Homoclinic and heteroclinic solutions for a class of secondorder nonautonomous ordinary differential equations: multiplicity results for stepwise potentials
 Elisa Ellero^{1} and
 Fabio Zanolin^{1}Email author
https://doi.org/10.1186/168727702013167
© Ellero and Zanolin; licensee Springer. 2013
Received: 30 November 2012
Accepted: 11 June 2013
Published: 15 July 2013
Abstract
We prove some multiplicity results for a class of onedimensional nonlinear Schrödingertype equations of the form
where $k>0$ and the weight $g(t)$ is a positive stepwise function. Instead of the cubic term, more general nonlinearities can be considered as well.
MSC: 34C37, 34B15.
Keywords
Dedication
Dedicated to Professor Jean Mawhin
1 Introduction
we suppose that
(∗) $f:\mathbb{R}\to {\mathbb{R}}^{+}:=[0,+\mathrm{\infty})$ is a locally Lipschitz function which is even, strictly increasing on $[0,+\mathrm{\infty})$ such that $f(0)=0$ and $f(+\mathrm{\infty})=+\mathrm{\infty}$.
Examples of functions f which are suitable for our considerations are, for instance, $f(s)={s}^{2}$ as in [1], or $f(s)={s}^{2}+b{s}^{4}$ ($b>0$) as in [2]. In both these cases, the function f is even. In Section 3 we briefly describe how this restriction could be avoided.
(see [3]). Usually, the Planck constant and the mass are omitted in (1.3) after rescaling.
which is a particular case of (1.3) with $N=1$, $p=3$ and ${V}_{1}=0$. In equation (1.4) the nonlinearity is called inhomogeneous due to the presence of a nonconstant weight ${V}_{2}(x)=g(x)$, which in [1] is assumed to be positive. The sign condition on $g(x)$ implies that nontrivial bounded solutions of (1.4) can exist only for $\lambda <0$ (see [[1], Theorem 1]). For this reason, we prefer to set $\lambda =k$ with $k>0$.
which belongs to the same class of (1.1).
In spite of the apparent simplicity of equation (1.5), a throughout study of its solutions may be a rather difficult task for a general nonconstant weight function $g(t)$. Looking for homoclinic and heteroclinic solutions of (1.5), it will be natural to focus on the behavior of $g(t)$ at ±∞. In similar situations, various authors have confined their study to the case in which $g(t)$ is asymptotically constant [4–6], or eventually constant [7, 8]. These assumptions are also justified by the analysis of some physical underlying models, in which a layered structure is present. With this respect, see the introduction in [8], where different eventually autonomous cases are listed for related NLSEs arising in nonlinear optics. Examples in which the nonlinear term presents a piecewise constant weight function have been studied also in biological and chemical models. In particular, these situations occur in the theory of wave propagation for reactiondiffusion systems; see, for instance, [9–11]. In the context of equation (1.5), examples in which $g(t)$ is a piecewise constant function have been considered as well (see [12]).
The aim of the present paper is to provide multiplicity results regarding homoclinic and heteroclinic solutions for equation (1.1) under particular assumptions on the weight function $a(t)$. Actually, we suppose that $a(t)$ is an eventually constant piecewise function with only two steps.
Our approach combines phaseplane analysis with timemapping estimates. As in [5, 7, 8, 10, 11, 13], the solutions are obtained by connecting the unstable and stable manifolds of the equilibrium points of the asymptotically autonomous equations. Such connections are performed by means of orbit paths of an intermediate equation, which represents the behavior of the system during a suitable interval of transition $[{t}_{},{t}_{+}]$ between the asymptotic states. Multiple connecting solutions arise when such interval length is sufficiently large. Lower estimates for ${t}_{+}{t}_{}$ will be provided in terms of time mappings, which can be expressed by Abelian integrals. This kind of approach is also reminiscent of some topological methods for the study of SturmLiouville boundary value problems. Indeed, a solution that satisfies the SturmLiouville boundary conditions can be interpreted as a trajectory in the phaseplane that connects two lines (see [14–17]). Generalized SturmLiouville solutions, which connect the graphs of two functions or given planar continua, have been considered as well (see [18–20]). Often these problems can be settled in the framework of the theory of ODEs with nonlinear boundary conditions (see [21, 22]).
