(t)$ is piecewise
continuous. In such case, our model can be viewed as a sort of generalized
Moore–Nehari prototype. The main findings of this section have been already
packaged in Theorem 1.1. According to it, for every $\lambda\in\mathbb{R}$,
the problem (1.6) admits infinitely many oscillatory solutions. We will come
back to consider this problem again in Section 7, where it will be used as the
simplest prototype model for testing the abstract results developed in
Sections 3–6. Naturally, the abstract results will provide us with finitely
many nodal solutions under the appropriate circumstances, because, at least
when dealing with the mean curvature operator, sufficiently large nodal
solutions will not provide us with classical solutions anymore. In some sense,
these highly oscillatory solutions are lost when $-D^{2}$ is inter-exchanged
by a quasilinear operator. Adopting this perspective the abstract results of
this paper for the dynamical system (1.9) are rather satisfactory.

In Section 3 we fix the class of weight functions considered in Sections 3–7.
For simplicity and following also previous works in this area, like those of
Moore and Nehari [35], Kajikiya [13] and Cubillos et al. [8], we confine
ourselves to deal with stepwise function coefficients, by taking
$a(t)\equiv\mu_{i}>0$ if $t_{i}\leq t\leq s_{i}$ for all $i\in\\{0,...,m\\}$.
From the method of the proofs, it will become apparent that our results are
stable with respect to sufficiently small perturbations of the coefficients in
the $L^{1}$-norm on $[0,L]$. For such choices of $a(t)$ it is natural to
describe the dynamics associated with the system (1.9) as the superposition of
those associated to the fully nonlinear system

$x^{\prime}=h(y),\quad y^{\prime}=-\lambda x-\mu_{i}g(x)$ (1.10)

coupled with the linear one in the second component

$x^{\prime}=h(y),\quad y^{\prime}=-\lambda x.$ (1.11)

For both of these systems we analyze their most significant arc-deformation
properties related to the corresponding Poincaré maps in Section 4. Recall
that the Poincaré map associated with system (1.9), on a time-interval
$[\tau_{0},\tau_{1}]$, is the application which maps a point
$p_{0}\in\mathbb{R}^{2}$ to the point
$p_{1}=(\hat{x}(\tau_{1}),\hat{y}(\tau_{1}))$, where
$(\hat{x}(\cdot),\hat{y}(\cdot))$ is the unique solution of (1.9) with initial
value $p_{0}$ at the time $\tau_{0}$. The uniqueness of the solution for the
associated Cauchy problems on the given intervals is implicitly assumed. By
the continuous dependence of the solutions with respect to the initial data,
the Poincaré map is a homeomorphism on its domain. For our choice of the
weight function, leading to a sequence of autonomous systems, each Poincaré
map on an interval $[t_{i},s_{i}]$ (resp. $[s_{i},t_{i+1}]$) is equivalent to
the map on $[0,\tau_{i}]$ with $\tau_{i}:=s_{i}-t_{i}$ (resp.
$[0,\varsigma_{i}]$ with $\varsigma_{i}:=t_{i+1}-s_{i}$). Thus, we can denote
by $\Phi_{i}$ and $\Psi_{i}$ the Poincaré maps associated with (1.10) and
(1.11) on the intervals of length $\tau_{i}$ and $\varsigma_{i}$,
respectively. Since we study (1.9) for every $\lambda\in\mathbb{R}$, the
system (1.11) exhibits different dynamics as $\lambda$ varies. Precisely, the
dynamics consists of a shift parallel to the $x$-axis when $\lambda=0$, like
in Moore and Nehari [35], a saddle point at the origin when $\lambda<0$, and a
center at the origin when $\lambda>0$. For this reason, we have to study
throughout this paper these three cases separately. We conclude Section 3 by
analyzing how acts the Poincaré map $\Psi_{i}$ according to the sign of
$\lambda$.

As already commented above, Section 4 analyzes the effect of the
transformation of certain arcs through the Poincaré maps $\Phi_{i}$,
$i\in\\{0,\ldots,m\\}$. Similarly, Section 5 focuses attention into the arc-
transformation properties of $\Psi_{i}$ and, hence, $\Psi_{i}\circ\Phi_{i}$.
These results are pivotal to get our main abstract results, delivered in
Section 6, concerning the multiplicity of nodal solutions for the generalized
Sturm–Liouville problem associated to (1.1). Essentially, in Section 6 we have
established that, for every $(n_{0},n_{1},...,n_{m})\in\mathbb{N}^{m+1}$, the
system (1.9) possesses a solution $(x,y)$ such that $x(t)$ has $n_{i}$ nodes
in the interval $[t_{i},s_{i}]$ for every $i\in\\{0,...,m\\}$. Naturally, the
number of zeroes of $x(t)$ in the vanishing intervals of $a(t)$,
$[s_{i},t_{i}+1]$, $i\in\\{1,...,m\\}$, where the system is linear in $x$,
depends on the sign of $\lambda$. The most delicate part the abstract analysis
in Section 6 is getting the appropriate twist conditions of the autonomous
flow on each of the intervals $[t_{i},s_{i}]$, as well as their balances with
respect to the lengths of the vanishing intervals of $a(t)$, measured by
$\varsigma_{i}:=t_{i+1}-s_{i}$, which should be sufficiently large to get a
maximal number of nodal solutions. Finally, in Section 7, we illustrate the
applicability of these results to the generalized Moore–Nehari model that we
have studied previously in Section 2. This is a good model for testing the
flexibility of the abstract results of Section 6, as it admits infinitely many
oscillatory solutions for every $\lambda\in\mathbb{R}$.

It should be emphasized that incorporating the parameter $\lambda$ to the
previous studies on the Moore–Nehari model adds a number of different
phenomenologies to the underlying theory that have not been previously
documented. In particular, we have found that, for $\lambda<0$ and
$t_{i+1}-s_{i}$ sufficiently large, we have more solutions than a priori
expected, presumably by the higher complexity of the Markov-type transition
diagram sketched in Figure 9, while, for $\lambda>0$, in order to get nodal
solutions, one must impose an additional compatibility condition requiring an
upper bound for the lengths of the vanishing intervals of $a(t)$, the
$\varsigma_{i}$’s. Some numerical examples establishing the compatibility of
these restrictions will be delivered in Section 7. As a matter of fact, by
simply having a glance at the bifurcation diagrams of our semilinear
Moore–Nehari model in Section 2, it is easily realized that the larger is
$\lambda>0$ the lower is the complexity of the solution set of (1.6). As
Kajikiya [13, 14] has used phase-portrait techniques in the special case when
$\lambda=0$, our results seem to be new even for the simplest semilinear
prototype model (1.6).

##  2 A pivotal paradigmatic semilinear problem

This section analyzes the structure of the set of solutions of the semilinear
problem

$\begin{cases}-u^{\prime\prime}=\lambda
u+a(t)|u|^{p-1}u\quad\hbox{in}\;\;(0,L),\\\\[3.0pt] u(0)=u(L)=0,\end{cases}$
(2.1)

where $L>0$, $p>0$ with $p\neq 1$, $a\gneq 0$, and $\lambda\in\mathbb{R}$ is
regarded as a bifurcation parameter. When $p=1$, (2.1) becomes a linear
problem. In such case, if we denote by

$\sigma(-D^{2}-a(t))=\Big{\\{}\sigma_{n}[-D^{2}-a(t)]\;:\;n\geq 1\Big{\\}}$

the spectrum of $-D^{2}-a(t)$ in $(0,L)$ under Dirichlet boundary conditions,
then, the solution set of (2.1) consists of the trivial solutions,
$(\lambda,0)$ with $\lambda\in\mathbb{R}$, plus a numerable union of straight
lines,

$\bigcup_{n\geq
1}\Big{\\{}\left(\sigma_{n}[-D^{2}-a(t)],u\right)\;:\;u\in\hbox{Ker\,}\Big{(}-D^{2}-a(t)-\sigma_{n}[-D^{2}-a(t)]\Big{)}\Big{\\}},$

which has been sketched in Figure 1.

Figure 1: Bifurcation diagram for $p=1$, where
$\sigma_{n}\equiv\sigma_{n}[-D^{2}-a(t)]$.

In Figure 1, we are plotting the parameter $\lambda$ in abscissas versus the
values

$M_{+}:=\max_{t\in[0,L]}|u(t)|,\quad\hbox{or}\quad
M_{-}:=-\max_{t\in[0,L]}|u(t)|,$ (2.2)

in ordinates, according to whether $u^{\prime}(0)>0$, or $u^{\prime}(0)<0$,
respectively, as in the remaining bifurcation diagrams of this paper. Since

$\lim_{n\to\infty}\sigma_{n}[-D^{2}-a(t)]=\infty,$

these eigenvalues are positive, except for, at most, finitely many among them.
Note that, in the special case when $a(t)\equiv\mu>0$ is a positive constant,
then

$\sigma_{n}[-D^{2}-a(t)]=\sigma_{n}[-D^{2}]-\mu=\left(\frac{n\pi}{L}\right)^{2}-\mu\quad\hbox{for
all integer}\;\;n\geq 1.$

Throughout this paper, we will simply denote

$\sigma_{n}:=\sigma_{n}[-D^{2}]=\left(\frac{n\pi}{L}\right)^{2},\qquad n\geq
1.$

In Figure 1, all the bifurcations from $u=0$ are vertical.

When $p>1$, the nonlinearity has a superlinear growth at zero and at infinity,
as in Moore and Nehari and López-Gómez and Rabinowitz [35, 30], while if
$p\in(0,1)$, then it inherits a sublinear growth at zero and at infinity, as
in Kajikiya [13].

The first part of this section analyzes, very briefly, the special case when
$a(t)$ is a positive constant, $\mu>0$, for all $t\in[0,L]$. Then, we will
study (2.1) for a general class of piece-wise continuous weight functions
$a(t)$ that remain bounded away from zero in the intervals where it is
positive.

An important feature of the problem (2.1) is the fact that, since the
nonlinearity is an odd function of $u$, the solutions arise by pairs:
$(\lambda,u)$ and $(\lambda,-u)$, which facilitates the construction of the
set of solutions of the problem. Another pivotal feature of (2.1) is that, for
every solution $(\lambda,u)$ with $u\neq 0$ and any $z\in[0,L]$ such that
$u(z)=0$, necessarily $u^{\prime}(z)\neq 0$. Thus, the zeroes of any non
trivial solution are simple and isolated. Therefore, any non-trivial solution
of (2.1) has, at most, finitely many zeroes in $(0,L)$.

###  2.1 The special case when $a\equiv\mu>0$ in $(0,L)$

In such case, the planar system associated to the differential equation of
(2.1) becomes

$(\mathcal{N}_{\lambda})\quad\left\\{\begin{array}[]{ll}x^{\prime}&=y,\\\
y^{\prime}&=-\lambda x-\mu|x|^{p-1}x.\end{array}\right.$ (2.3)

When $p\in(0,1)$, $|x|^{p-1}x$ admits a continuous extension up to $x=0$ and
the uniqueness of a solution for the associated Cauchy problem follows from
Rebelo [43, Th. 1]; the existence being a direct consequence of the Peano’s
theory. When $p>1$ the nonlinearity is Lipschitz continuous and these results
are a direct consequence from the Cauchy–Lipschitz theorem.

Our construction of the set of solutions of (2.3) is based on the several
phase portraits of the model and hence, it relies on the period function
associated to the periodic orbits of (2.3) surrounding the origin. By the
oddity of $y$ and $|x|^{p-1}x$, the necessary time to cross each quadrant in
the phase-plane is the same. Thus, it is the quarter of the period. Let denote
by $\mathcal{H}(x,y)$ the associated Hamiltonian

$\mathcal{H}(x,y)=\frac{1}{2}y^{2}+\frac{\lambda}{2}x^{2}+\frac{\mu}{p+1}|x|^{p+1}.$

Then, for the appropriate range of $x_{+}>0$, the periodic orbit of (2.3),
$(x(t),y(t))$, $t\in\mathbb{R}$, through the point $(x_{+},0)$ satisfies

$\mathcal{H}(x(t),y(t))=\frac{1}{2}y^{2}(t)+\frac{\lambda}{2}x^{2}(t)+\frac{\mu}{p+1}|x(t)|^{p+1}=\frac{\lambda}{2}x_{+}^{2}+\frac{\mu}{p+1}x_{+}^{p+1}$
(2.4)

for all $t\in\mathbb{R}$. Therefore, after some straightforward manipulations,
it is apparent that the period of the orbit trough $(x_{+},0)$ can be
expressed in the form

$\mathcal{T}(x_{+})\equiv\mathcal{T}_{\lambda}(x_{+}):=4\int_{0}^{1}\frac{ds}{\sqrt{\lambda(1-s^{2})+\frac{2\mu}{p+1}x_{+}^{p-1}(1-s^{p+1})}}.$
(2.5)

The precise range of $x_{+}$’s for which $\mathcal{T}(x_{+})$ is well defined,
as well as the precise behavior of the map $\mathcal{T}$ depends on $\lambda$
and the energy level $\mathcal{H}=c$ of the Hamiltonian. Indeed, when
$\lambda\geq 0$, the origin, which is the unique equilibrium of (2.3), is a
global nonlinear center and (2.5) makes sense for all $x_{+}>0$. However, when
$\lambda<0$, then the situation is a bit more subtle, as (2.3) possesses two
additional equilibria at the points

$\omega_{\pm}:=\pm\left(\frac{-\lambda}{\mu}\right)^{\frac{1}{p-1}},$ (2.6)

and the global structure of the phase portrait depends on the size of $p>0$.
Suppose $p>1$. Then, the origin is a saddle point whose stable and unstable
manifolds consist of two homoclinic connections surrounding $\omega_{\pm}$,
with zero energy level, passing through $(\pm x^{*},0)$, where we are setting

$x^{*}:=\left(\frac{-\lambda(p+1)}{2\mu}\right)^{\frac{1}{p-1}}.$ (2.7)

In this case, the integral curves ${\mathcal{H}}(x,y)=c$ with $c>0$ are
periodic orbits surrounding the origin with period given by (2.5) for all
$x_{+}>x^{*}$, while, for every $\mathcal{H}(\omega_{\pm},0)<c<0$,
$\mathcal{H}^{-1}(c)$ consists of two periodic orbits surrounding each of the
equilibria $(\omega_{\pm},0)$ in the phase portrait. Instead, when
$p\in(0,1)$, the origin is a local center whose periodic orbits fill the
region enclosed by two heteroclinic connections linking the equilibria
$(\omega_{\pm},0)$ at the energy level

$c^{*}:={\mathcal{H}}(\omega_{\pm},0)=\left(\frac{-\lambda}{\mu}\right)^{\frac{2}{p-1}}\frac{\lambda(p-1)}{2(p+1)}.$
(2.8)

Thus, for every $x_{+}\in(0,\omega_{+})$, $\mathcal{H}(x_{+},0)<c^{*}$ and the
solution through $(x_{+},0)$ is a periodic orbit surrounding $(0,0)$ whose
period is given by (2.5).

Based on (2.5) and on some well known properties of planar phase-portraits,
the period map $\mathcal{T}$ behaves like sketched in Table 1 according to the
sign of $\lambda$ and the size of $p$. Table 1 provides us with the
monotonicity properties of $\mathcal{T}$ and its limiting behavior at the ends
of its interval of definition.

$\lambda$ | behavior of $x_{+}$ | case $p>1$ | case $0<p<1$
---|---|---|---
$\lambda\geq 0$ |  $x_{+}$ increases |  ${\mathcal{T}}(x_{+})$ decreases |  ${\mathcal{T}}(x_{+})$ increases
$x_{+}\downarrow 0$ | ${\mathcal{T}}(x_{+})\uparrow 2\pi/\sqrt{\lambda}$ | ${\mathcal{T}}(x_{+})\downarrow 0$
$x_{+}\uparrow+\infty$ | ${\mathcal{T}}(x_{+})\downarrow 0$ | ${\mathcal{T}}(x_{+})\uparrow 2\pi/\sqrt{\lambda}$
$\lambda<0$ |  $x_{+}$ increases |  ${\mathcal{T}}(x_{+})$ decreases |  ${\mathcal{T}}(x_{+})$ increases
$x_{+}\downarrow x^{*}$ | ${\mathcal{T}}(x_{+})\uparrow+\infty$ |
$x_{+}\downarrow 0$ |  | ${\mathcal{T}}(x_{+})\downarrow 0$
| $x_{+}\uparrow+\infty$ | ${\mathcal{T}}(x_{+})\downarrow 0$ |
| $x_{+}\uparrow\omega_{+}$ |  | ${\mathcal{T}}(x_{+})\uparrow+\infty$
Table 1: Behavior of the period map $\mathcal{T}(x_{+})$

By the results collected in Table 1, the period map $\mathcal{T}$ has the
graph plotted in Figure 2 according to the sign of $\lambda$ and the size of
$p$. In Table 1 and in Figure 2, we agree that

$\frac{2\pi}{\sqrt{\lambda}}\equiv+\infty\qquad\hbox{if}\;\;\lambda=0.$

Based on these properties, the set of solutions of (2.1) can be easily
constructed in the special case when $a(t)$ is a positive constant $\mu>0$.
Subsequently, we use the notation (2.2) and take into account that if
$(u,u^{\prime})$ is a periodic solution of (2.3) with energy

$c=\frac{\lambda}{2}x_{+}^{2}+\frac{\mu}{p+1}x_{+}^{p+1},\qquad x_{+}>0,$

then $M_{+}=x_{+}\equiv x_{+}(c)$. The following result holds.

Figure 2: Plots of $\mathcal{T}$ according to Table 1.

######  Proposition 2.1.

Suppose $a(t)\equiv\mu>0$ in $[0,L]$. Then, for every $p>0$, with $p\neq 1$,
and any integer $n\geq 1$, the problem (2.1) admits a (unique) solution
$(\lambda,u)$ with $u^{\prime}(0)>0$ and $n-1$ interior zeros in $(0,L)$ if,
and only if,

$\lambda<\sigma_{n}:=\left(\frac{n\pi}{L}\right)^{2}.$ (2.9)

In addition, when $n\geq 2$ the nodes are equidistant, i.e., they are located
at the points

$t_{i}:=\frac{i}{n}L,\qquad i\in\\{1,...,n-1\\}.$

Moreover, if, for every $\lambda<\sigma_{n}$, we denote by $u_{n-1,\lambda}$
the unique solution of (2.1) with $n-1$ interior nodes such that
$u^{\prime}(0)>0$, then the map $(-\infty,\sigma_{n})\to\mathcal{C}[0,L]$
defined by $\lambda\mapsto u_{n-1,\lambda}$ is continuous. Thus,

$M_{+}(\lambda)\equiv
M_{+}(n,\lambda):=\max_{t\in[0,L]}|u_{n-1,\lambda}(t)|,\qquad\lambda<\sigma_{n},$
(2.10)

is also continuous and, actually, independent of $n\geq 1$. Furthermore, it
satisfies:

  1. (a)

In case $p>1$, $M_{+}(\lambda)$ decreases with respect to
$\lambda<\sigma_{n}$, $M_{+}(\lambda)>x^{*}$ if $\lambda<0$, and

$\lim_{\lambda\uparrow\sigma_{n}^{-}}M_{+}(\lambda)=0,\qquad\lim_{\lambda\downarrow-\infty}M_{+}(\lambda)=+\infty.$

  2. (b)

In case $p\in(0,1)$, $M_{+}(\lambda)$ increases with respect to
$\lambda<\sigma_{n}$, $M_{+}(\lambda)<\omega_{+}$ if $\lambda<0$, and

$\lim_{\lambda\uparrow\sigma_{n}^{-}}M_{+}(\lambda)=+\infty,\quad\lim_{\lambda\downarrow-\infty}M_{+}(\lambda)=0.$

###### Proof.

Note that a solution $(u,u^{\prime})$ of (2.3) passing through $(x_{+},0)$ has
$n-1$ interior zeroes if, and only if, $\mathcal{T}(x_{+})=\frac{2L}{n}$.
Moreover, thanks to Table 1, we have that

$\left(0,\tfrac{2\pi}{\sqrt{\lambda}}\right)=\mathcal{T}((0,+\infty))\;\,\hbox{if}\;\;\lambda\geq
0,\qquad\left(0,+\infty\right)=\left\\{\begin{array}[]{ll}\mathcal{T}((x^{*},+\infty))&\;\,\hbox{if}\;\;\lambda<0,\;p>1,\\\
\mathcal{T}((0,\omega_{+}))&\;\,\hbox{if}\;\;\lambda<0,\;p\in(0,1).\end{array}\right.$

Thus, a solution with $n-1$ zeros exists if, and only if,
$\frac{2L}{n}<\frac{2\pi}{\sqrt{\lambda}}$, which is equivalent to (2.9). The
uniqueness follows from the fact that, for any fixed $\lambda<\sigma_{n}$, the
period $\mathcal{T}(x_{+})$ is monotone with respect to $x_{+}$. The
distribution of the zeroes is a direct consequence of the symmetries of the
problem, because the solution with $u^{\prime}(0)>0$ and $n-1$ interior nodes
in $(0,L)$ can be easily constructed from the positive solution in
$(0,\frac{L}{n})$, by recursive translation and reflection. This explains also
why the values of $M_{+}$ in (2.10) are independent of $n\geq 1$.

The continuity of the map $\lambda\mapsto u_{n-1,\lambda}$ is a byproduct of
the uniqueness, by the continuous dependence of the solutions with respect to
the parameter $\lambda$.

Suppose $p>1$. Then, thanks to (2.5), for any fixed $\lambda<\sigma_{n}$,
$\mathcal{T}_{\lambda}(x_{+})$ is decreasing with respect to $x_{+}$.
Moreover, fixing $x_{+}>x^{*}$, $\mathcal{T}_{\lambda}(x_{+})$ is decreasing
with respect to $\lambda$. Thus, for any given
$\lambda_{1}<\lambda_{2}<\sigma_{n}$ and $c_{1},c_{2}>0$,
$\mathcal{T}_{\lambda_{1}}(x_{+}(c_{1}))=\mathcal{T}_{\lambda_{2}}(x_{+}(c_{2}))$
implies that $x_{+}(c_{1})>x_{+}(c_{2})$. Since $x_{+}(c)=M_{+}$, it becomes
apparent that $M_{+}$ is decreasing in $(-\infty,\sigma_{n})$. Moreover, since
$\lim_{\lambda\uparrow\sigma_{n}^{-}}\frac{2\pi}{\sqrt{\lambda}}=\frac{2L}{n}$,
it is easily seen that

$\lim_{\lambda\uparrow\sigma_{n}^{-}}M_{+}(\lambda)=0,$

and that, for every $\lambda<0$, $0<x^{*}<M_{+}(\lambda)$. This inequality
implies, in particular, that $M_{+}(\lambda)\uparrow+\infty$ if
$\lambda\downarrow-\infty$, which concludes the proof of Part (a).

Suppose $p\in(0,1)$. Then, $\mathcal{T}_{\lambda}(x_{+})$ is increasing with
respect to $x_{+}$ for each fixed $\lambda$. Moreover, for every (fixed)
$x_{+}<\omega_{+}$, $\mathcal{T}_{\lambda}(x_{+})$ is decreasing with respect
to $\lambda$. Thus, as soon as $\lambda_{1}<\lambda_{2}<\sigma_{n}$ and
$c_{1},c_{2}>0$, the identity
$\mathcal{T}_{\lambda_{1}}(x_{+}(c_{1}))=\mathcal{T}_{\lambda_{2}}(x_{+}(c_{2}))$
implies that $x_{+}(c_{1})<x_{+}(c_{2})$. Hence, $M_{+}(\lambda)$ is
increasing with respect to $\lambda\in(-\infty,\sigma_{n})$. Moreover, since
$\lim_{\lambda\uparrow\sigma_{n}^{-}}\frac{2\pi}{\sqrt{\lambda}}=\frac{2L}{n}$,
we have that $\lim_{\lambda\uparrow\sigma_{n}^{-}}M_{+}(\lambda)=+\infty$.
Also, for every $\lambda<0$, it is easily seen that $0<M_{+}<\omega_{+}$,
which implies that $M_{+}(\lambda)\rightarrow 0^{+}$ if
$\lambda\downarrow-\infty$ and ends the proof of the theorem. $\square$

Figure 3 shows the global bifurcation diagrams of the non-zero solutions of
(2.1) for the special case when $a(t)\equiv\mu>0$ in $[0,L]$. The case $p>1$
has been plotted above, and the case $p\in(0,1)$ below. Precisely, we are
plotting the value of the parameter $\lambda$, in abscissas, versus the value
of $M_{+}$ if $u^{\prime}(0)>0$, or $M_{-}$ if $u^{\prime}(0)<0$. As the
solutions do arise by pairs, $(\lambda,\pm u)$, the two bifurcations diagrams
are symmetric with respect to the horizontal axis.

Suppose $p>1$. Then, by Proposition 2.1, for every integer $n\geq 1$, the set
of solutions with $n-1$ zeroes in $(0,L)$ consists of a continuous curve
bifurcating from $u=0$ subcritically at $\lambda=\sigma_{n}$. Moreover, the
upper curve increases up to $+\infty$ as $\lambda\downarrow-\infty$. Each
point on the superior curves represents a single solution, $(\lambda,u)$, of
(2.1) with $u^{\prime}(0)>0$, while the opposite solutions, $(\lambda,-u)$,
fill the reflected inferior curves of the bifurcation diagram, providing us
with all the solutions $(\lambda,w)$ such that $w^{\prime}(0)<0$.

Suppose $p\in(0,1)$. Then, by Proposition 2.1, for every integer $n\geq 1$,
the set of solutions with $n-1$ zeroes in $(0,L)$ consists of two continuous
curves bifurcating from $\pm\infty$ subcritically at $\lambda=\sigma_{n}$. The
upper curve increases from zero, while the inferior one is decreasing. As in
the previous case, the solutions of the form $(\lambda,u)$ with
$u^{\prime}(0)>0$ fill the superior curves, while the inferior ones consist of
the solutions of the form $(\lambda,-u)$.

In Figure 3, the curves of positive solutions have been plotted by using red
color, the solutions with one interior node fill the blue curves, the
solutions with two interior nodes fill the orange curves, and the solutions
with $n-1$ interior nodes have been plotted using green color.

Figure 3: Bifurcation diagrams for $a\equiv\mu>0$

Since the continuous curves plotted in Figure 3 consist of solutions with a
prescribed number of zeroes, two different curves cannot meet. Actually, even
projecting the solutions over $M_{\pm}$, the intersections are excluded by the
construction of the nodal solutions from the positive solutions on smaller
intervals.

According to Figure 3, for every $\lambda<\sigma_{1}$ and any integer $n\geq
1$, the problem (2.1) has a unique solution $(\lambda,u)$ with
$u^{\prime}(0)>0$ having $n-1$ zeroes in $(0,L)$. Similarly, for every
$\lambda\in[\sigma_{1},\sigma_{2})$ and any integer $n\geq 2$, (2.1) has a
unique solution $(\lambda,u)$ with $u^{\prime}(0)>0$ having $n-1$ zeroes in
$(0,L)$, while it does not admit a positive solution. More generally, for
every integer $\kappa\geq 1$, $\lambda\in[\sigma_{\kappa},\sigma_{\kappa+1})$
and $n\geq\kappa+1$, (2.1) possesses a unique solution $(\lambda,u)$ with
$u^{\prime}(0)>0$ having $n-1$ zeroes in $(0,L)$, while it does not admit any
solution with $\ell$ zeroes in $(0,L)$ if $\ell\leq\kappa-1$.

The bifurcation diagrams of Figure 3 should be compared with those of Figure 2
in Cubillos et al. [8], for the special case when $a(t)$ is a negative
constant and $p>1$, which are the classical bifurcation diagrams of Rabinowitz
[39, 42].

###  2.2 A generalized Moore–Nehari model

In this section we consider a piecewise continuous weight function $a(t)\gneq
0$ defined in $[0,L]$, which is divided into finitely many disjoint intervals
where, alternately, either $a$ is positive (not necessarily constant) or
$a\equiv 0$. Precisely, we assume that $[0,L]$ splits out into

$I^{+}_{i}=[t_{i},s_{i}]\quad\text{and }\;I^{0}_{j}=(s_{j},t_{j+1}),\quad
i\in\\{0,....,m\\},\;\;j\in\\{0,...,m-1\\},$

where

$0=t_{0}<s_{0}<t_{1}<\dots<t_{i}<s_{i}<\dots<s_{m-1}<t_{m}<s_{m}=L,$ (2.11)

in such a way that $a\in\mathcal{C}[t_{i},s_{i}]$ satisfies $a(t)>0$ for all
$t\in[t_{i},s_{i}]$ and $i\in\\{0,...,m\\}$, while $a\equiv 0$ in $I^{0}_{j}$
for all $j\in\\{0,...,m\\}$. Thus, there are $m+1$ intervals where the weight
function $a(t)$ is positive, separated away by $m$ intervals where $a\equiv
0$. Our class of weight functions extends considerably the one introduced by
Moore and Nehari [35], later considered by Kajikiya [13, 14], where
$\lambda=0$ and, for some $\mu\in(0,\frac{L}{2})$,

$a=1-\chi_{(\frac{L}{2}-\mu,\frac{L}{2}+\mu)},$

where $\chi_{J}$ stands for the characteristic function of the interval
$J\subset[0,L]$, i.e., $a(t)=0$ if $|t-\frac{L}{2}|<\mu$, while $a(t)=1$ if
$|t-\frac{L}{2}|\geq\mu$.

Essentially, the results of this section establish that the global bifurcation
diagrams already found for the autonomous problem when $a\equiv\mu>0$,
collected in Figure 3, are “preserved” for the general class of weight
functions considered in this section, though for some values of
$\lambda\in\mathbb{R}$ and some integer $n\geq 1$, the problem might have an
arbitrarily large number of solutions with $n-1$ interior nodes (see Cubillos
et al. [8] for some examples in this direction when $a\lneq 0$). The main
technical tool to get these results is bifurcation theory with the a priori
bounds of Amann and López-Gómez [1]. As it will become apparent from the
proofs in this section, the general bifurcation approach combined with [1],
applies to a broader class of weight functions than the ones in which the
intervals of positivity and degeneracy follow the split of (2.11). Indeed, the
location of the intervals of positivity is not important; our only assumption
here is that $\min a(t)>0$ in each interval where $a(t)>0.$ In any case, it is
convenient to adopt (2.11) in order to have a comparison with the results in
the other sections as well as with respect to other cited articles.

Subsequently, we adapt the analysis of Fencl and López-Gómez [10] to our
setting here. The next result is Lemma 2.1 of [10]. As usual,
$W^{k,p}_{0}(0,L)$ denotes the closed subspace of the Sobolev space
$W^{k,p}(0,L)$ consisting of the functions $u\in W^{k,p}(0,L)$ such that
$u(0)=0=u(L)$. Similarly, $C^{k}_{0}[0,L]$ stands for the set of functions of
class $\mathcal{C}^{k}[0,L]$ vanishing at $0$ and $L$.

######  Lemma 2.1.

For every $f\in L^{2}(0,L)$, the function

$u(t)\equiv
Kf(t):=\int_{0}^{t}(s-t)f(s)\,ds-\frac{t}{L}\int_{0}^{L}(s-L)f(s)\,ds$

provides us with the unique solution of

$\left\\{\begin{array}[]{ll}-u^{\prime\prime}=f\quad\hbox{in}\;\,(0,L),\\\
u(0)=u(L)=0,\end{array}\right.$

in $W^{2,2}(0,L)$. Moreover, the solution operator $K:L^{2}(0,L)\rightarrow
W^{2,2}_{0}(0,L)$ is linear and continuous. Thus, when considering the compact
embedding $\iota:W^{2,2}_{0}(0,L)\hookrightarrow C^{1}_{0}[0,L]$, the
composition

$\mathcal{K}:=\iota\circ K|_{C^{1}_{0}[0,L]}:C^{1}_{0}[0,L]\rightarrow
C^{1}_{0}[0,L]$

is a linear, continuous and compact operator.

Thanks to Lemma 2.1, the problem (2.1) can be expressed as a fixed point
equation for a compact operator. Indeed, a function $u$ piecewise
$\mathcal{C}^{2}$ in $[0,L]$ solves (2.1) if, and only if,

$u=\mathcal{K}(\lambda u+a(t)|u|^{p-1}u).$

Thus, the solutions of (2.1) are the zeroes of the nonlinear operator

$\begin{array}[]{rcl}\mathfrak{F}:\mathbb{R}\times
C^{1}_{0}[0,L]&\longrightarrow&C^{1}_{0}[0,L]\\\\[5.0pt]
(\lambda,u)&\longmapsto&\mathfrak{F}(\lambda,u):=u-\mathcal{K}(\lambda
u+a(t)|u|^{p-1}u),\end{array}$ (2.12)

which can be decomposed in the form

$\mathfrak{F}(\lambda,u)=\mathfrak{L}(\lambda)u+\mathfrak{N}(\lambda,u),$

where

$\mathfrak{L}(\lambda)u:=u-\lambda\mathcal{K}u\quad\hbox{and}\quad\mathfrak{N}(\lambda,u):=-\mathcal{K}(a(t)|u|^{p-1}u).$
(2.13)

By (2.13), $\mathfrak{L}(\lambda)$ is a compact perturbation of the identity
$I$ in $C^{1}_{0}[0,L]$ and, hence, a Fredholm operator of index $0$.
Moreover,

$\mathfrak{N}(\lambda,u)=o(\|u\|)\quad\hbox{as}\quad\left\\{\begin{array}[]{ll}\|u\|\rightarrow
0&\quad\hbox{if}\;p>1,\\\
\|u\|\rightarrow+\infty&\quad\hbox{if}\;0<p<1.\end{array}\right.$

Thus, we can apply the abstract theory of [16, Ch. 6] for $p>1$, going back to
Rabinowitz [40], as well as Theorem 1.6 of Rabinowitz [41] for $0<p<1$.

The generalized spectrum, $\Sigma(\mathfrak{L})$, of the Fredholm curve
$\mathfrak{L}(\lambda)$ is the set of $\lambda\in\mathbb{R}$ satisfying
$u=\lambda\mathcal{K}u$ for some $u\neq 0$, $u\in C^{1}_{0}[0,L]$. It is
folklore that

$\Sigma(\mathfrak{L})=\left\\{\sigma_{n}=\left(\frac{n\pi}{L}\right)^{2}\,:\,n\geq
1\right\\}.$

As in the general case when $p>1$, the nonlinearity is not of class
$\mathcal{C}^{2}$, unless $p>2$, the local theorem of Crandall and Rabinowitz
[7] cannot be applied to show the existence of a smooth curve of nodal
solutions with $n-1$ interior nodes bifurcating from $u=0$ at
$\lambda=\sigma_{n}$. Consequently, to establish the existence of a component
of nodal solutions bifurcating from $u=0$ at $\lambda=\sigma_{n}$, we must
find out the Leray–Schauder degree
$\mathrm{Deg\,}(\mathfrak{L}(\lambda),B_{R})$ for every
$\lambda\in\mathbb{R}\setminus\Sigma(\mathfrak{L}(\lambda))$, where $B_{R}$
denotes the ball of radius $R>0$ centered at the origin of $C^{1}_{0}[0,L]$.
The degree can be computed through the Schauder’s formula

${\rm Deg}(\mathfrak{L}(\lambda),B_{R})=(-1)^{m(\mathfrak{L}(\lambda))},$

where $m(\mathfrak{L}(\lambda))$ is the sum of the algebraic multiplicities of
the negative eigenvalues of $\mathfrak{L}(\lambda)$. According to the Schauder
formula, the next result holds.

######  Lemma 2.2.

Setting $\sigma_{0}\equiv-\infty$, it turns out that, for every integer $n\geq
0$,

$\mathrm{Deg}(\mathfrak{L}(\lambda),B_{R})=\left\\{\begin{array}[]{ll}1&\quad\hbox{if}\;\,\lambda\in(\sigma_{2n},\sigma_{2n+1}),\\\
-1&\quad\hbox{if}\;\,\lambda\in(\sigma_{2n+1},\sigma_{2n+2}).\end{array}\right.$
(2.14)

###### Proof.

To determine $m(\mathfrak{L}(\lambda))$, we should construct the set of
$\mu\in\mathbb{R}$ such that, for some $u\in C^{1}_{0}[0,L]\setminus\\{0\\}$,

$\mathfrak{L}(\lambda)u=u-\lambda\mathcal{K}u=\mu u.$ (2.15)

Setting $\sigma_{0}:=-\infty$, for every $n\geq 1$, $\mathfrak{L}(\lambda)$ is
invertible for $\lambda\in(\sigma_{n-1},\sigma_{n})$. Thus, by the homotopy
invariance of the degree,

$d_{n}:=\mathrm{Deg}(\mathfrak{L}(\lambda),B_{R})\;\;\hbox{is constant for
all}\;\;\lambda\in(\sigma_{n-1},\sigma_{n}).$

In particular, $\mathrm{Deg}(\mathfrak{L}(\lambda),B_{R})$ is constant in
$(-\infty,\sigma_{1})$. Thus, to determine $d_{1}$ we can assume, without lost
of generality, that $\lambda>0$. Then, by (2.15), $\mu\neq 1$ and hence, the
equation (2.15) can be equivalently written as

$\left\\{\begin{array}[]{ll}-u^{\prime\prime}=\frac{\lambda}{1-\mu}u&\quad\hbox{in}\;\;[0,L],\\\\[2.0pt]
u(0)=u(L)=0.&\end{array}\right.$

Thus, the set of classical eigenvalues of $\mathfrak{L}(\lambda)$ is

$\sigma(\mathfrak{L}(\lambda)):=\left\\{\mu_{n}:=1-\frac{\lambda}{\sigma_{n}}\;:\;n\geq
1\right\\}.$

The $\mu_{n}$’s are algebraic simple eigenvalues of $\mathfrak{L}(\lambda)$
because, for every $n\geq 1$, setting $\psi_{n}(t):=\sin(\frac{n\pi}{L}t)$,
$t\in[0,L]$, we have that

$N[I-\lambda\mathcal{K}-\mu_{n}I]=N[\mathfrak{L}(\sigma_{n})]={\rm
span}[\psi_{n}],\qquad\psi_{n}\notin R[I-\lambda\mathcal{K}-\mu_{n}I].$

Indeed, if there exists $u\in C^{1}_{0}[0,L]$ such that

$\psi_{n}=(1-\mu_{n})u-\lambda\mathcal{K}u\Longleftrightarrow(1-\mu_{n})u=\lambda\mathcal{K}u+\psi_{n}\in
C^{2}_{0}[0,L],$

differentiating twice with respect to $t$, it follows from the definition of
$\mu_{n}$ that

$\displaystyle-(1-\mu_{n})u^{\prime\prime}=\lambda
u-\psi_{n}^{\prime\prime}=\lambda
u+\sigma_{n}\psi_{n}\Longleftrightarrow-u^{\prime\prime}-\sigma_{n}u=\frac{\sigma_{n}^{2}}{\lambda}\psi_{n}.$

Thus, multiplying by $\psi_{n}$ and integrating in $(0,L)$ yields

$0<\frac{\sigma_{n}^{2}}{\lambda}\int_{0}^{L}\psi_{n}^{2}=\int_{0}^{L}(-u^{\prime\prime}-\sigma_{n}u)\psi_{n}=0,$

which is impossible. Therefore, $m(\lambda)$ equals the number of negative
eigenvalues of $\mathfrak{L}(\lambda)$. Consequently, since for every $n\geq
1$ and any given $\sigma_{n}<\lambda<\sigma_{n+1}$ we have that

$\mu_{1}=1-\frac{\lambda}{\sigma_{1}}<\cdots<\mu_{n}=1-\frac{\lambda}{\sigma_{n}}<0<\mu_{n+1}=1-\frac{\lambda}{\sigma_{n+1}}<\cdots,$

the identity (2.14) holds. $\square$

Let $\mathcal{S}$ denote the set of non-trivial solutions of (2.1), i.e.,

$\mathcal{S}\equiv\\{(\lambda,u)\in\mathfrak{F}^{-1}(0):u\neq
0\\}\cup\left\\{\begin{array}[]{ll}\\{(\sigma_{n},0)\;:\;n\geq
1\\}&\quad\hbox{if}\;\,p>1,\\\\[5.0pt] \\{(\sigma_{n},\pm\infty)\;:\;n\geq
1\\}&\quad\hbox{if}\;\,0<p<1,\end{array}\right.$

where $\mathfrak{F}$ is given by (2.12). In the case $0<p<1$ we are
identifying, from a projective perspective, $(\sigma_{n},+\infty)$ and
$(\sigma_{n},-\infty)$ to emphasize that solutions losing their bounds at
$\lambda=\sigma_{n}$ belong also to $\mathcal{S}$. With this convention in
mind, the next result holds. By a component we mean a closed and connected
subset maximal for the inclusion.