In order to make our approach more transparent, we are going to perform our analysis for equation (1.5). This choice is motivated by the sake of avoiding unnecessary technicalities. Our arguments can be modified in a straightforward manner in the case of more general equation (1.1), with f satisfying (∗) (see Section 3). Homoclinic and heteroclinic orbits can be interpreted as solutions for some boundary value problems on unbounded intervals. In the last section we also outline possible applications of our approach to boundary value problems on a compact interval (like the SturmLiouville one).
Besides his manifold achievements in different areas of mathematics, Professor Jean Mawhin is one of the pioneers in the study of topological methods for nonlinear boundary value problems. It is a pleasure and an honor to have the possibility to dedicate our work to his important contributions in this area.
2 Homoclinic and heteroclinic solutions: multiplicity results
2.1 General setting
where $k>0$ is a fixed coefficient and $g:\mathbb{R}\to {\mathbb{R}}_{0}^{+}:=\phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty})$ is a bounded measurable function. Solutions of (2.1) are meant in the generalized (i.e., Carathéodory) sense (see [23]). Actually, in our results a step function $g(t)$ with only two jumps is considered and the solutions are piecewise smooth.
In [1] BelmonteBeitia and Torres supposed that the weight function $g(t)$ is even and Tperiodic and proved the existence of a heteroclinic solution connecting two periodic stationary states. Such a result generalizes to the case of periodic coefficients the situation described above for the autonomous system.
As already observed in the introduction, equation (2.1) has been already studied by various authors for its relevance in many applicative models. The aim of the present paper is to provide multiplicity results for heteroclinic and also homoclinic solutions of equation (2.1). A possible way to obtain this goal is to assume that the weight function $g(t)$ has a different behavior at infinity and in some intermediate time interval. In this setting, a first step consists of analyzing the situation in which $g(t)$ is a stepwise function assuming only two values. We believe that the study of such a simplified case may lead to more general considerations, in which, for instance, $g(t)$ is eventually periodic. In this way, multiplicity results for BelmonteBeitia and Torres’s model can be obtained (see Section 3 for a brief overview concerning possible applications of our approach). Equation (2.1) with a periodic stepwise coefficient $g(t)$ has recently been studied in [24] in the context of chaoticlike dynamics.
From the fundamental theory of differential equations it follows that the domain ${D}_{\mathrm{\Phi}}$ of Φ is an open subset of the plane and Φ is an orientation preserving homeomorphism of ${D}_{\mathrm{\Phi}}$ onto its image $\mathrm{\Phi}({D}_{\mathrm{\Phi}})$.
2.2 Analysis of the equation
The solutions of system (2.2) with $g(t)$ defined as in (2.5) will be obtained by connecting suitably chosen parts of stable and unstable manifolds of (2.6) with trajectories of (2.7). More in detail, we proceed as follows.
where $sn(\cdot ,k)$ is the Jacobi elliptic sine function of modulus $k:=a/b$ (see [25, 26]).
the energy level line ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$ contains the saddle points ${P}_{\pm}$ of system (2.6).
In order to obtain homoclinic or heteroclinic solutions for system (2.2) with $g(t)$ defined as in (2.5), we connect the unstable manifold of the point $A\in \{{P}_{},{P}_{+}\}$ to the stable manifold of $B\in \{{P}_{},{P}_{+}\}$ via an orbit path of ${\mathrm{\Gamma}}^{c}$ for $c\in [{\ell}_{1},{\ell}_{0}[$. Actually, this is the only way to get the desired solutions. If $c\le {\ell}^{\ast}$, such connection will follow a trajectory $(x(t),y(t))$ with ${\xi}_{0}\le x(t)\le {\xi}_{0}$, so it will lie in the invariant annular region ℰ bounded by ${\mathrm{\Gamma}}^{{\ell}_{1}}$ and ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$. On the contrary, for $c>{\ell}^{\ast}$ the connection lies in the set ℱ between ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$ and ${\mathrm{\Gamma}}^{{\ell}_{0}}$.