######  Theorem 2.1.

For every $n\geq 1$, $\mathcal{S}$ possesses a component, $\mathfrak{C}_{n}$,
such that

$\left\\{\begin{array}[]{ll}(\sigma_{n},0)\in\mathfrak{C}_{n}&\quad\hbox{if}\;\,p>1,\\\\[5.0pt]
(\sigma_{n},\pm\infty)\in\mathfrak{C}_{n}&\quad\hbox{if}\;\,0<p<1,\end{array}\right.$

consisting of solutions $(\lambda,u)$ with $n-1$ interior zeroes. Thus,

$\mathfrak{C}_{n}\cap\mathfrak{C}_{m}=\emptyset\quad\hbox{if}\;\,n\neq m.$
(2.16)

Moreover, since $\mathfrak{F}$ is odd in $u$, for every $n\geq 1$,

$\mathfrak{C}_{n}=\left\\{\begin{array}[]{ll}\mathfrak{C}^{+}_{n}\cup\mathfrak{C}^{-}_{n}\cup\\{(\sigma_{n},0)\\}&\quad\hbox{if}\;\,p>1,\\\\[5.0pt]
\mathfrak{C}^{+}_{n}\cup\mathfrak{C}^{-}_{n}\cup\\{(\sigma_{n},\pm\infty)\\}&\quad\hbox{if}\;\,0<p<1,\end{array}\right.$

where

$\mathfrak{C}^{\pm}_{n}:=\\{(\lambda,u)\in\mathfrak{C}_{n}\;:\;\pm
u^{\prime}(0)>0\\}.$

Note that
$\mathfrak{C}^{-}_{n}:=\\{(\lambda,-u)\;:\;(\lambda,u)\in\mathfrak{C}_{n}^{+}\\}$.

###### Proof.

The existence of the component $\mathfrak{C}_{n}$ in the case $p>1$ follows
from Lemma 2.2 and [16, Th. .2.1]. Once proven the existence of
$\mathfrak{C}_{n}$, by [16, Le. 6.4.1], we have that, for every
$(\lambda,u)\in\mathcal{S}\setminus\\{(\sigma_{n},0)\\}\cap
B_{\delta}(\sigma_{n},0)$ with sufficiently small $\delta>0$,

$\lambda=\sigma_{n}+o(1)\;\;\hbox{and}\;\;u(s)=s[\psi_{n}+o(1)]\;\;\hbox{as}\;\,s\rightarrow
0.$ (2.17)

Thus, the solutions of $\mathfrak{C}_{n}$ in a neighborhood of
$(\sigma_{n},0)$ have $n-1$ interior zeroes. Since the zeroes of the solutions
of (2.1) are simple, they vary continuously in $\mathbb{R}\times
C^{1}_{0}[0,L]$. Therefore, $\mathfrak{C}_{n}\setminus\\{(\sigma_{n},0)\\}$
consists of solutions with $n-1$ interior nodes. This implies also (2.16).
Finally, [16, Pr. 6.4.2] guarantees the existence of the subcomponents
$\mathfrak{C}_{n}^{\pm}$.

Similarly, the result for $p\in(0,1)$ is a consequence of Rabinowitz [41, Le.
1.3] and the first part of [41, Th. 1.6]. $\square$

######  Remark 2.1.

In the proof of Lemma 2.2 we have actually shown that the transversality
condition of Crandall and Rabinowitz [7] holds for
$\mathfrak{L}(\sigma_{n})=I-\sigma_{n}\mathcal{K}$ and
$\mathfrak{L}_{1}:=\mathfrak{L}^{\prime}(\sigma_{n})=-\mathcal{K}$. Hence, in
order to apply [7, Th. 1.7], it suffices that the operator $\mathfrak{F}$
defined in (2.12) be of class $\mathcal{C}^{r}$ with $r\geq 2$. This holds
provided $p>2$. In such case, $\mathfrak{C}_{n}$ is a curve of class
$\mathcal{C}^{r-1}$ in a neighborhood of $(\sigma_{n},0)$ (see [16, Sect. 2.2]
if necessary).

Next, we will ascertain the global behavior of the components
$\mathfrak{C}_{n}$ for $n\geq 1$. Essentially, we will show that
$\mathfrak{C}_{n}$ behaves much like the curve bifurcating from
$(\sigma_{n},0)$ in Figure 3, though in the general case covered in this
section the uniqueness of the nodal solution with $u^{\prime}(0)>0$ might
fail. Our main result will follow after a series of technical lemmas. The next
one shows that (2.1) cannot admit a solution $(\lambda,u)$ with $n-1$ interior
zeroes if $\lambda\geq\sigma_{n}$.

######  Lemma 2.3.

Suppose $p>1$, or $p\in(0,1)$, and (2.1) has a solution with $n-1$ interior
zeroes, $(\lambda,u)$. Then, $\lambda<\sigma_{n}$.

###### Proof.

Let $(\lambda,u)$ be a solution with $n-1$ zeroes in $(0,L)$. Then,

$\left\\{\begin{array}[]{l}\left(-D^{2}-a(t)|u|^{p-1}\right)u=\lambda
u\quad\hbox{in}\;\;[0,L],\\\ u(0)=u(L)=0,\end{array}\right.$

and hence, $u$ is an eigenfunction with $n-1$ zeroes associated with the
eigenvalue $\lambda$ of the differential operator $-D^{2}-a(t)|u|^{p-1}$ under
homogeneous Dirichlet boundary conditions. Consequently, by the uniqueness of
these eigenvalues,

$\lambda=\sigma_{n}\left[-D^{2}-a(t)|u|^{p-1}\right]$

and, thanks to the monotonicity with respect to the potential (see Butazzo et
al. [4]), we find that

$\lambda=\sigma_{n}\left[-D^{2}-a(t)|u|^{p-1}\right]<\sigma_{n}[-D^{2}]=\sigma_{n}.$

This ends the proof. $\square$

As a byproduct of Lemma 2.3,
$\mathcal{P}_{\lambda}(\mathfrak{C}_{n})\subset(-\infty,\sigma_{1}]$ for all
$p>0$, where $\mathcal{P}_{\lambda}$ stands for the $\lambda$-projection
operator, $\mathcal{P}_{\lambda}(\lambda,u):=\lambda$.

The following lemma establishes that $(\sigma_{n},0)$ is the unique
bifurcation point of $\mathfrak{C}_{n}$ from $u=0$ if $p>1$, and that
$(\sigma_{n},\pm\infty)$ is the unique bifurcation point from infinity of
$\mathfrak{C}_{n}$ if $p\in(0,1)$. As in (2.2) for the constant case, we will
denote

$M_{\pm}(\lambda,u):=\pm\max_{t\in[0,L]}|u(t)|$

for every solution, $(\lambda,u)$, of (2.1).

######  Lemma 2.4.

Let $\mathfrak{C}_{\kappa}$ be the component of nodal solutions of (2.1) with
$\kappa-1$ interior nodes whose existence is guaranteed by Theorem 2.1. Then,

  1. (i)

Whenever $p>1$, for every sequence of solutions of (2.1),
$\\{(\lambda_{n},u_{n})\\}_{n\geq 1}$, such that
$\lim_{n\to\infty}\lambda_{n}=\lambda_{*}\in\mathbb{R}$, one has that

$\lim_{n\to\infty}M_{+}(\lambda_{n},u_{n})=0$ (2.18)

if, and only if, $\lambda_{*}=\sigma_{\kappa}$ for some $\kappa\geq 1$.

  2. (ii)

Similarly, when $p\in(0,1)$, for every sequence of solutions of (2.1),
$\\{(\lambda_{n},u_{n})\\}_{n\geq 1}$, such that
$\lim_{n\to\infty}\lambda_{n}=\lambda_{*}\in\mathbb{R}$, one has that

$\lim_{n\to\infty}M_{+}(\lambda_{n},u_{n})=+\infty$ (2.19)

if, and only if, $\lambda_{*}=\sigma_{\kappa}$ for some $\kappa\geq 1$.

###### Proof.

Let $\\{(\lambda_{n},u_{n})\\}_{n\geq 1}$ be a sequence of solutions in
$\mathfrak{C}_{k}$ such that
$\lim_{n\to\infty}\lambda_{n}=\lambda_{*}\in\mathbb{R}$ and

$\lim_{n\to\infty}M_{+}(\lambda_{n},u_{n})=\left\\{\begin{array}[]{ll}0&\quad\hbox{if}\;\,p>1,\\\\[5.0pt]
+\infty&\quad\hbox{if}\;\,0<p<1.\end{array}\right.$ (2.20)

Then, dividing by $M_{+}(\lambda_{n},u_{n})$ the differential equation
satisfied by $(\lambda_{n},u_{n})$ and inverting $-D^{2}$, we find that, for
every $n\geq 1$,

$\frac{u_{n}}{M_{+}(\lambda_{n},u_{n})}=\mathcal{K}\left(\lambda_{n}\frac{u_{n}}{M_{+}(\lambda_{n},u_{n})}+a(t)\frac{|u_{n}|^{p-1}u_{n}}{M_{+}(\lambda_{n},u_{n})}\right).$
(2.21)

Since $\lim_{n\to\infty}\lambda_{n}=\lambda_{*}\in\mathbb{R}$, by (2.20), the
sequence

$\lambda_{n}\frac{u_{n}}{M_{+}(\lambda_{n},u_{n})}+a(t)\frac{|u_{n}|^{p-1}u_{n}}{M_{+}(\lambda_{n},u_{n})},\qquad
n\geq 1,$

is bounded. Thus, since $\mathcal{K}$ is compact, there exists a convergent
subsequence

$\lim_{m\to\infty}\frac{u_{n_{m}}}{M_{+}(\lambda_{n_{m}},u_{n_{m}})}=\psi$

to some function $\psi$ with $\kappa-1$ interior zeroes such that
$\|\psi\|_{\infty}=1$. Moreover, letting $m\to\infty$ in

$\frac{u_{n_{m}}}{M_{+}(\lambda_{n_{m}},u_{n_{m}})}=\mathcal{K}\left(\lambda_{n_{m}}\frac{u_{n_{m}}}{M_{+}(\lambda_{n_{m}},u_{n_{m}})}+a(t)\frac{|u_{n_{m}}|^{p-1}u_{n_{m}}}{M_{+}(\lambda_{n_{m}},u_{n_{m}})}\right)$

it becomes apparent that

$\left\\{\begin{array}[]{ll}&-\psi^{\prime\prime}=\lambda_{*}\psi,\\\
&\psi(0)=0=\psi(L).\end{array}\right.$

Therefore, $\lambda_{*}=\sigma_{k}=\left(\frac{k\pi}{L}\right)^{2}$. The
converses follow straight ahead from Theorem 2.1. $\square$

The next lemma provides us with a priori bounds in both cases, $p>1$ and
$p\in(0,1)$, ensuring that the solutions of (2.1) never blow up if $p>1$, and
that they cannot emanate from $u=0$ if $p\in(0,1)$. These a priori bounds
follow from Amann and López-Gómez [1, Le. 4.2] for the case $p>1$, and from a
counterpart of that result for the case $p\in(0,1)$.

######  Lemma 2.5.

Let $\mathfrak{C}_{\kappa}$ be the component of nodal solutions of (2.1) with
$\kappa-1$ interior nodes whose existence is guaranteed by Theorem 2.1. Then,
for every compact subinterval $J\subset(-\infty,\sigma_{\kappa})$, there exist
$C=C(J)>0$ and $\omega=\omega(J)>0$ such that

  1. (i)

For every $(\lambda,u)\in\mathfrak{C}_{\kappa}$ with $\lambda\in J$,
$M(\lambda,u)\leq C$ if $p>1$.

  2. (ii)

For every $(\lambda,u)\in\mathfrak{C}_{\kappa}$ with $\lambda\in J$,
$M(\lambda,u)\geq\omega$ if $p\in(0,1)$.

###### Proof.

Assume that there exists a sequence $\\{(\lambda_{n},u_{n})\\}_{n\geq
1}\subset\mathfrak{C}_{\kappa}^{+}$ such that in some interval
$(\alpha,\beta)$ where $a(t)>0$ there is a further sequence
$\\{x_{n}\\}_{n\geq 1}\subset(\alpha,\beta)$ such that

$\lim_{n\to+\infty}x_{n}=x_{\infty}\in(\alpha,\beta),\qquad\lim_{n\to\infty}\lambda_{n}=\lambda_{\infty}\in(-\infty,\sigma_{\kappa}),$

and

$M(\lambda_{n},u_{n})=u_{n}(x_{n}):=\max_{t\in[\alpha,\beta]}u_{n}(t)\overset{n\uparrow\infty}{\longrightarrow}\left\\{\begin{array}[]{ll}+\infty&\quad\hbox{if}\;\,p>1,\\\
0&\quad\hbox{if}\;\,0<p<1.\end{array}\right.$

Since $u_{n}(x_{n})>0$, there exists $r_{n}>0$ such that

$J_{n}:=(x_{n}-r_{n},x_{n}+r_{n})\subset(\alpha,\beta)$

and $u_{n}(t)>0$ for all $t\in J_{n}$. Then, setting
$\varrho_{n}:=M_{n}^{-\frac{p-1}{2}}$, $n\geq 1$, it is apparent that, for
every $p>0$, $p\neq 1$,

$\lim_{n\to+\infty}\varrho_{n}=0.$ (2.22)

Now, for every $t\in J_{n}$, by performing the change of variables

$y:=\frac{t-x_{n}}{\varrho_{n}},\qquad\quad
v_{n}(y):=\varrho_{n}^{\frac{2}{p-1}}u_{n}(t),\qquad n\geq 1,$ (2.23)

it becomes apparent that, for every $n\geq 1$,

$-\ddot{v}_{n}(y)=\lambda_{n}\varrho_{n}^{2}v_{n}(y)+a(x_{n}+\varrho_{n}y)|v_{n}(y)|^{p-1}v_{n}(y).$
(2.24)

By (2.23) and the definitions of $M_{n}$ and $\varrho_{n}$, it is easily seen
that, for every $n\geq 1$,

$|v_{n}(y)|\leq 1\quad\hbox{and}\quad|v_{n}(0)|=1.$

Thus, $v_{n}\not\equiv 0$ is bounded for all $n\geq 1$. Consequently, by
(2.22) and arguing as in [1, Le. 4.2], along some subsequence, relabeled by
$n\geq 1$, we have that $v_{n}$ converges weakly to some positive function $w$
such that

$-\ddot{w}(y)=a(x_{\infty})|w(y)|^{p-1}w(y),\qquad w(0)=1.$ (2.25)

Since $a(x_{\infty})>0$, the origin of (2.25) is a center. Thus, all non-
trivial solutions have infinitely many zeroes in $\mathbb{R}$, contradicting
the fact that $w$ is a non-trivial positive solution defined in $\mathbb{R}$.
Adapting this argument, one can also get a contradiction when
$x_{\infty}\in\\{\alpha,\beta\\}$. In such case, (2.25) is not defined in
$\mathbb{R}$ but only in $(-\infty,0]$, or $[0,\infty)$. This argument works
out because we are assuming that $a(t)>0$ for all $t\in[\alpha,\beta]$. The
general case when $a(t)$ is continuous in $[0,L]$ will be treated in a
forthcoming paper.

This concludes the proof of the theorem if the maximum of the $u_{n}$’s is
attained in some of the finite intervals of $a^{-1}((0,+\infty))$, which is
always the case if $\lambda_{n}<0$. Indeed, if $u_{n}$ reaches its maximum at
$x_{n}\in(0,L)$, then

$0\leq-u^{\prime\prime}_{n}(x_{n})=\lambda_{n}u_{n}(x_{n})+a(x_{n})|u_{n}(x_{n})|^{p-1}u_{n}(x_{n}).$
(2.26)

Thus, $a(x_{n})>0$ if $\lambda_{n}<0$.

To complete the proof, suppose that $\lambda_{n}\geq 0$ and that the maximum
of $u_{n}$ is attained in some subinterval of $a^{-1}(0)$. If $0<p<1$ and
$\lim_{n\to\infty}M_{+}(\lambda_{n},u_{n})=0$, then on each subinterval of
$a^{-1}((0,+\infty))$ we also have that

$\lim_{n\to\infty}\max_{t\in[t_{i},s_{i}]}|u_{n}(t)|=0$

and the result holds very easily by adapting the argument already given
before.

Finally, suppose that $p>1$ and that the $u_{n}$’s are bounded above in the
intervals where $a>0$. Then, along some subsequence, labeled again by $n$, we
have that

$\lim_{n\to\infty}\max_{t\in[s_{i},t_{i+1}]}|u_{n}(t)|=\infty$

for some $i\in\\{0,...,m-1\\}$. Thus, since

$-u_{n}^{\prime\prime}(t)=\lambda_{n}u_{n}(t),$

with $\lambda_{n}\geq 0$, it is easily seen that also in a neighborhood of
some point $x_{n}\in\\{t_{i},s_{i}:0\leq i\leq m\\}$ the solutions go to
infinity. According to the first part of the proof, the result holds true.
This ends the proof. $\square$

######  Remark 2.2.

In the special case when the weight function $a(t)$ is piecewise constant, the
proof of Lemma 2.5 can be simplified very substantially. Indeed, assume the
existence of some sequence $(\lambda_{n},u_{n})\in\mathfrak{C}_{\kappa}^{+}$,
$n\geq 1$, such that

$\lim_{n\to\infty}\lambda_{n}=\lambda_{*}<\sigma_{\kappa}$

and

$\lim_{n\to\infty}M_{+}(\lambda_{n},u_{n})=\left\\{\begin{array}[]{ll}+\infty&\quad\hbox{if}\;\,p>1,\\\\[5.0pt]
0&\quad\hbox{if}\;\,0<p<1.\end{array}\right.$

Let $[t_{i},s_{i}]$ be any interval of length $\ell=s_{i}-t_{i}>0$ where $a$
is a constant $\mu>0$. By (2.5), the period of the orbit through
$(M_{+}(\lambda_{n},u_{n}),0)$ in $[t_{i},s_{i}]$ is given, for every $n\geq
1$, by

$\mathcal{T}(M_{+}(\lambda_{n},u_{n}))=4\int_{0}^{1}\frac{ds}{\sqrt{\lambda_{n}(1-s^{2})+\frac{2\mu}{p+1}M_{+}^{p-1}(\lambda_{n},u_{n})(1-s^{p+1})}}.$
(2.27)

Since $(\lambda_{n},u_{n})\in\mathfrak{C}_{\kappa}$, for every $n\geq 1$, we
have that

$\mathcal{T}(M_{+}(\lambda_{n},u_{n}))>\frac{2\ell}{\kappa+1}.$

Indeed, if $\mathcal{T}(M_{+}(\lambda_{n},u_{n}))\leq\frac{2\ell}{\kappa+1}$
for some $n\geq 1$, then $(\lambda_{n},u_{n})$ must be a solution with more
than $\kappa$ nodes and, hence,
$(\lambda_{n},u_{n})\notin\mathfrak{C}_{\kappa}$. However, letting
$n\to\infty$ in (2.27) yields to

$\lim_{n\to\infty}\mathcal{T}(M_{+}(\lambda_{n},u_{n}))=0<\frac{2\ell}{\kappa+1},$

which is a contradiction.

According to Lemmas 2.4 and 2.5, for every integer $\kappa\geq 1$, it follows
from the global alternatives of Rabinowitz (see [40, Th. 1.3] and [41, Th.
1.6] for $p>1$ and $0<p<1$, respectively) that the component
$\mathfrak{C}_{\kappa}$ must be unbounded in
$\mathbb{R}\times\mathcal{C}^{1}_{0}[0,L]$. Thus, thanks to Lemma 2.3, it
becomes apparent that

$\mathcal{P}_{\lambda}\left(\mathfrak{C}_{\kappa}\setminus\\{(\sigma_{\kappa},0)\\}\right)=(-\infty,\sigma_{\kappa}).$
(2.28)

Note that $(\sigma_{\kappa},0)\in\mathfrak{C}_{\kappa}$ if $p>1$, while
$(\sigma_{\kappa},0)\notin\mathfrak{C}_{\kappa}$ if $p\in(0,1)$. The next
results collects these findings in a compact way.

######  Theorem 2.2.

For every integer $\kappa\geq 1$, the component $\mathfrak{C}_{\kappa}$ of
nodal solutions with $\kappa-1$ interior nodes satisfies (2.28). In
particular, it is unbounded in $\mathbb{R}\times\mathcal{C}_{0}^{1}[0,L]$.
Moreover, $\mathfrak{C}_{\kappa}$ bifurcates from $(\sigma_{\kappa},0)$ if
$p>1$, and from $(\sigma_{\kappa},\pm\infty)$ if $p\in(0,1)$, and it satisfies
Lemmas 2.4 and 2.5. Furthermore, for every
$(\lambda,u)\in\mathfrak{C}_{\kappa}$ with $\lambda<0$, we have that

$M_{+}(\lambda,u)\geq\left(\frac{-\lambda}{\|a\|_{\infty}}\right)^{\frac{1}{p-1}}\quad\hbox{if}\;\;p>1,$
(2.29)

while

$M_{+}(\lambda,u)\leq\left(\frac{-\lambda}{\|a\|_{\infty}}\right)^{\frac{1}{p-1}}\quad\hbox{if}\;\;p\in(0,1).$
(2.30)

###### Proof.

The first assertions have been already proven. Suppose $\lambda<0$ and
$(\lambda,u)\in\mathfrak{C}_{\kappa}^{+}$ attains its maximum, which is
positive, at $x_{0}\in(0,L)$. Then, since

$0\leq-u^{\prime\prime}(x_{0})=\lambda
u(x_{0})+a(x_{0})|u(x_{0})|^{p-1}u(x_{0})=\left(\lambda+a(x_{0})|u(x_{0})|^{p-1}\right)u(x_{0}),$

we find that

$\lambda+a(x_{0})|u(x_{0})|^{p-1}=\lambda+a(x_{0})M_{+}^{p-1}(\lambda,u)\geq
0.$

Therefore, $a(x_{0})>0$ and

$M_{+}^{p-1}(\lambda,u)=u^{p-1}(x_{0})\geq\frac{-\lambda}{a(x_{0})}.$

From this estimate, (2.29) and (2.30) follow readily. $\square$

Figure 4: Two minimal global bifurcation diagrams

In Figure 4 we have plotted the _minimal_ global bifurcation diagram of nodal
solutions of (2.1) for the generalized Moore–Nehari model analyzed in this
section. It is minimal in the sense, that, besides the components
$\mathfrak{C}_{\kappa}$, $\kappa\geq 1$, represented in it, the model might
possess some additional components, separated away from the
$\mathfrak{C}_{\kappa}$’s, that has not been plotted on it, because its
existence cannot be guaranteed unless some additional work is done. Actually,
the results of the next sections suggest the existence of a more complicated
bifurcation diagram for a piecewise constant weight satisfying (2.11). Indeed,
at least for $n$ sufficiently large we can provide several solutions with a
fixed number of zeros which are differently distributed in the intervals
$I^{+}_{i}$ and $I^{0}_{i}$. Moreover, the existence of an arbitrarily large
number of turning points along some of the components $\mathfrak{C}_{\kappa}$,
$\kappa\geq 1$, cannot be excluded unless $a\equiv\mu>0$ in $[0,L]$. Indeed,
for the classical choice of Moore and Nehari [35], there are some negative
ranges of the parameters where (2.1) has three positive solutions. And, at
least when $a(t)<0$ for all $t\in[0,L]$, one can construct examples of weight
functions, as close as wanted to some constant, for which (2.1) has an
arbitrarily large number of solutions with one interior node, distributed in
finitely many isolas (see Cubillos et al. [8]). Whether or not this
multiplicity result holds also when $a\gneq 0$ remains an open problem yet.

More precisely, in Figure 4, for any given solution $(\lambda,u)$, we are
plotting $M_{+}(\lambda,u)$ versus $\lambda$ if $u^{\prime}(0)>0$, while
$\lambda$ is plotted versus $M_{-}(\lambda,u)$ if $u^{\prime}(0)<0$.
Naturally, the “fictitious” intersections between the components
$\mathfrak{C}_{\kappa}$’s cannot be excluded to occur, neither the existence
of turning points along the components $\mathfrak{C}_{\kappa}$’s. However, as
far as concerns the minimal structure of those global bifurcation diagrams,
Figure 4 preserves the main topological features of the global bifurcation
diagrams already plotted in Figure 3 for the special case when
$a(t)\equiv\mu>0$ is a positive constant. The dashed black curves in Figure 4
represent the limiting curves of the estimates (2.29) and (2.30).

##  3 Some preliminaries for the quasilinear prototype

Throughout the rest of this paper we will focus attention into the quasilinear
equation

$-(\phi(u^{\prime}))^{\prime}=\lambda u+a(t)g(u),\quad\lambda\in{\mathbb{R}},$
(3.1)

already introduced in Section 1, where $a(t)$ is a piece-wise constant
function in the class of (1.2). Thus,

$a(t)\equiv\mu_{i}>0,\;\;\forall\,t\in[t_{i},s_{i}],\quad i\in\\{0,...,m\\}.$
(3.2)

So, $a(t)$ is a stepwise function with constant values $\mu_{i}>0$ on the
$I^{+}_{i}$-intervals. In the special case when $\phi(s)=s$ for all
$s\in\mathbb{R}$, (3.1) is a semilinear equation of the form (2.1). Thus,
(3.1) can be regarded as a quasilinear generalized prototype of the
generalized Moore–Nehari equation (2.1).

Our study of (3.1) in this paper will be performed by analyzing the dynamical
system associated with the associated first order system

$\begin{cases}x^{\prime}=h(y),\\\ y^{\prime}=-\lambda x-a(t)g(x),\end{cases}$
(3.3)

where $h(y):=\phi^{-1}(y)$, whose general properties were fixed in Section 1.
Under assumption (3.2), the system (3.3) can be seen as a switched system,
i.e., a system made of a superposition of a fully nonlinear system,

$\begin{cases}x^{\prime}=h(y),\\\ y^{\prime}=-f_{i,\lambda}(x),\end{cases}$
$None$

where, for every $i\in\\{0,...,m\\}$,

$f_{i,\lambda}(x):=\lambda x+\mu_{i}g(x),\quad\lambda,x\in\mathbb{R},$ (3.4)

together with a linear system in the $x$-variable

$\begin{cases}x^{\prime}=h(y),\\\ y^{\prime}=-\lambda x.\end{cases}$ $None$

Since both $({\mathcal{N}}_{i,\lambda})$ and $({\mathcal{L}}_{\lambda})$ are
autonomous systems, we can equivalently assume that
$({\mathcal{N}}_{i,\lambda})$ acts on a time-interval $[0,\tau_{i}]$ with
$\tau_{i}=s_{i}-t_{i}$ and its effect is followed by that of the system
$({\mathcal{L}}_{\lambda})$ acting on a time-interval $[0,\varsigma_{i}]$ with
$\varsigma_{i}=t_{i+1}-s_{i}$. In this framework, if we denote by $\Phi_{i}$
the Poincaré map associated with the system $({\mathcal{N}}_{i,\lambda})$ on
the interval $[0,\tau_{i}]$ and by $\Psi_{i}$ that one associated with system
$({\mathcal{L}}_{\lambda})$ on the interval $[0,\varsigma_{i}]$, we have that
the Poincaré map of (3.3) in $[0,L]$ splits as

${\mathscr{P}}_{L}:=\Phi_{m}\circ\Psi_{m-1}\circ\Phi_{m-1}\circ\dots\circ\Psi_{0}\circ\Phi_{0}.$

To make sure that the Poincaré map is well defined, we impose the uniqueness
of the solutions for the initial value problems associated with
$({\mathcal{N}}_{i,\lambda})$ and $({\mathcal{L}}_{\lambda})$. Since we are
dealing with planar Hamiltonian systems, the uniqueness holds under very
general conditions of $h$ and $g$. If some extra assumptions would be
required, we will highlight them when needed. Actually, we can directly apply
a uniqueness result from Rebelo [43] if $\lambda>0$, and the same result
applies when $\lambda=0$ for the system $({\mathcal{N}}_{i,\lambda})$, while
for the Cauchy problem associated to $({\mathcal{L}}_{\lambda})$ the
uniqueness holds from an elementary direct argument. The situation might be
more subtle when $\lambda<0$, where we need to prevent the possibility that
nontrivial solutions reach the origin in finite time. For instance, this could
happen for the system $({\mathcal{L}}_{\lambda})$ for $\lambda=-1$ and
$h(y)=|y|^{q}{\rm sgn}(y)$ with $0<q<1$. In the case $\lambda<0$ the
uniqueness can be inferred via the strong maximum principle for
$\phi$-Laplacian operators (see Pucci and Serrin [38, Th. 1.1.1]).

In this section, we are going to describe some general properties of the
systems $({\mathcal{N}}_{i,\lambda})$ and $({\mathcal{L}}_{\lambda})$.
Throughout the rest of this paper, we will consider the primitives

$H(y):=\int_{0}^{y}h(s)\,ds,\qquad
F_{i,\lambda}(x):=\int_{0}^{x}f_{i,\lambda}(s)\,ds=\frac{\lambda}{2}x^{2}+\mu_{i}G(x),$

where

$G(x):=\int_{0}^{x}g(s)\,ds,$

and we define

$H(\varrho_{\pm})_{=}\lim_{y\to\varrho_{\pm}}H(y),$

where $(\varrho_{-},\varrho_{+})$ is the interval of definition of $h$. Note
that $H(\varrho_{\pm})=+\infty$ if $\varrho_{\pm}=\pm\infty$.

###  3.1 A first glance at the system $({\mathcal{N}}_{i,\lambda})$

Observe that $({\mathcal{N}}_{i,\lambda})$ is an Hamiltonian system with

${\mathcal{H}}(x,y)={\mathcal{H}}_{i,\lambda}(x,y):=H(y)+F_{i,\lambda}(x),\quad\text{for
}\;(x,y)\in{\mathbb{R}}\times(\varrho_{-},\varrho_{+}).$

Hamiltonian functions “with separable variables” of this form have been
already studied in Schaaf [44, 45], Cima et al. [6], and, more recently, in
Villadelprat and Zhang [47], where the reader is sent for some recent studies
in this area. Given $c\in\mathbb{R}$ such that

$0<c<H^{*}:=\min\\{H(\varrho_{-}),H(\varrho_{+})\\},$ (3.5)

assume that there are constants $x_{\pm}(c)$, with $x_{-}(c)<0<x_{+}(c)$, such
that

$\begin{split}&F_{i,\lambda}(x_{-}(c))=F_{i,\lambda}(x_{+}(c))=c,\\\
&F_{i,\lambda}(x)<c\;\;\hbox{for all}\;\;x\in(x_{-}(c),x_{+}(c)),\\\
&F^{\prime}_{i,\lambda}(x_{-}(c))<0<F^{\prime}_{i,\lambda}(x_{+}(c))\end{split}$
(3.6)

Thus, the level line ${\mathcal{H}}(x,y)=c$ is a simple closed curve around
the origin intersecting the $x$-axis at the points $(x_{\pm}(c),0)$ and the
$y$-axis at the points $(0,y_{\pm}(c))$, where $y_{-}(c)<0<y_{+}(c)$ and
$H(y_{\pm}(c))=c$. Since $\mathcal{H}^{-1}(c)$ does not contain equilibrium
points of the system $({\mathcal{N}}_{i,\lambda})$, it is a periodic orbit
running clockwise. The next result shows that the existence of a
$c\in(0,H^{*})$ for which (3.6) holds is ensured by a more general condition.

######  Lemma 3.1.

Assume that $F_{i,\lambda}(x^{\pm})>0$ for some $x^{-}<0<x^{+}$. Then, there
exists an interval $[c_{1},c_{2}]\subset(0,H^{*})$ such that, for every
$c\in[c_{1},c_{2}]$, there are $x_{\pm}(c)$ satisfying (3.6).

###### Proof.

Since $F_{i,\lambda}(0)=0$ and $F_{i,\lambda}$ is continuous, for every
$c\in(0,\min\\{H^{*},F_{i,\lambda}(x^{\pm})\\})$, there exist $x_{\pm}(c)$
satisfying the first two relations of (3.6). Moreover, since $F_{i,\lambda}$
is of class $C^{1}$, by applying Sard’s theorem, the existence of a regular
value $\tilde{c}$ of $F_{i,\lambda}$ in
$(0,\min\\{H^{*},F_{i,\lambda}(x^{\pm})\\})$ holds. Thus, there exist
$x_{\pm}(\tilde{c})$ satisfying (3.6). Finally, since $F_{i,\lambda}$ is
$C^{1}$, there exists a closed neighborhood $[c_{1},c_{2}]$ of $\tilde{c}$
where (3.6) holds. $\square$

To determine the period of the orbit $\mathcal{H}^{-1}(c)$, we can sum up the
times needed to cross each quadrant. First, from the relation
$H(y)+F_{i,\lambda}(x)=c$, since $x^{\prime}=h(y)$, we find that

$x^{\prime}=\left\\{\begin{array}[]{ll}(h\circ
H_{+}^{-1})(c-F_{i,\lambda}(x))&\text{if}\;x>0,\\\\[3.0pt] (h\circ
H_{-}^{-1})(c-F_{i,\lambda}(x))&\text{if}\;x<0,\end{array}\right.$

where $H_{+}^{-1}$ and $H_{-}^{-1}$ are the inverse functions of $H$
restricted to $[0,\varrho_{+})$ and $(\varrho_{-},0]$, respectively. From
this, we obtain that the necessary time to move from $(0,y_{+}(c))$ to
$(x_{+}(c),0)$ in the first quadrant is given by

${\mathcal{T}}_{i,I}(c):=\int_{0}^{x_{+}(c)}\frac{dx}{(h\circ
H_{+}^{-1})(c-F_{i,\lambda}(x))}.$

Similarly, the times to cross the fourth, third and second quadrants, can be
expressed as

${\mathcal{T}}_{i,IV}(c):=\int_{0}^{x_{+}(c)}\frac{dx}{-(h\circ
H_{-}^{-1})(c-F_{i,\lambda}(x))},$
${\mathcal{T}}_{i,III}(c):=\int_{x_{-}(c)}^{0}\frac{dx}{-(h\circ
H_{-}^{-1})(c-F_{i,\lambda}(x))}$

and

${\mathcal{T}}_{i,II}(c):=\int_{x_{-}(c)}^{0}\frac{dx}{(h\circ
H_{+}^{-1})(c-F_{i,\lambda}(x))},$

respectively. Obviously, the period of the closed orbit corresponding to the
level line ${\mathcal{H}}=c$ is given by

${\mathcal{T}}_{i}(c)={\mathcal{T}}_{i,I}(c)+{\mathcal{T}}_{i,IV}(c)+{\mathcal{T}}_{i,III}(c)+{\mathcal{T}}_{i,II}(c).$

Observe that

$\displaystyle{\mathcal{T}}_{i,I}(c)={\mathcal{T}}_{i,IV}(c)\;\;\text{and
}\;\;{\mathcal{T}}_{i,II}(c)={\mathcal{T}}_{i,III}(c)\quad\text{if $h$ is
odd},$
$\displaystyle{\mathcal{T}}_{i,I}(c)={\mathcal{T}}_{i,II}(c)\;\;\text{and
}\;\;{\mathcal{T}}_{i,III}(c)={\mathcal{T}}_{i,IV}(c)\quad\text{if $g$ is
odd},$

so that all the quarter-laps times coincide with ${\mathcal{T}}_{i,I}(c)$ if
both $h$ and $g$ are odd. The formulas for the times ${\mathcal{T}}_{i,j}(c)$
have been already given in [6, Pr. 2.4].