To provide more explicit details, we consider the case $A={P}_{+}$. A solution of (2.2), which is homoclinic to ${P}_{+}$, can be produced in two ways. One consists in connecting $\mathcal{O}({P}_{+})$ to $\mathcal{I}({P}_{+})$, using an orbit path of ${\mathrm{\Gamma}}^{c}$ for $c\in \phantom{\rule{0.2em}{0ex}}]{\ell}^{\ast},{\ell}_{0}[$. Another possibility is given by connecting a point of $\mathcal{L}({P}_{+},{P}_{})$ to a point of $\mathcal{L}({P}_{},{P}_{+})$ via an orbit path of ${\mathrm{\Gamma}}^{c}$ for $c\in [{\ell}_{1},{\ell}^{\ast}[$. For $c={\ell}^{\ast}$, we can obtain our solution if and only if ${t}_{+}{t}_{}$ is an integer multiple of the fundamental period ${\tau}_{{\ell}^{\ast}}$ of the orbit ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$. Indeed, such a solution will be constantly equal to ${P}_{+}$ for $t\notin ]{t}_{},{t}_{+}[$, and for $t\in [{t}_{},{t}_{+}]$ it coincides with the periodic solution of (2.7) such that $x({t}_{})={\xi}_{1}$ and $y({t}_{})=0$. This solution makes $({t}_{+}{t}_{})/{\tau}_{{\ell}^{\ast}}$ turns around the origin for $t\in [{t}_{},{t}_{+}]$.
Similar considerations can be developed with respect to heteroclinic solutions. For instance, a heteroclinic orbit from ${P}_{+}$ to ${P}_{}$ can be obtained as follows: by connecting a point of $\mathcal{O}({P}_{+})$ with one of $\mathcal{I}({P}_{})$ using an orbit path of ${\mathrm{\Gamma}}^{c}$ for $c\in \phantom{\rule{0.2em}{0ex}}]{\ell}^{\ast},{\ell}_{0}[$, or by a connection of two different points of $\mathcal{L}({P}_{+},{P}_{})$ via an orbit path of ${\mathrm{\Gamma}}^{c}$ for $c\in [{\ell}_{1},{\ell}^{\ast}[$. For $c={\ell}^{\ast}$, we can obtain our solution if and only if ${t}_{+}{t}_{}$ is an odd multiple of ${\tau}_{{\ell}^{\ast}}/2$, where ${\tau}_{{\ell}^{\ast}}$ is the period of the orbit ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$. Indeed, such a solution will be constantly equal to ${P}_{+}$ for $t\in (\mathrm{\infty},{t}_{}]$ and constantly equal to ${P}_{}$ for $t\in [{t}_{+},+\mathrm{\infty})$. Moreover, it coincides for $t\in [{t}_{},{t}_{+}]$ with the solution of (2.7) such that $x({t}_{})={\xi}_{1}$ and $y({t}_{})=0$. This solution makes $2({t}_{+}{t}_{})/{\tau}_{{\ell}^{\ast}}$ halfturns around the origin for $t\in [{t}_{},{t}_{+}]$.
In general, the connections through the trajectories of system (2.7) will either involve only an arc of the closed curve ${\mathrm{\Gamma}}^{c}$, or they will require to perform a certain number of winds around the origin, depending on the available time. If the length of the time interval $[{t}_{},{t}_{+}]$ is small, only the first case is possible. However, if ${t}_{+}{t}_{}$ is large enough, more choices arise. For instance, looking at the orbit δ in Figure 4, we could make a certain number of loops from the black circle before reaching the grey square on the same orbit. Similar considerations apply to the heteroclinic trajectory described above with reference to γ.
Until now we have described the ‘external connections’, namely those which lie in the region ℱ. Further possibilities appear if we consider ‘internal connections’ by means of orbit paths contained in ℰ. For example, if we look at the orbit β in Figure 4, we can obtain trajectories homoclinic to ${P}_{+}$ by choosing one of the two black circles as a starting point from $\mathcal{L}({P}_{+},{P}_{})$ and one of the two grey squares as an end point on $\mathcal{L}({P}_{},{P}_{+})$. Loops along β will be permitted if the time is sufficiently large. The construction of heteroclinic orbits from ${P}_{+}$ to ${P}_{}$ via the line α follows a similar procedure.