Observe that in the case $\lambda\geq 0$ there is always an interval
$[c_{1},c_{2}]\subset(0,H^{*})$ for which (3.6) holds on $x_{\pm}(c)$ for
every $c\in[c_{1},c_{2}]$. The case $\lambda<0$ is more delicate. First, as
the existence of such an interval is not a priori ensured, the existence of
$x^{\pm}$ in the statement of Lemma 3.1 should be imposed when working in the
interval $[t_{i},s_{i}]$ with the problem $(\mathcal{N}_{i,\lambda})$. Second,
the geometric structure of the orbits can change completely depending on the
sublinear/superlinear behavior of $g(x)$ near zero and/or infinity, as one can
see, by analyzing the function

$f_{i,\lambda}(x)=\lambda x+\mu_{i}|x|^{p-1}x$

for $p>1$ or $0<p<1$.

###  3.2 A first glance at the system $({\mathcal{L}}_{\lambda})$

In this section, we evaluate the necessary time to move along the trajectories
of the $x$-linear system $({\mathcal{L}}_{\lambda})$. The trajectories lie on
the level lines of the Hamiltonian function

${\mathcal{E}}(x,y)={\mathcal{E}}_{\lambda}(x,y):=H(y)+\frac{\lambda}{2}x^{2},\quad\text{for
}\;(x,y)\in{\mathbb{R}}\times(\varrho_{-},\varrho_{+}).$

The dynamics changes substantially according to the sign of the parameter
$\lambda$. Thus, we will consider three different cases.

  * •

The case $\lambda<0$.

Then, the origin is a saddle point whose stable and unstable manifolds lie on
the level line of zero-energy, given by $2H(y)=-\lambda x^{2}$. The level
lines with positive energy correspond to trajectories in the upper half-plane
moving from the second towards the first quadrant, and to trajectories in the
lower half-plane moving from the fourth towards the third quadrant. Let us
consider now the energy level lines passing through the points $(0,y_{+})$ and
$(0,y_{-})$ with $\varrho_{-}<y_{-}<0<y_{+}<\varrho_{+}$, namely

$H(y)=H(y_{\pm})-\frac{\lambda}{2}x^{2}.$

Then, either

$x^{\prime}=(h\circ
H_{+}^{-1})\left(H(y_{+})-\frac{\lambda}{2}x^{2}\right),\quad\hbox{or}\quad
x^{\prime}=(h\circ H_{-}^{-1})\left(H(y_{-})-\frac{\lambda}{2}x^{2}\right).$

Thus, for any given $\ell<r$, the transition times from the straight line
$x=\ell$ to the straight line $x=r$, with $y>0$, and from the straight line
$x=r$ to the straight line $x=\ell$, with $y<0$, are given by

${\mathcal{S}}^{\lambda}(\ell,r;(0,y_{+})):=\int_{\ell}^{r}\frac{dx}{(h\circ
H_{+}^{-1})\left(H(y_{+})-\frac{\lambda}{2}x^{2}\right)}$ (3.7)

and

${\mathcal{S}}^{\lambda}(r,\ell;(0,y_{-})):=\int_{\ell}^{r}\frac{dx}{-(h\circ
H_{-}^{-1})\left(H(y_{-})-\frac{\lambda}{2}x^{2}\right)},$ (3.8)

respectively. In order to have the arc of trajectory well defined between
these two straight lines, it is required that

$\max\left\\{H(y_{\pm})-\frac{\lambda}{2}\ell^{2},H(y_{\pm})-\frac{\lambda}{2}r^{2}\right\\}<H(\varrho_{\pm}).$

The level lines with negative energy correspond to trajectories on the right
half-plane moving from the fourth towards the first quadrant, and to
trajectories on the left half-plane moving from the second towards the third
quadrant. Figure 5 shows the deformation of some regions under the flow
associated with $({\mathcal{L}}_{\lambda})$ for $\lambda<0$.

Figure 5: The action of the Poicaré map.

Precisely, Figure 5 shows the deformation of two trapezoidal regions under the
action of the Poincaré map associated with $({\mathcal{L}}_{\lambda})$. For
this particular example we have chosen $h(y)=y$ and $\lambda=-1/2$, for a time
interval of length $\varsigma=1.5$. The two trapezoids are originally
supported on each of the two components of the stable manifold through the
origin, which is the straight line $y=-\sqrt{|\lambda|}x$. As an effect of the
Poincaré map, the segments on the components of the stable manifold are
shifted towards the origin, while the remaining points of the trapezoids move
towards the unstable manifold $y=\sqrt{|\lambda|}x$.

Next, we will consider the energy level lines passing through the points
$(x_{\pm},0)$, with $x_{-}<0<x_{+}$, namely

$H(y)=\frac{\lambda}{2}(x_{\pm}^{2}-x^{2}).$

Then, either

$x^{\prime}=(h\circ
H_{\pm}^{-1})\left(\frac{\lambda}{2}(x_{+}^{2}-x^{2})\right),\quad\hbox{or}\quad
x^{\prime}=(h\circ
H_{\pm}^{-1})\left(\frac{\lambda}{2}(x_{-}^{2}-x^{2})\right),$

and hence, for any given $r_{1},r_{2}>x_{+}$ and $\ell_{1},\ell_{2}<x_{-}$,
the transition time between the straight lines $x=r_{1}$, with $y<0$, and
$x=r_{2}$, with $y>0$, along the respective energy level lines, is given by

${\mathcal{S}}^{\lambda}(r_{1},r_{2};(x_{+},0)):=\\!\\!\int_{x_{+}}^{r_{2}}\frac{dx}{(h\circ
H_{+}^{-1})\left(\frac{\lambda}{2}(x_{+}^{2}-x^{2})\right)}\\!+\\!\int_{x_{+}}^{r_{1}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(\frac{\lambda}{2}(x_{+}^{2}-x^{2})\right)}.$ (3.9)

Similarly, the transition time between the straight line $x=\ell_{1}$, with
$y>0$, and $x=\ell_{2}$, with $y>0$, is given through

${\mathcal{S}}^{\lambda}(\ell_{1},\ell_{2};(x_{-},0)):=\\!\\!\int_{\ell_{1}}^{x_{-}}\frac{dx}{(h\circ
H_{+}^{-1})\left(\frac{\lambda}{2}(x_{-}^{2}-x^{2})\right)}\\!+\\!\int_{\ell_{2}}^{x_{-}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(\frac{\lambda}{2}(x_{-}^{2}-x^{2})\right)}.$ (3.10)

The next compatibility condition

$\max\left\\{\frac{\lambda}{2}(x_{+}^{2}-r_{j}^{2}),\frac{\lambda}{2}(x_{-}^{2}-\ell_{j}^{2})\right\\}<H^{*},\quad\text{for
}\;j=1,2,$

makes sure that the arc of trajectory between each of these two lines is
contained in the strip ${\mathbb{R}}\times(\varrho_{-},\varrho_{+})$.

  * •

The case $\lambda=0$.

In this case, the system $({\mathcal{L}}_{0})$ reduces to $x^{\prime}=h(y)$,
$y^{\prime}=0$. Thus, the $x$-axis consists of equilibria. Moreover, the
motion has a constant speed, $h(y)$, on parallel lines to the $x$-axis from
left to the right, or the right to the left, according to whether $y>0$, or
$y<0$. Figure 6 shows the deformation of some rectangles by the action of the
flow induced by $({\mathcal{L}}_{0})$.

Figure 6: The action of the flow induced by $({\mathcal{L}}_{0})$.

Figure 6 shows the deformation of two rectangles under the action of the
Poincaré map associated with $({\mathcal{L}}_{0})$. For this particular
example we have chosen $h(y)=y/\sqrt{1+y^{2}}$, which comes from inverting the
relativistic operator
$\frac{d}{dt}\bigl{(}\frac{x^{\prime}}{\sqrt{1-(x^{\prime})^{2}}}\bigr{)}$,
for a time interval of length $\varsigma=6$. The rectangles are
$[-2,-1]\times[0,4]$ and $[2,3]\times[-5,0]$.

Let us consider the orbits passing through the points $(0,y_{+})$ and
$(0,y_{-})$ with $\varrho_{-}<y_{-}<0<y_{+}<\varrho_{+}$. From
$x^{\prime}=h(y_{\pm})$ we find that, for any given $\ell<r$, the transition
times from the straight line $x=\ell$ to the straight line $x=r$ with $y>0$,
and from $x=r$ to $x=\ell$ with $y<0$, are given by

${\mathcal{S}}^{0}(\ell,r;(0,y_{+})):=\frac{r-\ell}{h(y_{+})}\quad\text{and
}\quad{\mathcal{S}}^{0}(r,\ell;(0,y_{-})):=\frac{r-\ell}{-h(y_{-})},$

respectively.

  * •

The case $\lambda>0$.

This situation is reminiscent of the case already analyzed for the system
$({\mathcal{N}}_{\lambda})$. Indeed, the origin is a center because, for every
$c\in(0,H^{*})$, the integral curve ${\mathcal{E}}_{\lambda}=c$ is a periodic
orbit surrounding the origin that is symmetric with respect to the $y$-axis.
This curve intersects the $x$-axis at the points $(x_{\pm}(c),0)$ with
$x_{\pm}(c)=\pm(2c/\lambda)^{1/2}$, and the $y$-axis at $(0,H^{-1}_{+}(c))$
and $(0,H^{-1}_{-}(c))$. Thus, by adapting the formulas for the times
${\mathcal{T}}_{i,j}$, $j\in\\{I,II,III,IV\\}$, needed to cross each quadrant,
after a simple change of variable, we find that

${\mathcal{S}}^{\lambda}_{I}(c):=(2c/\lambda)^{1/2}\int_{0}^{1}\frac{dx}{(h\circ
H_{+}^{-1})(c(1-x^{2}))}$

is the crossing time of the first quadrant, and

${\mathcal{S}}^{\lambda}_{IV}(c):=(2c/\lambda)^{1/2}\int_{0}^{1}\frac{dx}{-(h\circ
H_{-}^{-1})(c(1-x^{2}))}$

is the crossing time of the fourth quadrant. By the symmetry with respect to
the $y$-axis, the crossing times of the third and second quadrants are given
by

${\mathcal{S}}^{\lambda}_{II}(c)={\mathcal{S}}^{\lambda}_{I}(c)\quad\text{ and
}\;\;{\mathcal{S}}^{\lambda}_{III}(c)={\mathcal{S}}^{\lambda}_{IV}(c),$

respectively.

##  4 Transformation of arcs in $({\mathcal{N}}_{i,\lambda})$

This section analyzes the effect of the transformation of certain arcs through
the Poincaré map $\Phi_{i}$, $i\in\\{0,\ldots,m\\}$. To describe the movement
of those arcs, we need to introduce some regions and a few notations. As these
regions are always defined in the domain
${\mathbb{R}}\times(\varrho_{-},\varrho_{+})$, throughout this section (3.5)
is required to be satisfied.

Let $0<c_{1,i}<c_{2,i}$ be such that, for every $c\in[c_{1,i},c_{2,i}]$, the
condition (3.6) is satisfied. Then, the annulus

${\mathcal{A}}_{i}:=\\{(x,y):c_{1,i}\leq{\mathcal{H}}(x,y)\leq c_{2,i}\\}$

consists of the union of the periodic orbits ${\mathcal{H}}(x,y)=c$ of the
system $({\mathcal{N}}_{i,\lambda})$ for $c\in[c_{1,i},c_{2,i}]$. As these
orbits surround the origin, also ${\mathcal{A}}_{i}$ is “centered ” at the
origin. Next, we split ${\mathcal{A}}_{i}$ into four rectangular regions,
${\mathcal{A}}_{i,j}$, with $j\in\\{I,II,III,IV\\}$, where the index $j$
corresponds to the intersection of ${\mathcal{A}}_{i}$ with the closed
$j$-quadrant. For each of these regions, we select an orientation, by fixing
two opposite sides, ${\mathcal{A}}_{i,j}^{-}$, that we denote as follows:

$\displaystyle\mathcal{A}^{\rm
energy}_{i,j}:=({\mathcal{A}}_{i,j},{\mathcal{A}}_{i,j}^{-}),\quad\hbox{where}\;\;{\mathcal{A}}_{i,j}^{-}:={\mathcal{A}}_{i,j}\cap\left(\\{{\mathcal{H}}=c_{1,i}\\}\cup\\{{\mathcal{H}}=c_{2,i}\\}\right),$
$\displaystyle\mathcal{A}^{\rm
axis}_{i,j}:=({\mathcal{A}}_{i,j},{\mathcal{A}}_{i,j}^{-}),\quad\hbox{where}\;\;{\mathcal{A}}_{i,j}^{-}:={\mathcal{A}}_{i,j}\cap\bigl{(}\\{xy=0\\}\bigr{)}.$

For each of the four regions represented on each picture of Figure 7 we have
drawn also an arc connecting the two opposite sides of
${\mathcal{A}}_{i,j}^{-}$.

Figure 7: The oriented regions $\mathcal{A}^{\rm energy}_{i,j}$ (left) and
$\mathcal{A}^{\rm axis}_{i,j}$ (right).

To generate Figure 7, we we have taken
$h(y)=\log(\varrho_{+}/(\varrho_{+}-y))$, with $\varrho_{+}=4$, which is an
increasing homeomorphism $(\varrho_{-},\varrho_{+})\to{\mathbb{R}}$, for
$\varrho_{-}=-\infty$, with $h(0)=0$, $g(x)=-1+e^{x}$, and
$f_{\lambda}(x):=\lambda x+\mu g(x)$ with $\lambda=-1/20$ and $\mu=4/5$. For
this particular choice of $h$, $g$ and $\lambda$, the integral curves of
${\mathcal{H}}$ are closed trajectories around the origin contained in the
domain ${\mathbb{R}}\times(\varrho_{-},\varrho_{+})$ only for sufficiently
small $c>0$.

Throughout the rest of this paper, given two oriented rectangular regions,
$({\mathcal{R}},{\mathcal{R}}^{-})$ and $({\mathcal{Q}},{\mathcal{Q}}^{-})$,
and a homeomorphism $\eta$ between these two _rectangles_ , by setting

$\eta:({\mathcal{R}},{\mathcal{R}}^{-})\overset{\mathfrak{m}}{\Bumpeq{\\!\\!\\!\\!\\!\\!\\!\\!{\longrightarrow}}}({\mathcal{Q}},{\mathcal{Q}}^{-})\quad(\mathfrak{m}\geq
1)$

we mean that there are $\mathfrak{m}$ compact disjoint sets
$K_{1},\ldots,K_{\mathfrak{m}}$ contained in ${\mathcal{R}}$ such that any arc
$\gamma$ contained in ${\mathcal{R}}$ and linking the two opposite sides of
${\mathcal{R}}^{-}$ contains $\mathfrak{m}$ sub-arcs $\gamma_{j}\subset
K_{j}$, $j\in\\{1,...,\mathfrak{m}\\}$, such that, for every
$j\in\\{1,...,\mathfrak{m}\\}$, $\eta(\gamma_{j})$ is an arc contained in
${\mathcal{Q}}$ connecting the two opposite sides of ${\mathcal{Q}}^{-}$.
Thus, $\mathfrak{m}$ can be regarded as a sort of crossing number, as
discussed by Kennedy and Yorke [15]. When $\mathfrak{m}=1$, we omit it and
simply set $\Bumpeq{\\!\\!\\!\\!\\!\\!\\!\\!{\longrightarrow}}$.

Once guaranteed that the annulus ${\mathcal{A}}_{i}$ is properly defined as a
subset of ${\mathbb{R}}\times(\varrho_{-},\varrho_{+})$, the Poincaré map
$\Phi_{i}$ of $({\mathcal{N}}_{i,\lambda})$ induces an homeomorphism of the
annulus onto itself. Now, we will impose a suitable twist condition on the
boundaries of ${\mathcal{A}}_{i}$ in order to prove that an arc connecting the
inner with the outer boundaries of the annulus stretches into a spiral like
curve winding a certain number of times across the annulus itself, in order to
prove a property of the form

$\mathcal{A}^{\rm
energy}_{i,j^{\prime}}\overset{\mathfrak{m}}{\Bumpeq{\\!\\!\\!\\!\\!\\!\\!\\!{\longrightarrow}}}\,\mathcal{A}^{\rm
axis}_{i,j^{\prime\prime}}$

for a suitable choice of $\mathfrak{m}$, $j^{\prime}$ and $j^{\prime\prime}$.
This is the main result of this section. Precisely, the following result
holds. Recall that the Poincaré map $\Phi_{i}$ acts on a time-interval of
length $\tau_{i}=s_{i}-t_{i}$ for system $({\mathcal{N}}_{i,\lambda})$.

######  Proposition 4.1.

Given the cyclic clockwise order $IV\prec III\prec II\prec I\prec IV$ and
$d,e\in\\{c_{1,i},c_{2,i}\\}$ with $d\not=e$, assume that, for every
$\kappa\in\\{IV,III,II,I\\}$, and, for some non-negative integers
$0\leq\alpha_{i}<\beta_{i}$, the following twist condition is satisfied

$\begin{split}&{\mathcal{T}}_{i,\kappa+1}(d)+{\mathcal{T}}_{i,\kappa+2}(d)+\alpha_{i}{\mathcal{T}}_{i}(d)>\tau_{i},\\\
&{\mathcal{T}}_{i,\kappa}(e)+{\mathcal{T}}_{i,\kappa+1}(e)+\beta_{i}{\mathcal{T}}_{i}(e)<\tau_{i}.\end{split}$
(4.1)

Then,

$\Phi_{i}:\mathcal{A}^{\rm
energy}_{i,\kappa}\overset{\beta_{i}-\alpha_{i}}{\Bumpeq{\\!\\!\\!\\!\\!\\!\\!\\!{\longrightarrow}}}\,\mathcal{A}^{\rm
axis}_{i,\kappa-1}\cup\mathcal{A}^{\rm axis}_{i,\kappa+1}.$ (4.2)

Furthermore, fixing $\kappa=I\;\hbox{or}\;III$, for every
$j=\\{1,2,\ldots,2(\beta_{i}-\alpha_{i})\\}$, there is at least one solution
of $(\mathcal{N}_{i,\lambda})$ starting at $\mathcal{A}_{i,\kappa}$ and having
exactly

  * (i)

$2\alpha_{i}+j$ zeroes in the $x$-component in the interval $(0,\tau_{i})$,
and

  * (ii)

$2\alpha_{i}+j+1$ zeroes in the $y$-component in $[0,\tau_{i}]$.

The rest of this section is devoted to the proof of Proposition 4.1, which is
divided into two parts. The first part shows (4.2). The second one holds from
a technical lemma where the nodal behavior of the solutions starting at
$\mathcal{A}_{i,\kappa}$ for $\kappa\in\\{I,III\\}$ is described. This
provides us with the multiplicity result of the zeroes in the $x$ and $y$
components stated in the proposition. A similar result holds for solutions
starting at $\mathcal{A}_{i,\kappa}$ with $\kappa\in\\{II,IV\\}$. However, as
it will become apparent in later sections, we are just interested in the nodal
behavior of the solutions starting at $\mathcal{A}_{i,I}$ or
$\mathcal{A}_{i,III}$.