2.3 Study of the Poincaré map
With reference to Figure 5, we observe that the intersection point of $\mathcal{O}({P}_{+})$ with ${\mathrm{\Gamma}}^{{\ell}_{0}}$ (indicated by a small circle) is moved by Φ along ${\mathrm{\Gamma}}^{{\ell}_{0}}$ to a point very close to ${Q}_{+}$. On the other hand, during the time interval $[{t}_{},{t}_{+}]$, the point ${P}_{+}$ makes a little more than a complete turn around the origin; the final position $\mathrm{\Phi}({P}_{+})$ is indicated by a cross. The arc $\mathrm{\Phi}(\mathcal{Y})$ is a spirallike curve with one end near ${Q}_{+}$ and the other on ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$. For the time length ${t}_{+}{t}_{}=5.5$, considered in Figure 5, there are precisely two intersections of $\mathrm{\Phi}(\mathcal{Y})$ with $\mathcal{I}({P}_{+})$, corresponding to external homoclinic solutions. Meanwhile, we have also an external heteroclinic solution due to the intersection of $\mathrm{\Phi}(\mathcal{Y})$ with $\mathcal{I}({P}_{})$. All these three intersection points have been indicated by a grey square (following the same convention used in Figure 4).
In order to describe further these solutions, let us consider their energy at the time ${t}_{+}$ that we denote by ${k}_{1}$, ${k}_{2}$, ${k}_{3}$, with ${\ell}^{\ast}<{k}_{3}<{k}_{2}<{k}_{1}<{\ell}_{0}$. Let us define the points ${H}_{i}$ as $\{{H}_{i}\}=\mathcal{O}({P}_{+})\cap {\mathrm{\Gamma}}^{{k}_{i}}$ for $i=1,2,3$. The solution $\zeta (t)=(x(t),y(t))$ of (2.2) with $\zeta ({t}_{})={H}_{1}$ or $\zeta ({t}_{})={H}_{3}$ has the following behavior: $\zeta (t)\to {P}_{+}$ for $t\to \pm \mathrm{\infty}$, with $x(t)>{\xi}_{1}$ and convex for $t\notin \phantom{\rule{0.2em}{0ex}}]{t}_{},{t}_{+}[$. Moreover, $x(t)$ is increasing on $(\mathrm{\infty},{t}_{}]$ and decreasing on $[{t}_{+},+\mathrm{\infty})$. On the interval $[{t}_{},{t}_{+}]$, if $i=1$, the solution $x(t)$ is concave, while if $i=3$, the solution has two maxima and one (negative) minimum separated by two simple zeros. The solution of system (2.2) with $\zeta ({t}_{})={H}_{2}$ is such that $\zeta (t)\to {P}_{+}$ for $t\to \mathrm{\infty}$ with $x(t)>{\xi}_{1}$ and $\zeta (t)\to {P}_{}$ for $t\to +\mathrm{\infty}$ with $x(t)<{\xi}_{1}$; $x(t)$ is convex for $t\notin \phantom{\rule{0.2em}{0ex}}]{t}_{},{t}_{+}[$. Moreover, the solution has one maximum and one minimum separated by one simple zero.
Even if the time interval length ${t}_{+}{t}_{}$ is small, we always have at least one intersection of $\mathrm{\Phi}(\mathcal{Y})$ with $\mathcal{I}({P}_{+})$ and thus a homoclinic solution. On the other hand, if ${t}_{+}{t}_{}$ grows, the spiral curve $\mathrm{\Phi}(\mathcal{Y})$ will wind more times around the origin, hence more homoclinic/heteroclinic solutions will appear.
If we look for the internal connections (made by orbit paths lying in the region ℰ), we proceed as follows.
Summarizing the above information, we conclude that, for every time interval $[{t}_{},{t}_{+}]$, we have at least two solutions which are homoclinic to ${P}_{+}$ (one external and one internal) and one internal heteroclinic solution from ${P}_{+}$ to ${P}_{}$.
For the external connections, if the time interval length ${t}_{+}{t}_{}$ grows, the situation can be summarized as follows. The number of external homoclinic/heteroclinic solutions increases, depending on the number of winds around the origin of the curve $\mathrm{\Phi}(\mathcal{Y})$. Indeed, the end point of such a curve on ${\mathrm{\Gamma}}^{{\ell}_{0}}$ cannot go beyond ${Q}_{+}$, while the other end point $\mathrm{\Phi}({P}_{+})$ is free to move on the periodic orbit ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$. If we denote by ${\eta}_{\mathrm{ext}}$ and ${\nu}_{\mathrm{ext}}$ the number of external homoclinic and heteroclinic solutions respectively, it holds that $0\le {\eta}_{\mathrm{ext}}{\nu}_{\mathrm{ext}}\le 1$. From the above discussion, we can conclude that if, for some nonnegative integer n, we have ${t}_{+}{t}_{}>n{\tau}_{{\ell}^{\ast}}$ (where ${\tau}_{{\ell}^{\ast}}$ is the period of ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$), then ${\eta}_{\mathrm{ext}}\ge n+1$. An analogous lower bound can be provided for ${\nu}_{\mathrm{ext}}$. For a formal proof, see Theorem 2.1.
In preparation for this theorem, we introduce the following notation. Given a positive real number S and an energy level $\ell \in [{\ell}_{1},{\ell}^{\ast}]$, we denote by $\underline{n}(S,\ell ):=\lfloor \frac{2S}{{\tau}_{\ell}}\rfloor $ the lower integer part of $S/({\tau}_{\ell}/2)$, and by $\overline{n}(S,\ell ):=\lceil \frac{2S}{{\tau}_{\ell}}\rceil $ the upper integer part of $S/({\tau}_{\ell}/2)$. By definition, during a time interval of length S, a point on the orbit ${\mathrm{\Gamma}}^{\ell}$ makes more that $\overline{n}(S,\ell )1$ halfturns around the origin, but less that $\underline{n}(S,\ell )+1$ halfturns. Thus the rotational gap defined above can be written as $w(T)=\overline{n}(T,{\ell}_{1})\underline{n}(T,{\ell}^{\ast})2$ for $T={t}_{+}{t}_{}$. By definition, in the open interval $]2T/{\tau}_{{\ell}^{\ast}},2T/{\tau}_{{\ell}_{1}}[$, the smallest integer is $\underline{n}(T,{\ell}^{\ast})+1$, while the largest integer is $\overline{n}(T,{\ell}^{\ast})1$. Hence, the open interval $]2T/{\tau}_{{\ell}^{\ast}},2T/{\tau}_{{\ell}_{1}}[$ contains $w(T)+1$ positive integers.
2.4 Conclusion
In order to make the exposition more clear, we have decided to consider separately the cases of external and internal connections.
Theorem 2.1 (External connections)

there always exist a (positive) solution homoclinic to${P}_{+}$and a (negative) solution homoclinic to${P}_{}$;

if${t}_{+}{t}_{}>{\tau}_{{\ell}^{\ast}}/2$, there always exist a heteroclinic solution from${P}_{+}$to${P}_{}$and a heteroclinic solution from${P}_{}$to${P}_{+}$, both with exactly one zero;

if${t}_{+}{t}_{}>n{\tau}_{{\ell}^{\ast}}$ (for some integer$n\ge 1$) then, for each integer j with$1\le j\le n$, there exist at least one solution homoclinic to${P}_{+}$and one solution homoclinic to${P}_{}$, both with exactly 2j zeros;

if${t}_{+}{t}_{}>(n+1){\tau}_{{\ell}^{\ast}}/2$ (for some integer$n\ge 1$) then, for each integer j with$1\le j\le n$, there exist at least one heteroclinic solution from${P}_{+}$to${P}_{}$and one from${P}_{}$to${P}_{+}$, both with exactly$2j+1$zeros.
Proof As a first step, we focus our attention on the search of solutions homoclinic to the point ${P}_{+}$. The case of homoclinics to ${P}_{}$ is analogous, thus it will be omitted.
Recalling that $T={t}_{+}{t}_{}$, our goal is to find points on $\mathcal{O}({P}_{+})$ which are moved by the Poincaré map ${\mathrm{\Phi}}_{0}^{T}$ on $\mathcal{I}({P}_{+})$.
We introduce in the phaseplane for (2.7), a system of polar coordinates $(\theta ,\rho )$, with center in the origin. The initial points in the arc (which is the closure of the intersection of $\mathcal{O}({P}_{+})$ with the external region ℱ) are parameterized as $\gamma (s):=(\rho (s)cos\theta (s),\rho (s)sin\theta (s))$, for $s\in [0,1]$, with $\gamma (0)={P}_{+}$, $\gamma (1)\in {\mathrm{\Gamma}}^{{\ell}_{0}}$ and $E(\gamma (s))$ strictly increasing with s. The target set is the symmetric of with respect to the xaxis. It will be denoted by ${\mathcal{Y}}^{\prime}$ and parameterized by reversing the angle $\theta (s)$.