###### Proof.

Suppose (4.1). We will detail the proof of (4.2) in the case $\kappa=I$, as in
the remaining cases follows easily by symmetry. Just to fix ideas, suppose
that $d=c_{1,i}$ and $e=c_{2,i}$ (the other case is analogous).

Using a system of polar coordinates with the angle counted in the clockwise
sense starting from the positive $y$-axis, we can associate to the solutions
of $(\mathcal{N}_{i,\lambda})$ an angular component $\theta(t;z)$, with
$t\in[0,\tau_{i}]$ and $z\in{\mathcal{A}}_{i,I}$ representing the angular
coordinate of the solution at time $t$, starting at the point $z$ at time $0$.
The map $\theta$ is continuous, and $0\leq\theta(0,z)\leq\pi/2$, for all
$z\in{\mathcal{A}}_{i,I}$.

Let $\gamma$ be a continuous arc of curve contained in ${\mathcal{A}}_{i,I}$
linking the two sides of ${\mathcal{A}}_{i,I}^{-}$ which are the part of the
level lines ${\mathcal{H}}=c_{1,i}$ and ${\mathcal{H}}=c_{2,i}$ in the first
quadrant. The arc $\gamma$ is the homeomorphic image of the compact interval
$[0,1]$ through a homeomorphism, still denoted by
$\gamma:[0,1]\to{\mathcal{A}}_{i,I}$, such that
$\gamma(0)=z_{0}\in\\{{\mathcal{H}}=c_{1,i}\\}$ and
$\gamma(1)=z_{1}\in\\{{\mathcal{H}}=c_{2,i}\\}$. Without loss of generality,
we can assume that $c_{1,i}<{\mathcal{H}}(\gamma(s))<c_{2,i}$ for all
$s\in(0,1)$. Now, consider the image of $\gamma=\gamma([0,1])$ through the
Poincaré map $\Phi_{i}$. Then,

$\vartheta(t,s):=\theta(t;\gamma(s)),\quad t\in[0,\tau_{i}],\;\;s\in[0,1],$

represents the angular coordinate of the solution of
$({\mathcal{N}}_{i,\lambda})$ at time $t$, starting at $\gamma(s)$ (at time
$0$). The map $\vartheta$ is continuous and satisfies
$0\leq\vartheta(0,s)\leq\pi/2$ for all $s\in[0,1]$. Moreover, for each
nonnegative integer $k$, as soon as $c=\mathcal{H}(\gamma(s))$, we have that

${\mathcal{T}}_{i,IV}(c)+{\mathcal{T}}_{i,III}(c)+k{\mathcal{T}}_{i}(c)>t\;\Longrightarrow\;\vartheta(t,s)<\frac{3}{2}\pi+2k\pi,$
(4.3)
${\mathcal{T}}_{i,I}(c)+{\mathcal{T}}_{i,IV}(c)+k{\mathcal{T}}_{i}(c)<t\;\Longrightarrow\;\vartheta(t,s)>\pi+2k\pi.$
(4.4)

Thus, from (4.3) and the first condition of (4.1), we find that

$\vartheta(\tau_{i},0)<\frac{3}{2}\pi+2\alpha_{i}\pi.$

On the other hand, from (4.4) and the second condition of (4.1), it becomes
apparent that

$\vartheta(\tau_{i},1)>\pi+2\beta_{i}\pi.$

Thus, the continuous map $[0,1]\ni s\mapsto\vartheta(\tau_{i},s)$ covers the
whole compact interval

$\left[\tfrac{3}{2}\pi+2\alpha_{i}\pi,\pi+2\beta_{i}\pi\right].$

Hence, there are $2(\beta_{i}-\alpha_{i})$ pairwise disjoint compact
subintervals of $[0,1]$ such that this map covers, for every
$k\in\\{\alpha_{i},\dots,\beta_{i}-1\\}$, the intervals

$\left[\tfrac{3}{2}\pi+2k\pi,2(k+1)\pi\right]\quad\text{and}\quad\left[\tfrac{1}{2}\pi+2(k+1)\pi,\pi+2(k+1)\pi\right].$

Now, we introduce the sets

$\begin{split}&K_{j,II}:=\left\\{z\in{\mathcal{A}}_{i,I}:\,\theta(\tau_{i};z)\in\left[\tfrac{3}{2}\pi+2(\alpha_{i}+j-1)\pi,2(\alpha_{i}+j)\pi\right]\right\\},\\\\[3.0pt]
&K_{j,IV}:=\left\\{z\in{\mathcal{A}}_{i,I}:\,\theta(\tau_{i};z)\in\left[\tfrac{1}{2}\pi+2(\alpha_{i}+j)\pi,\pi+2(\alpha_{i}+j)\pi\right]\right\\},\end{split}$
(4.5)

for $j=1,\dots,\beta_{i}-\alpha_{i},$ which are compact, nonempty and pairwise
disjoint. By a simple continuity argument, given an arbitrary arc $\gamma$ in
${\mathcal{A}}_{i,I}$ connecting the two opposite sides of $\mathcal{A}^{\rm
energy}_{i,I}$, it is easily seen that, for every $\kappa\in\\{II,IV\\}$,
there are $\beta_{i}-\alpha_{i}$ sub-arcs $\gamma_{j,\kappa}\subset
K_{j,\kappa}$ such that $\Phi_{i}(\gamma_{j,\kappa})$ is an arc contained in
${\mathcal{A}}_{i,\kappa}$ linking the two sides $y=0$ and $x=0$ of
$\mathcal{A}^{\rm axis}_{i,\kappa}$. This proves that

$\Phi_{i}:\mathcal{A}^{\rm
energy}_{i,I}\overset{\beta_{i}-\alpha_{i}}{\Bumpeq{\\!\\!\\!\\!\\!\\!\\!\\!{\longrightarrow}}}\,\mathcal{A}^{\rm
axis}_{i,II}\cup\mathcal{A}^{\rm axis}_{i,IV}$

according to our definition. The multiplicity results of the zeroes of the
solutions stated in the second part of the proposition holds as a direct
consequence of the next technical lemma, which completes the proof of the
proposition. $\square$

######  Lemma 4.1.

For every $\kappa\in\\{I,III\\}$, and any arc
$\gamma\subset\mathcal{A}_{i,\kappa}^{\rm energy}$ joining the opposite sides
of $\mathcal{A}_{i,\kappa}^{-}$, the solutions of
$({\mathcal{N}}_{i,\lambda})$ with initial value on any sub-arc
$\gamma_{j,\tilde{\kappa}}\subset K_{j,\tilde{\kappa}}$ of $\gamma$ have
exactly, for every $j=1,2,\ldots,\beta_{i}-\alpha_{i}$,

$\left.\begin{array}[]{ll}2(\alpha_{i}+j)-1&\;\hbox{zeroes in the x-component
in}\;\;(0,\tau_{i})\\\ 2(\alpha_{i}+j)&\;\hbox{zeroes in the y-component
in}\;\;[0,\tau_{i}]\end{array}\right\\}\;\hbox{if}\;\;\tilde{\kappa}=\kappa-1$

and

$\left.\begin{array}[]{ll}2(\alpha_{i}+j)&\;\hbox{zeroes in the x-component
in}\;\;(0,\tau_{i})\\\ 2(\alpha_{i}+j)+1&\;\hbox{zeroes in the y-component
in}\;\;[0,\tau_{i}]\end{array}\right\\}\;\hbox{if}\;\;\tilde{\kappa}=\kappa+1.$

###### Proof.

We focus our attention on case $\kappa=I$, since the proof for $\kappa=III$ is
completely analogous. By the definition of $\vartheta(t,s)$, we have that
$0\leq\vartheta(0,s)\leq\frac{\pi}{2}$ for all $s\in[0,1]$. Moreover, if the
arc $\gamma_{j,II}\subset K_{j,II}$ is parameterized on a subinterval
$[s^{0}_{j,II},s^{1}_{j,II}]$ of $[0,1]$, then

$\tfrac{3}{2}\pi+2(\alpha_{i}+j-1)\pi\leq\vartheta(\tau_{i},s)\leq
2(\alpha_{i}+j)\pi,$

with

$\vartheta(\tau_{i},s^{0}_{j,II})=\tfrac{3}{2}\pi+2(\alpha_{i}+j-1)\pi,\qquad\vartheta(\tau_{i},s^{1}_{j,II})=2(\alpha_{i}+j)\pi.$

Thus, for every $z\in\gamma_{j}$, the solution $(x(t;z),y(t;z))$ satisfies:

  * •

$x(t;z)$ has exactly $2(\alpha_{i}+j)-1$ zeroes in the interval
$(0,\tau_{i})$,

  * •

$y(t,z)$, as well as $x^{\prime}(t;z)$, has exactly $2(\alpha_{i}+j)$ zeroes
in the interval $[0,\tau_{i}]$.

Similarly, if the arc $\gamma_{j,IV}\subset K_{j,IV}$ is parameterized on a
subinterval $[s^{0}_{j,IV},s^{1}_{j,IV}]$ of $[0,1]$,

$\tfrac{1}{2}\pi+2(\alpha_{i}+j)\pi\leq\vartheta(\tau_{i},s)\leq\pi+2(\alpha_{i}+j)\pi,$

with

$\vartheta(\tau_{i},s^{0}_{j,IV})=\tfrac{1}{2}\pi+2(\alpha_{i}+j)\pi,\qquad\vartheta(\tau_{i},s^{1}_{j,IV})=\pi+2(\alpha_{i}+j)\pi.$

Therefore, for every $z\in\gamma_{j}$, the solution $(x(t;z),y(t;z))$
satisfies:

  * •

$x(t,z)$ has exactly $2(\alpha_{i}+j)$ zeroes in the interval $(0,\tau_{i})$,

  * •

$y(t,z)$, as well as $x^{\prime}$, has exactly $2(\alpha_{i}+j)+1$ zeroes in
the interval $[0,\tau_{i}]$.

This end the proof. $\square$

Adapting the proof of Proposition 4.1, the next result holds. It will be
invoked at the final stages of our analysis for the associated boundary value
problem in $[0,L]$.

######  Corollary 4.1.

Under the same assumptions of Proposition 4.1, assume that, for every
$\kappa\in\\{IV,III,II,I\\}$, the stronger twist condition

$\begin{split}&{\mathcal{T}}_{i,\kappa+1}(d)+\alpha_{i}{\mathcal{T}}_{i}(d)>\tau_{i},\\\
&{\mathcal{T}}_{i,\kappa}(e)+{\mathcal{T}}_{i,\kappa+1}(e)+\beta_{i}{\mathcal{T}}_{i}(e)<\tau_{i},\end{split}$
(4.6)

holds. Then, for every $\kappa^{\prime}\in\\{IV,III,II,I\\}$,

$\Phi_{i}:\mathcal{A}^{\rm
energy}_{i,\kappa}\overset{\beta_{i}-\alpha_{i}}{\Bumpeq{\\!\\!\\!\\!\\!\\!\\!\\!{\longrightarrow}}}\,\mathcal{A}^{\rm
axis}_{i,\kappa^{\prime}}.$ (4.7)

######  Remark 4.1.

By slightly changing the twist conditions of Proposition 4.1, we can get some
different results for the Poincaré map $\Phi_{i}$. For instance, imposing the
twist condition

${\mathcal{T}}_{i,IV}(d)+{\mathcal{T}}_{i,III}(d)+\alpha_{i}{\mathcal{T}}_{i}(d)>\tau_{i},\quad\beta_{i}{\mathcal{T}}_{i}(e)<\tau_{i},$

it follows that

$\Phi_{i}:\mathcal{A}^{\rm
energy}_{i,I}\overset{\beta_{i}-\alpha_{i}}{\Bumpeq{\\!\\!\\!\\!\\!\\!\\!\\!{\longrightarrow}}}\,\mathcal{A}^{\rm
axis}_{i,II}.$

However, in our application, it suffices to apply Proposition 4.1.

##  5 Transformation of arcs in $(\mathcal{L}_{\lambda})$

This section considers the Poincaré map $\Psi_{i}$ associated with the
$x$-linear system $({\mathcal{L}}_{\lambda})$ in order to link the dynamics in
${\mathcal{A}}_{i}$ and ${\mathcal{A}}_{i+1}$. The map $\Psi_{i}$ moves the
points along the level lines of the Hamiltonian ${\mathcal{E}}(x,y)$. As the
dynamics vary with the sign of $\lambda$, we will discuss each of these cases
separately.

###  5.1 The case $\lambda<0$

Throughout this section, for every $i\in\\{0,...,m\\}$, we denote by
$y_{+}(c_{\kappa,i})>0$, $\kappa=1,2$, the unique positive value of $y$ for
which $\mathcal{H}(0,y_{+}(c_{\kappa,i}))=c_{\kappa,i}$, $\kappa=1,2$, and set

$y_{+,i}:=\min\\{y_{+}(c_{1,i}),y_{+}(c_{1,i+1})\\}>0,\qquad
i\in\\{0,...,m-1\\}.$ (5.1)

Then, we consider the energy level line
$\mathcal{E}(x,y)=\mathcal{E}(0,y_{+,i})$ and its intersections with the outer
boundaries of ${\mathcal{A}}_{i,II}$ and ${\mathcal{A}}_{i+1,I}$, whose
abscissas are

$\displaystyle x_{\ell,i}$
$\displaystyle:=G^{-1}_{-}(\mu_{i}^{-1}[H(y_{+}(c_{2,i}))-H(y_{+,i})])<0,$
$\displaystyle x_{r,i+1}$
$\displaystyle:=G^{-1}_{+}(\mu_{i+1}^{-1}[H(y_{+}(c_{2,i+1}))-H(y_{+,i})])>0,$

respectively, where $G^{-1}_{-}$ and $G^{-1}_{+}$ are the inverse functions of
$G$ restricted to $(-\infty,0)$ and $(0,+\infty)$, respectively. Then, by
(3.7), the transition time between the straight lines $x=x_{\ell,i}$ and
$x=x_{r,i+1}$ along $\mathcal{E}(x,y)=\mathcal{E}(0,y_{+,i})$ is given by

${\mathcal{S}}_{i,II\to
I}:={\mathcal{S}}^{\lambda}(x_{\ell,i},x_{r,i+1};(0,y_{+,i}))=\int_{x_{\ell,i}}^{x_{r,i+1}}\frac{dx}{(h\circ
H_{+}^{-1})\left(H(y_{+,i})-\frac{\lambda}{2}x^{2}\right)}\,.$ (5.2)

Similarly, for every $i\in\\{0,...,m\\}$, we denote by
$y_{-}(c_{\kappa,i})<0$, $\kappa=1,2$, the unique negative value of $y$ for
which $\mathcal{H}(0,y_{-}(c_{\kappa,i}))=c_{\kappa,i}$, $\kappa=1,2$, and set

$y_{-,i}:=\max\\{y_{-}(c_{1,i}),y_{-}(c_{1,i+1})\\}<0,\qquad
i\in\\{0,...,m-1\\}.$ (5.3)

Arguing as before, the energy level line passing through $(0,y_{-,i})$
intersects the outer boundaries of ${\mathcal{A}}_{i,IV}$ and
${\mathcal{A}}_{i+1,III}$ at the points with abscissas

$\displaystyle x_{r,i}$
$\displaystyle:=G^{-1}_{+}(\mu_{i}^{-1}[H(y_{-}(c_{2,i}))-H(y_{-,i})])>0,$
$\displaystyle x_{\ell,i+1}$
$\displaystyle:=G^{-1}_{-}(\mu_{i+1}^{-1}[H(y_{-}(c_{2,i+1}))-H(y_{-,i})])<0,$

respectively. Thus, due to (3.8), the transition time between the straight
lines $x=x_{r,i}$ and $x=x_{\ell,i+1}$ along
$\mathcal{E}(x,y)=\mathcal{E}(0,y_{-,i})$ on the lower half-plane is given by

${\mathcal{S}}_{i,IV\to
III}:={\mathcal{S}}^{\lambda}(x_{r,i},x_{\ell,i+1};(0,y_{-,i}))=\int_{x_{\ell,i+1}}^{x_{r,i}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(H(y_{-,i})-\frac{\lambda}{2}x^{2}\right)}\,.$ (5.4)

Analogously, for every $i\in\\{0,...,m\\}$, we denote by
$x_{+}(c_{\kappa,i})>0$, $\kappa=1,2$, the unique positive value of $x$ for
which $\mathcal{H}(x_{+}(c_{\kappa,i}),0)=c_{\kappa,i}$, $\kappa=1,2$, set

$x_{+,i}:=\min\\{x_{+}(c_{1,i}),x_{+}(c_{1,i+1})\\}>0,\qquad
i\in\\{0,...,m-1\\},$

and consider the energy level line passing through $(x_{+,i},0)$, as well as
its intersections with the outer boundaries of ${\mathcal{A}}_{i,IV}$ and
${\mathcal{A}}_{i+1,I}$, whose abscissas are

$\displaystyle x_{r_{1},i}$
$\displaystyle:=G^{-1}_{+}\left(G(x_{+}(c_{2,i}))+\frac{\lambda}{2\mu_{i}}[x_{+}^{2}(c_{2,i})-x_{+,i}^{2}]\right)>0,$
$\displaystyle x_{r_{2},i+1}$
$\displaystyle:=G^{-1}_{+}\left(G(x_{+}(c_{2,i+1}))+\frac{\lambda}{2\mu_{i+1}}[x_{+}^{2}(c_{2,i+1})-x_{+,i}^{2}]\right)>0,$

respectively. Thanks to (3.9), the transition time between the straight lines
$x=x_{r_{1},i}$ and $x=x_{r_{2},i+1}$ along the level line
$\mathcal{E}(x,y)=\mathcal{E}(x_{+,i},0)$ on the right half-plane is given by

$\begin{split}{\mathcal{S}}_{i,IV\to
I}:=&{\mathcal{S}}^{\lambda}(x_{r_{1},i},x_{r_{2},i+1};(x_{+,i},0))\\\
=&\int_{x_{+,i}}^{x_{r_{2},i+1}}\frac{dx}{(h\circ
H_{+}^{-1})\left(\frac{\lambda}{2}(x_{+,i}^{2}-x^{2})\right)}+\int_{x_{+,i}}^{x_{r_{1},i}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(\frac{\lambda}{2}(x_{+,i}^{2}-x^{2})\right)}.\end{split}$
(5.5)

Finally, for every $i\in\\{0,...,m\\}$, we denote by $x_{-}(c_{\kappa,i})>0$,
$\kappa=1,2$, the unique negative value of $x$ for which
$\mathcal{H}(x_{-}(c_{\kappa,i}),0)=c_{\kappa,i}$, $\kappa=1,2$, set

$x_{-,i}:=\max\\{x_{-}(c_{1,i}),x_{-}(c_{1,i+1})\\}<0,\qquad
i\in\\{0,...,m-1\\},$

and consider the energy level line passing through $(x_{-,i},0)$, and its
intersections with the outer boundaries of ${\mathcal{A}}_{i,II}$ and
${\mathcal{A}}_{i+1,III}$, whose abscissas are

$\displaystyle x_{\ell_{1},i}$
$\displaystyle:=G^{-1}_{-}\left(G(x_{-}(c_{2,i}))+\frac{\lambda}{2\mu_{i}}[x_{-}^{2}(c_{2,i})-x_{-,i}^{2}]\right),$
$\displaystyle x_{\ell_{2},i+1}$
$\displaystyle:=G^{-1}_{-}\left(G(x_{-}(c_{2,i+1}))+\frac{\lambda}{2\mu_{i+1}}[x_{-}^{2}(c_{2,i+1})-x_{-,i}^{2}]\right),$

respectively. Then, by (3.10), we obtain that the transition time between the
straight lines $x=x_{\ell_{1},i}$ and $x=x_{\ell_{2},i+1}$ along the integral
curve $\mathcal{E}(x,y)=\mathcal{E}(x_{-,i},0)$ on the left half-plane is
given by

$\begin{split}{\mathcal{S}}_{i,II\to
III}:=&{\mathcal{S}}^{\lambda}(x_{\ell_{1},i},x_{\ell_{2},i+1};(x_{-,i},0))\\\
=&\int_{x_{\ell_{1},i}}^{x_{-,i}}\frac{dx}{(h\circ
H_{+}^{-1})\left(\frac{\lambda}{2}(x_{-,i}^{2}-x^{2})\right)}+\int_{x_{\ell_{2},i+1}}^{x_{-,i}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(\frac{\lambda}{2}(x_{-,i}^{2}-x^{2})\right)}.\end{split}$
(5.6)

###  5.2 The case $\lambda=0$

The abscissas in the upper half-plane of the intersection points of the
straight line $y=y_{+,i}$ (see (5.1)) with the outer boundaries of
${\mathcal{A}}_{i,II}$ and ${\mathcal{A}}_{i+1,I}$ are

$\displaystyle x_{\ell,i}$
$\displaystyle:=G^{-1}_{-}(\mu_{i}^{-1}[c_{2,i}-H(y_{+,i})])<0,$
$\displaystyle x_{r,i+1}$
$\displaystyle:=G^{-1}_{+}(\mu_{i+1}^{-1}[c_{2,i+1}-H(y_{+,i})])>0,$

respectively. Thus, the transition time between $x=x_{\ell,i}$ and
$x=x_{r,i+1}$ along $y=y_{+,i}$ is

${\mathcal{S}}_{i,II\to
I}:={\mathcal{S}}^{0}(x_{\ell,i},x_{r,i+1},(0,y_{+,i})):=\frac{x_{r,i+1}-x_{\ell,i}}{h(y_{+,i})}<\frac{x_{+}(c_{2,i+1})-x_{-}(c_{2,i})}{h(y_{+,i})}.$
(5.7)

Similarly, the abscissas in the lower half-plane of the intersection points of
the straight line $y=y_{-,i}$ (see (5.3)) with the outer boundaries of
${\mathcal{A}}_{i,IV}$ and ${\mathcal{A}}_{i+1,III}$ are

$\displaystyle x_{r,i}$
$\displaystyle:=G^{-1}_{+}(\mu_{i}^{-1}[c_{2,i}-H(y_{-,i})])>0,$
$\displaystyle x_{\ell,i+1}$
$\displaystyle:=G^{-1}_{-}(\mu_{i+1}^{-1}[c_{2,i+1}-H(y_{-,i})])<0,$

respectively. Hence, the transition time between $x=x_{r,i}$ and
$x=x_{\ell,i+1}$ along $y=y_{-,i}$ equals

${\mathcal{S}}_{i,IV\to
III}:={\mathcal{S}}^{0}(x_{r,i},x_{\ell,i+1},(0,y_{-,i})):=\frac{x_{r,i}-x_{\ell,i+1}}{-h(y_{-,i})}<\frac{x_{+}(c_{2,i})-x_{-}(c_{2,i+1})}{-h(y_{-,i})}\,.$
(5.8)

###  5.3 The case $\lambda>0$

To carry out our analysis in this case, we must previously guarantee the
existence of a closed orbit of $({\mathcal{L}}_{\lambda})$ crossing the two
annuli ${\mathcal{A}}_{i}$ and ${\mathcal{A}}_{i+1}$ as described in Figure 8.
This figure shows an admissible configuration of two annuli
${\mathcal{A}}_{i}$ and ${\mathcal{A}}_{i+1}$ which are crossed by a periodic
orbit of the $x$-linear system $({\mathcal{L}}_{\lambda})$ represented by the
level line ${\mathcal{E}}(x,y)=c$. For the particular example generating
Figure 8 we have chosen $h(y)=y^{3}/(1+y^{2})$, which comes from the inverse
of a differential operator behaving like a $p$-Laplacian near the origin that
it is asymptotically linear at infinity, $g(x)=x|x|$, $\lambda=1$, $\mu_{i}=1$
and $\mu_{i+1}=2$. The annular regions are determined by the values of the
parameters

$c_{1,i}=H(6),\quad c_{2,i}=H(7),\quad c_{1,i+1}=H(10),\quad c_{2,i+1}=H(11),$

and we have taken $c=11/2$.

Figure 8: A closed orbit of $({\mathcal{L}}_{\lambda})$ crossing
${\mathcal{A}}_{i}$ and ${\mathcal{A}}_{i+1}$.

To accomplish such a configuration, let us consider the level line
${\mathcal{E}}(x,y)=c$ with $c>0$ such that

$0<c\leq
H_{*}:=\min_{j\in\\{i,i+1\\}}H(y_{\pm}(c_{1,j}))=\min_{j\in\\{i,i+1\\}}c_{1,j}.$
(5.9)

By the definition of $\mathcal{E}(x,y)$, we have that

$-\sqrt{\frac{2c}{\lambda}}\leq x\leq\sqrt{\frac{2c}{\lambda}}\quad\hbox{for
all}\;\;(x,y)\in\mathcal{E}^{-1}(c).$

Thus, assuming that

$\min\left\\{G(-\sqrt{2c/\lambda}),G(\sqrt{2c/\lambda})\right\\}>\max_{j\in\\{i,i+1\\}}\frac{c_{2,j}-c}{\mu_{j}},$
(5.10)

it becomes apparent that $\mathcal{E}^{-1}(c)$ intersects the outer boundaries
of ${\mathcal{A}}_{i,II}$ and ${\mathcal{A}}_{i+1,I}$, as illustrated in
Figure 8. Note that (5.10) is equivalent to

$-\sqrt{\frac{2c}{\lambda}}<\min_{j\in\\{i,i+1\\}}x_{-}(c_{2,j})<0<\max_{j\in\\{i,i+1\\}}x_{+}(c_{2,j})<\sqrt{\frac{2c}{\lambda}}.$
(5.11)

Indeed, since for every $x\in\mathbb{R}$

$\displaystyle xG^{\prime}(x)=xg(x)>0,$ $\displaystyle
x\left(\frac{\lambda}{2}x^{2}+\mu_{j}G(x)\right)^{\prime}=\lambda
x^{2}+x\mu_{j}g(x)>0,$
$\displaystyle\frac{\lambda}{2}x_{\pm}^{2}(c_{2,j})+\mu_{j}G(x_{\pm}(c_{2,j}))=c_{2,j}>c,\qquad\hbox{for}\;j\in\\{i,i+1\\},$

it becomes apparent that, under condition (5.10), the intersections of the
level line ${\mathcal{E}}^{-1}(c)$ with the $x$-axis are outside both annuli
${\mathcal{A}}_{i}$ and ${\mathcal{A}}_{i+1}$. Thus, they must across, as
claimed above. Moreover, the abscissas of the intersection points are

$\displaystyle x_{\ell,c_{2,i}}$
$\displaystyle:=G^{-1}_{-}((c_{2,i}-c)/\mu_{i})<0,$ $\displaystyle
x_{\ell,c_{1,i}}$ $\displaystyle:=G^{-1}_{-}((c_{1,i}-c)/\mu_{i})\leq 0,$
$\displaystyle x_{r,c_{2,i+1}}$
$\displaystyle:=G^{-1}_{+}((c_{2,i+1}-c)/\mu_{i+1})>0,$ $\displaystyle
x_{r,c_{1,i+1}}$ $\displaystyle:=G^{-1}_{+}((c_{1,i+1}-c)/\mu_{i+1})\geq 0,$

respectively. Thus, the transition times between the straight lines
$x=x_{\ell,c_{2,i}}$ and $x=x_{r,c_{2,i+1}}$, and between $x=x_{\ell,c_{1,i}}$
and $x=x_{r,c_{1,i+1}}$, along $\mathcal{E}^{-1}(c)$ in the upper half-plane
are given by

$\displaystyle{\mathcal{S}}^{c_{2,i\to i+1}}_{II\to I}(c)$
$\displaystyle:={\mathcal{S}}^{\lambda}(x_{\ell,c_{2,i}},x_{r,c_{2,i+1}},H_{+}^{-1}(c))=\int_{x_{\ell,c_{2,i}}}^{x_{r,c_{2,i+1}}}\frac{dx}{(h\circ
H_{+}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)},$
$\displaystyle{\mathcal{S}}^{c_{1,i\to i+1}}_{II\to I}(c)$
$\displaystyle:={\mathcal{S}}^{\lambda}(x_{\ell,c_{1,i}},x_{r,c_{1,i+1}},H_{+}^{-1}(c))=\int_{x_{\ell,c_{1,i}}}^{x_{r,c_{1,i+1}}}\frac{dx}{(h\circ
H_{+}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)},$

respectively. Similarly, the transition time from the straight line
$x=x_{\ell,c_{1,i}}$ to the point $(\sqrt{2c/\lambda},0)$ along the curve
${\mathcal{E}}^{-1}(c)$ in the upper half-plane is given by

${\mathcal{S}}^{c_{1,i}\to\sqrt{}}_{II\to
I}(c):={\mathcal{S}}^{\lambda}(x_{\ell,c_{1,i}},\sqrt{2c/\lambda},H_{+}^{-1}(c))=\int_{x_{\ell,c_{1,i}}}^{\sqrt{2c/\lambda}}\frac{dx}{(h\circ
H_{+}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)}.$ (5.12)

The same level line intersects the outer and inner boundaries of
${\mathcal{A}}_{i,IV}$ and ${\mathcal{A}}_{i+1,III}$ at the points with
abscissas

$\displaystyle x_{r,c_{2,i}}$
$\displaystyle:=G^{-1}_{+}((c_{2,i}-c)/\mu_{i})>0,$ $\displaystyle
x_{r,c_{1,i}}$ $\displaystyle:=G^{-1}_{+}((c_{1,i}-c)/\mu_{i})\geq 0,$
$\displaystyle x_{\ell,c_{2,i+1}}$
$\displaystyle:=G^{-1}_{-}((c_{2,i+1}-c)/\mu_{i+1})<0,$ $\displaystyle
x_{\ell,c_{1,i+1}}$ $\displaystyle:=G^{-1}_{-}((c_{1,i+1}-c)/\mu_{i+1})\leq
0,$

respectively. Hence, the transition times between the straight lines
$x=x_{r,c_{2,i}}$ and $x=x_{\ell,c_{2,i+1}}$, and between $x=x_{r,c_{1,i}}$
and $x=x_{\ell,c_{1,i+1}}$, along ${\mathcal{E}}^{-1}(c)$ in the lower half-
plane, are given by

$\displaystyle{\mathcal{S}}^{c_{2,i\to i+1}}_{IV\to III}(c)$
$\displaystyle:={\mathcal{S}}^{\lambda}(x_{r,c_{2,i}},x_{\ell,c_{2,i+1}},H_{-}^{-1}(c))=\int_{x_{\ell,c_{2,i+1}}}^{x_{r,c_{2,i}}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)},$
$\displaystyle{\mathcal{S}}^{c_{1,i\to i+1}}_{IV\to III}(c)$
$\displaystyle:={\mathcal{S}}^{\lambda}(x_{r,c_{1,i}},x_{\ell,c_{1,i+1}},H_{-}^{-1}(c))=\int_{x_{\ell,c_{1,i+1}}}^{x_{r,c_{1,i}}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)},$

respectively. Similarly, the transition time from the straight line
$x=x_{r,c_{1,i}}$ to the point $(-\sqrt{2c/\lambda},0)$, along the level line
${\mathcal{E}}^{-1}(c)$ in the lower half-plane, is given by

${\mathcal{S}}^{c_{1,i}\to\sqrt{}}_{IV\to
III}(c):={\mathcal{S}}^{\lambda}(x_{r,c_{1,i}},-\sqrt{2c/\lambda},H_{-}^{-1}(c))=\int_{-\sqrt{2c/\lambda}}^{x_{r,c_{1,i}}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)}.$ (5.13)

Finally, we need to introduce two further transition times as follows. Recall
that $x_{\ell,c_{2,i}}$ equals the abscissa of the intersections between
$\mathcal{E}^{-1}(c)$ and the outer boundary of ${\mathcal{A}}_{i,III}$. From
this, it is easily seen that the transition time between the straight lines
$x=x_{\ell,c_{2,i}}$ and $x=x_{r,c_{1,i+1}}$ passing through
$(-\sqrt{{2c}/{\lambda}},0)$ along ${\mathcal{E}}^{-1}(c)$ is given by

${\mathcal{S}}_{III\to
I}(c):=\int_{-\sqrt{2c/\lambda}}^{x_{\ell,c_{2,i}}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)}+\int_{-\sqrt{{2c}/{\lambda}}}^{x_{r,c_{1,i+1}}}\frac{dx}{(h\circ
H_{+}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)}.$ (5.14)

Similarly, since $x_{r,c_{2,i}}$ provides the abscissa of the intersections
between $\mathcal{E}^{-1}(c)$ and the outer boundary of ${\mathcal{A}}_{i,I}$,
the transition time between the straight lines $x=x_{r,c_{2,i}}$ and
$x=x_{\ell,c_{1,i+1}}$ passing through the point $(\sqrt{{2c}/{\lambda}},0)$
along $\mathcal{E}^{-1}(c)$ is given by

${\mathcal{S}}_{I\to
III}(c):=\int_{x_{r,c_{2,i}}}^{\sqrt{2c/\lambda}}\frac{dx}{(h\circ
H_{+}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)}+\int_{x_{\ell,c_{1,i+1}}}^{\sqrt{{2c}/{\lambda}}}\frac{dx}{-(h\circ
H_{-}^{-1})\left(c-\frac{\lambda}{2}x^{2}\right)}.$ (5.15)

This concludes our analysis in this case.

##  6 Linking dynamics in general Sturm–Liouville BVP’s

In this section we show some applications of the preceding results to the
existence of multiple solutions to (1.9) satisfying prescribed boundary
conditions. In our analysis, we will restrict ourselves to consider regular
Sturm–Liouville boundary value problems.

From a geometrical point of view, the homogeneous boundary conditions of
Sturm–Liouville type can be described by, e.g., assigning two lines $r_{0}$
and $r_{L}$ passing through the origin and looking for solutions of (1.9) such
that $(x(0),y(0))\in r_{0}$ and $(x(L),y(L))\in r_{L}$, according to the
particular boundary value problem considered. In this setting, also the case
of non-homogeneous boundary conditions can be treated by considering two
affine lines not passing through the origin provided they satisfy appropriate
compatibility conditions. Namely, the lines $r_{0}$ and $r_{L}$ must cross the
annuli ${\mathcal{A}}_{0}$ and ${\mathcal{A}}_{m}$, respectively. More
generally, we can also study problems associated with nonlinear boundary
conditions, by taking as $r_{0}$ and $r_{L}$ two general curves crossing
${\mathcal{A}}_{0}$ and ${\mathcal{A}}_{m}$, respectively. See Papini and
Zanolin [37] and the references therein for further details on generalized
boundary conditions of Sturm–Liouville type.

More precisely, we will study the boundary value problem

$\left\\{\begin{array}[]{ll}x^{\prime}=h(y),\\\ y^{\prime}=-\lambda
x-a(t)g(x),\\\ x(0)\in r_{0},\;\;x(L)\in r_{L},\end{array}\right.$ (6.1)

where $r_{0}$ is an arc contained in ${\mathcal{A}}_{0}$ and $r_{L}$ is an arc
contained in ${\mathcal{A}}_{m}$, both arcs connecting the inner with the
outer boundary of the corresponding annulus, while the weight function $a(t)$
satisfies the general properties of Section 3 in the interval $[0,L]$, as well
as the functions $h=\phi^{-1}:(\varrho_{-},\varrho_{+})\to{\mathbb{R}}$ and
$g:{\mathbb{R}}\to{\mathbb{R}}$. Moreover, we suppose the annular regions
considered to satisfy the compatibility conditions with respect to the domain
${\mathbb{R}}\times(\varrho_{-},\varrho_{+})$ and, when $\lambda>0$, with
respect to (5.10), as illustrated in Figure 8.

In order to analyze (6.1) under the above assumptions we will proceed as
follows. First, they are studied, for every $i\in\\{0,1,\ldots,m\\}$, the
linking dynamics between the Poincaré maps $\Phi_{i}$ and $\Psi_{i}$
associated to $(\mathcal{N}_{i,\lambda})$ and $(\mathcal{L}_{\lambda})$,
respectively. Then, when these linking dynamics hold, by applying a
combinatorial scheme, there will be obtained a number of multiplicity results
for solutions having a prescribed nodal behavior. As the sign of $\lambda$
changes completely the structure of $(\mathcal{N}_{i,\lambda})$ and
$(\mathcal{L}_{\lambda})$, as well as the linking between them, the cases
$\lambda\leq 0$ and $\lambda>0$ will be dealt with throughout separately.

###  6.1 The case $\lambda\leq 0$

We start analyzing the case $\lambda\leq 0$ by delivering a result in the vain
of Proposition 4.1. Namely, by using the times of transition calculated in
Section 5 for the linear problem $(\mathcal{L}_{\lambda})$, linking two non-
linear problems $(\mathcal{N}_{j,\lambda})$ for $j\in\\{i,i+1\\}$, we get a
property on the arc transformation of the composite map
$\Psi_{i}\circ\Phi_{i}$ as well as the nodal properties of such arcs.

######  Proposition 6.1.

Given the cyclic clockwise ordering $IV\prec III\prec II\prec I\prec IV$ we
fix $\kappa=I$ or $III$, pick $d,e\in\\{c_{1,i},c_{2,i}\\}$ with $d\not=e$,
and assume that there are two non-negative integers
$0\leq\alpha_{i}<\beta_{i}$ satisfying the twist condition (4.1). Suppose, in
addition, that the transition times satisfy either

  1. (1)

${\mathcal{S}}_{i,\kappa-1\to\kappa}\leq\varsigma_{i}$ and
${\mathcal{S}}_{i,\kappa+1\to\kappa+2}\leq\varsigma_{i}$, or

  2. (2)

${\mathcal{S}}_{i,\kappa+1\to\kappa}\leq\varsigma_{i}$ and
${\mathcal{S}}_{i,\kappa+3\to\kappa+2}\leq\varsigma_{i}$.

Then,

$\Psi_{i}\circ\Phi_{i}:\mathcal{A}^{\rm
energy}_{i,\kappa}\overset{\beta_{i}-\alpha_{i}}{\Bumpeq{\\!\\!\\!\\!\\!\\!\\!\\!{\longrightarrow}}}\,\mathcal{A}^{\rm
energy}_{i+1,\kappa}\cup\mathcal{A}^{\rm energy}_{i+1,\kappa+2}.$ (6.2)

Furthermore, depending on whether the assumption (1) or (2) occurs, for every
arc $\gamma\subset\mathcal{A}_{i,\kappa}^{\rm energy}$ joining the opposite
sides of $\mathcal{A}_{i,\kappa}^{-}$, the solutions of
$({\mathcal{N}}_{i,\lambda})$ with initial value on a sub-arc
$\gamma_{j,\kappa}\subset K_{j,\kappa}$ of $\gamma$ and ending at
$\mathcal{A}_{i,\tilde{\kappa}}$ have exactly, for every
$j=1,2,\ldots,\beta_{i}-\alpha_{i}$,

$(1)\left\\{\begin{array}[]{llll}(a)\;\tilde{\kappa}=\kappa&\left\\{\begin{array}[]{ll}2(\alpha_{i}+j)-1&\;\hbox{zeroes
in the x-component in}\;\;(0,\tau_{i})\equiv(t_{i},s_{i})\\\\[2.0pt]
1&\;\hbox{zeroes in the x-component
in}\;\;[\tau_{i},\tau_{i}+\varsigma_{i}]\equiv[t_{i},\varsigma_{i}]\\\\[10.0pt]
2(\alpha_{i}+j)&\;\hbox{zeroes in the y-component
in}\;\;[0,\tau_{i}]\equiv[t_{i},s_{i}]\\\\[2.0pt] 0&\;\hbox{zeroes in the
y-component
in}\;\;[\tau_{i},\tau_{i}+\varsigma_{i}]\equiv[t_{i},\varsigma_{i}]\end{array}\right.\\\\[5.0pt]
(b)\;\tilde{\kappa}=\kappa+2&\left\\{\begin{array}[]{ll}2(\alpha_{i}+j)&\;\hbox{zeroes
in the x-component in}\;\;(0,\tau_{i})\equiv(t_{i},s_{i})\\\\[2.0pt]
1&\;\hbox{zeroes in the x-component
in}\;\;[\tau_{i},\tau_{i}+\varsigma_{i}]\equiv[t_{i},\varsigma_{i}]\\\\[10.0pt]
2(\alpha_{i}+j)+1&\;\hbox{zeroes in the y-component
in}\;\;[0,\tau_{i}]\equiv[t_{i},s_{i}]\\\\[2.0pt] 0&\;\hbox{zeroes in the
y-component
in}\;\;[\tau_{i},\tau_{i}+\varsigma_{i}]\equiv[t_{i},\varsigma_{i}]\end{array}\right.\end{array}\right.$

and

$(2)\left\\{\begin{array}[]{llll}(a)\;\tilde{\kappa}=\kappa&\left\\{\begin{array}[]{ll}2(\alpha_{i}+j)&\;\hbox{zeroes
in the x-component in}\;\;(0,\tau_{i})\equiv(t_{i},s_{i})\\\\[2.0pt]
0&\;\hbox{zeroes in the x-component
in}\;\;[\tau_{i},\tau_{i}+\varsigma_{i}]\equiv[t_{i},\varsigma_{i}]\\\\[10.0pt]
2(\alpha_{i}+j)+1&\;\hbox{zeroes in the y-component
in}\;\;[0,\tau_{i}]\equiv[t_{i},s_{i}]\\\\[2.0pt] 1&\;\hbox{zeroes in the
y-component
in}\;\;[\tau_{i},\tau_{i}+\varsigma_{i}]\equiv[t_{i},\varsigma_{i}]\end{array}\right.\\\\[5.0pt]
(b)\;\tilde{\kappa}=\kappa+2&\left\\{\begin{array}[]{ll}2(\alpha_{i}+j)-1&\;\hbox{zeroes
in the x-component in}\;\;(0,\tau_{i})\equiv(t_{i},s_{i})\\\\[2.0pt]
0&\;\hbox{zeroes in the x-component
in}\;\;[\tau_{i},\tau_{i}+\varsigma_{i}]\equiv[t_{i},\varsigma_{i}]\\\\[10.0pt]
2(\alpha_{i}+j)&\;\hbox{zeroes in the y-component
in}\;\;[0,\tau_{i}]\equiv[t_{i},s_{i}]\\\\[2.0pt] 1&\;\hbox{zeroes in the
y-component
in}\;\;[\tau_{i},\tau_{i}+\varsigma_{i}]\equiv[t_{i},\varsigma_{i}]\end{array}\right.\end{array}\right.$

By the analysis already done in Section 5, it is clear that the case (1) can
occurs for $\lambda\leq 0$ with ${\mathcal{S}}_{i,j\to j+1}$ given by (5.2),
(5.4), (5.6), or (5.7), respectively. However, the case (2) can only occurs
for $\lambda<0$.

###### Proof.

We focus our efforts in proving the case (1) in the sub-case $\kappa=I$, since
the proofs of the remaining cases can be completed with a rather similar
argument. According to Proposition 4.1, there are $\beta_{i}-\alpha_{i}$
pairwise disjoint compact sets $K_{1,II},\dots,K_{\beta_{i}-\alpha_{i},II}$
such that each arc $\gamma$ in ${\mathcal{A}}_{i,I}$ connecting the level
lines $\\{{\mathcal{H}}=c_{i,1}\\}$ and $\\{{\mathcal{H}}=c_{i,2}\\}$ has
$\ell_{i}$ sub-arcs $\gamma_{j,II}\subset K_{j,II}$ such that
$\Phi_{i}(\gamma_{j,II})$ is an arc contained in ${\mathcal{A}}_{i,II}$ which
connects the $x$-axis with the $y$-axis.

We first discuss the case $\lambda<0$. For each of the arcs
$\Phi_{i}(\gamma_{j,II})$ we consider a suitable sub-arc
$\Phi_{i}(\gamma_{j})$, where $\gamma_{j}$ is a compact sub-arc of
$\gamma_{j,II}$ that intersects in the second quadrant at a unique point
$w_{1}$ the level line ${\mathcal{E}}=0$ and at a unique point $w_{2}$ the
level line ${\mathcal{E}}=H(y_{+,i})$. Furthermore, we suppose that
$\Phi_{i}(\gamma_{j})$ is contained in the region

$\\{(x,y):0\leq{\mathcal{E}}(x,y)\leq H(y_{+,i})\\}.$

By definition, $w_{1}$ lies on the component of the stable manifold of the
origin of $({\mathcal{L}}_{\lambda})$ in the second quadrant. Therefore,
$\Psi_{i}(w_{1})$ is moved toward the origin, but it remains in the second
quadrant. On the other hand, the condition ${\mathcal{S}}_{i,II\to
I}\leq\varsigma_{i}$ implies that the point $w_{2}$ moves under the action of
$\Psi_{i}$ along the level line ${\mathcal{E}}=H(y_{+,i})$ to the first
quadrant and outside the annulus ${\mathcal{A}}_{i+1}$. To be more precise,
one should prevent the possibility that there is a blow up at some time
$t\leq\varsigma_{i}$ in the sense that the $y$-component of the solutions
achieves the value $\varrho_{+}$, where we might abandon the domain of
definition of the differential equation. This, however, is not a difficulty as
we are interested only on the pieces of $\Phi_{i}(\gamma_{j})$ which stay in
the annulus ${\mathcal{A}}_{i+1}$ after applying $\Psi_{i}$. For this reason,
we can just repeat the same argument by restricting ourselves to a further
sub-arc of the form $\Phi_{i}(\tilde{\gamma}_{j})$ lying between
${\mathcal{E}}=0$ and ${\mathcal{E}}=c$ for a suitable $c\in(0,y_{+,i}))$.
Accordingly, from now on, we identify $\tilde{\gamma}_{j}$ with $\gamma_{j}$
and assume, without loss of generality, that $\Psi_{i}$ is well defined on the
arc $\Phi_{i}(\gamma_{j})$. In this manner, we can conclude that
$\Psi_{i}(\Phi_{i}(\gamma_{j}))$ is an arc connecting $\Psi_{i}(w_{1})$, which
is in the region $\\{{\mathcal{H}}_{i+1,\lambda}<c_{1,{i+1}}\\}$ with $x<0$,
to $\Psi_{i}(w_{2})$, which is in the region
$\\{{\mathcal{H}}_{i+1,\lambda}\geq c_{2,{i+1}}\\}$ with $x>0$. Moreover, all
the points of $\Psi_{i}(\Phi_{i}(\gamma_{j}))$ are contained in the set
$0\leq{\mathcal{E}}\leq H(y_{+,i})$ with $y>0$.

Next, we consider the case when $\lambda=0$. As before, for each of the arcs
$\Phi_{i}(\gamma_{j,II})$ we consider a suitable sub-arc
$\Phi_{i}(\gamma_{j})$, where $\gamma_{j}$ is a compact sub-arc of
$\gamma_{j,II}$, that intersects at a unique point $w_{1}$ the $x$-axis, which
is the level line ${\mathcal{E}}=0$, and at a unique point $w_{2}$ the level
line ${\mathcal{E}}=H(y_{+,i})$, which is the line $y=y_{+,i}$. Recalling that
$({\mathcal{L}}_{0})$ reduces to $x^{\prime}=h(y)$ and $y^{\prime}=0$, we have
that $\Psi_{i}(w_{1})=w_{1}$. On the other hand, the condition
${\mathcal{S}}_{i,II\to I}\leq\varsigma_{i}$ implies that $w_{2}$ moves under
the action of $\Psi_{i}$ along the line $y=y_{+,i}$ to the first quadrant
outside the annulus ${\mathcal{A}}_{i+1}$ (see Figure 6). Hence,
$\Psi_{i}(\Phi_{i}(\gamma_{j}))$ is an arc connecting $w_{1}$ to
$\Psi_{i}(w_{2})$ (which is in the region $\\{{\mathcal{H}}_{i+1,\lambda}\geq
c_{2,{i+1}}\\}$ with $x>0$). Moreover, all the points of
$\Psi_{i}(\Phi_{i}(\gamma_{j}))$ are contained in the set $0\leq y\leq
y_{+,i}$.

At this point, we have already proven that, in either case $\lambda<0$ and
$\lambda=0$, there is a sub-arc $\gamma_{j}$ of $\gamma_{j,II}$ such that
$\Psi_{i}(\Phi_{i}(\gamma_{j}))$ crosses the annulus ${\mathcal{A}}_{i+1}$ in
the fist quadrant. Passing to a further sub-arc $\hat{\gamma}_{j}$ of
$\gamma_{j}$ contained in $K^{\prime}_{j}$, we have that
$\Phi_{i}(\hat{\gamma}_{j})$ is a sub-arc of $\Phi_{i}(\gamma_{j})$ contained
in the set

$\mathcal{A}_{i,II}\cap\\{0\leq\mathcal{E}\leq H(y_{+,i})\\}$

such that $(\Psi_{j}\circ\Phi_{j})(\hat{\gamma}_{j})$ is an arc contained in
${\mathcal{A}}_{i+1,I}$ linking the two level lines
$\mathcal{H}^{-1}(c_{i+1,1})$ and $\mathcal{H}^{-1}(c_{i+1,2})$. This proves
(6.2) and shows how the corresponding solutions in the interval
$[\tau_{i},\tau_{i}+\varsigma_{i}]$ have exactly one (interior) zero in the
$x$-component, while the $y$-component remains positive.

The remaining nodal properties of the solutions come directly from Proposition
4.1. Precisely, for every $j=1,\dots,\beta_{i}-\alpha_{i}$, we have the
following classification, where we are taking $\kappa\in\\{I,III\\}$:

  * $(1.a)$

For the transition between ${\mathcal{A}}_{i,\kappa}$ and
${\mathcal{A}}_{i+1,\kappa}$:
There exists a solution with $2(\alpha_{i}+j)$ zeroes in the $x$-component in
the interval $(t_{i},t_{i+1})$. Moreover, this solution has $2(\alpha_{i}+j)$
zeroes in the $x^{\prime}/y$-component in the interval $[t_{i},t_{i+1}]$.
Thus, the number of zeroes of the $x$ and $y$ components of the solution,
indicated over the arrows, follow the next patterns, respectively:

${\mathcal{A}}_{i,\kappa}\overset{2(\alpha_{i}+j)-1}{\longrightarrow}{\mathcal{A}}_{i,\kappa-1}\overset{1}{\longrightarrow}{\mathcal{A}}_{i+1,\kappa}\quad\hbox{and}\quad{\mathcal{A}}_{i,\kappa}\overset{2(\alpha_{i}+j)}{\longrightarrow}{\mathcal{A}}_{i,\kappa-1}\overset{0}{\longrightarrow}{\mathcal{A}}_{i+1,\kappa}$

  * $(1.b)$

For the transition between ${\mathcal{A}}_{i,\kappa}$ and
${\mathcal{A}}_{i+1,\kappa+2}$:
There exists a solution with $2(\alpha_{i}+j)+1$ zeroes in the $x$-component
in the interval $(t_{i},t_{i+1})$, and this solution has $2(\alpha_{i}+j)+1$
zeroes in the $x^{\prime}/y$-component in the interval $[t_{i},t_{i+1}]$.
Thus, the number of zeroes of the $x$ and $y$ components of the solution,
indicated over the arrows, follow the next patterns, respectively:

${\mathcal{A}}_{i,\kappa}\overset{2(\alpha_{i}+j)}{\longrightarrow}{\mathcal{A}}_{i,\kappa+1}\overset{1}{\longrightarrow}{\mathcal{A}}_{i+1,\kappa+2}\quad\hbox{and}\quad{\mathcal{A}}_{i,\kappa}\overset{2(\alpha_{i}+j)+1}{\longrightarrow}{\mathcal{A}}_{i,\kappa+1}\overset{0}{\longrightarrow}{\mathcal{A}}_{i+1,\kappa+2}$

  * $(2.a)$

For the transition between ${\mathcal{A}}_{i,\kappa}$ and
${\mathcal{A}}_{i+1,\kappa}$:
There exists a solution with $2(\alpha_{i}+j)$ zeroes in the $x$-component in
the interval $(t_{i},t_{i+1})$. Moreover, this solution can be chosen to have
$2(\alpha_{i}+j)+2$ zeroes in the $x^{\prime}/y$-component in the interval
$[t_{i},t_{i+1}]$. Hence, the number of zeroes of the $x$ and $y$ components
of the solution, indicated over the arrows, follow the next patterns,
respectively:

${\mathcal{A}}_{i,\kappa}\overset{2(\alpha_{i}+j)}{\longrightarrow}{\mathcal{A}}_{i,\kappa+1}\overset{0}{\longrightarrow}{\mathcal{A}}_{i+1,\kappa}\quad\hbox{and}\quad{\mathcal{A}}_{i,\kappa}\overset{2(\alpha_{i}+j)+1}{\longrightarrow}{\mathcal{A}}_{i,\kappa+1}\overset{1}{\longrightarrow}{\mathcal{A}}_{i+1,\kappa}$

  * $(2.b)$

For the transition between ${\mathcal{A}}_{i,\kappa}$ and
${\mathcal{A}}_{i+1,\kappa+2}$:
There exists a solution with $2(\alpha_{i}+j)-1$ zeroes in the $x$-component
in the interval $(t_{i},t_{i+1})$ with $2(\alpha_{i}+j)+1$ zeroes in the
$x^{\prime}/y$-component in the interval $[t_{i},t_{i+1}]$. Hence, the number
of zeroes of the $x$ and $y$ components of the solution, indicated over the
arrows, follow the next patterns, respectively:

${\mathcal{A}}_{i,\kappa}\overset{2(\alpha_{i}+j)-1}{\longrightarrow}{\mathcal{A}}_{i,\kappa-1}\overset{0}{\longrightarrow}{\mathcal{A}}_{i+1,\kappa+2}\quad\hbox{and}\quad{\mathcal{A}}_{i,\kappa}\overset{2(\alpha_{i}+j)}{\longrightarrow}{\mathcal{A}}_{i,\kappa-1}\overset{1}{\longrightarrow}{\mathcal{A}}_{i+1,\kappa+2}$

This concludes the proof. $\square$

Figure 9 (left panel) illustrates the admissible transitions in the context of
Proposition 6.1 (above) when $\lambda\leq 0$ and Propositions 6.2 and 6.3
(further in this section) when $\lambda>0$. If $\lambda\leq 0$, the
intermediate blocks are given by ${\mathcal{B}}_{i,II}={\mathcal{A}}_{i,II}$
and ${\mathcal{B}}_{i,IV}={\mathcal{A}}_{i,IV}$, while, for $\lambda>0$, we
have that

${\mathcal{B}}_{i,II}={\mathcal{A}}_{i,II}\cup{\mathcal{A}}_{i,III},\quad{\mathcal{B}}_{i,IV}={\mathcal{A}}_{i,IV}\cup{\mathcal{A}}_{i,I}.$

When $\lambda\geq 0$ we have only two connections available for $\Psi_{i}$,
namely from ${\mathcal{B}}_{i,II}$ to ${\mathcal{A}}_{i+1,I}$ and from
${\mathcal{B}}_{i,IV}$ to ${\mathcal{A}}_{i+1,III}$ (the black arrows), while,
for $\lambda<0$ all four connections for $\Psi_{i}$ are allowed (black and red
arrows). In the horizontal and diagonal transitions, $\beta_{i}-\alpha_{i}$ is
the associated crossing number of the linking $\Phi_{i}$. Concerning the maps
$\Psi_{i}$, they have $1$ as crossing number, except if $\lambda>0$, where the
crossing number could equal $\kappa_{i}$ under the assumptions described in
Remark 6.3.

Figure 9: Admissible transitions of of $\Psi_{i}\circ\Phi_{i}$ (right panel).
In the left panel we have put in evidence the intermediate transitions across
the sets $\mathcal{B}^{\rm axis}$.

Now, by applying recursively Proposition 6.1 we get the next multiplicity
results of nodal solutions for the boundary value problem (6.1). By
simplicity, the case $\lambda=0$ is analyzed at the first place. The case
$\lambda<0$ holds in a very similar way.

######  Theorem 6.1.

Suppose that $\lambda=0$ and that either $r_{0}\in{\mathcal{A}}_{0,I}$, or
$r_{0}\in{\mathcal{A}}_{0,III}$ and either $r_{L}\in{\mathcal{A}}_{m,II}$ or
$r_{L}\in{\mathcal{A}}_{m,IV}$. Assume, in addition, that, for every
$i\in\\{0,...,m\\}$ and $\kappa\in\\{I,III\\}$, the twist conditions (4.1) are
satisfied together with

${\mathcal{S}}_{i,\kappa-1\to\kappa}\leq\varsigma_{i}\quad\text{and
}\;\;{\mathcal{S}}_{i,\kappa+1\to\kappa+2}\leq\varsigma_{i}\quad\hbox{for
all}\;\;i\in\\{0,\dots,m-1\\}.$

Then, for each choice of $r_{0}$ and $r_{L}$, the problem (6.1) has, at least,

$2^{m}\prod_{i=0}^{m}(\beta_{i}-\alpha_{i})$

nodal solutions. Moreover, each of these solutions can be classified by
assigning it a precise counter describing its nodal properties on each of the
intervals $(t_{i},s_{i})$ and $(s_{i},t_{i+1})$, according to any admissible
itinerary chosen in the Markov-type diagram of Figure 9 (left panel) at
$\lambda=0$.

###### Proof.

The proof employs a shooting type argument. Its goal is to prove the existence
of points $P_{0}\in r_{0}$ such that ${\mathscr{P}}_{L}(P_{0})\in r_{L}$,