is strictly decreasing on $[0,+\mathrm{\infty})$ (this is an equivalent way to express the fact that the solutions turn around the origin in the clockwise sense).
for some nonnegative integer j. In this case, if we denote by $\stackrel{\u02c6}{x}(\cdot )$ the corresponding homoclinic solution of (2.1) such that $(\stackrel{\u02c6}{x}({t}_{}),{\stackrel{\u02c6}{x}}^{\prime}({t}_{}))=\gamma (\stackrel{\u02c6}{s})$, we have that $E(\stackrel{\u02c6}{x}(t),{\stackrel{\u02c6}{x}}^{\prime}(t))\in \phantom{\rule{0.2em}{0ex}}]{\ell}^{\ast},{\ell}_{0}[$ for all $t\in \mathbb{R}$. Moreover, if $j=0$, $\stackrel{\u02c6}{x}(t)>{\xi}_{1}$ for all $t\in [{t}_{},{t}_{+}]$, while $\stackrel{\u02c6}{x}(t)$ has precisely 2j zeros in $]{t}_{},{t}_{+}[$ if $j\ge 1$.
is satisfied (for the same j).
As a consequence, $\theta (T,0)+\theta (0)<0<\theta (T,1)+\theta (1)$ and the intermediate value theorem ensures the existence of an $\stackrel{\u02c6}{s}$ such that $\theta (T,\stackrel{\u02c6}{s})=\theta (\stackrel{\u02c6}{s})$. In this way we have found a positive homoclinic solution to ${P}_{+}$, independently of the length T of the time interval $[{t}_{},{t}_{+}]$.
As a consequence, for every integer j with $1\le j\le n$, we have $\theta (T,0)+\theta (0)<2j\pi <0<\theta (T,1)+\theta (1)$, and again the intermediate value theorem ensures the existence of an $\stackrel{\u02c6}{s}$ such that $\theta (T,\stackrel{\u02c6}{s})=\theta (\stackrel{\u02c6}{s})2j\pi $. This ends the proof for homoclinic solutions.
for some integer $j\ge 1$. Then the proof can be concluded as above, via the intermediate value theorem. We observe that the case of heteroclinic solutions from ${P}_{}$ to ${P}_{+}$ is analogous, thus it will be omitted. □
The next theorem deals with the internal connections. For this result, it is useful to recall the rotational gap defined in (2.11).
Theorem 2.2 (Internal connections)

there always exist a solution homoclinic to${P}_{+}$and a solution homoclinic to${P}_{}$;

there always exist a heteroclinic solution from${P}_{+}$to${P}_{}$and a heteroclinic solution from${P}_{}$to${P}_{+}$;

if${t}_{+}{t}_{}>\frac{{\tau}_{{\ell}_{1}}{\tau}_{{\ell}^{\ast}}}{{\tau}_{{\ell}^{\ast}}{\tau}_{{\ell}_{1}}}$, there exist at least$2w({t}_{+}{t}_{})+1$solutions homoclinic to${P}_{+}$and$2w({t}_{+}{t}_{})+1$solutions homoclinic to${P}_{}$;

if${t}_{+}{t}_{}>\frac{{\tau}_{{\ell}_{1}}{\tau}_{{\ell}^{\ast}}}{{\tau}_{{\ell}^{\ast}}{\tau}_{{\ell}_{1}}}$, there exist at least$2w({t}_{+}{t}_{})+1$heteroclinic solutions from${P}_{+}$to${P}_{}$and$2w({t}_{+}{t}_{})+1$from${P}_{}$to${P}_{+}$.
Proof As the first step, we focus our attention on the search of solutions homoclinic to the point ${P}_{+}$. The case of homoclinics to ${P}_{}$ is analogous, thus it will be omitted.
Our goal is to find points on $\mathcal{L}=\overline{\mathcal{L}({P}_{+},{P}_{})}$ which are moved by the Poincaré map ${\mathrm{\Phi}}_{0}^{T}$ on ${\mathcal{L}}^{\prime}=\overline{\mathcal{L}({P}_{},{P}_{+})}$.
As a preliminary remark, we note that ${\mathrm{\Phi}}_{0}^{T}(\mathcal{L})$ is a simple arc contained in ℰ, connecting its end points ${\mathrm{\Phi}}_{0}^{T}({P}_{+})$ and ${\mathrm{\Phi}}_{0}^{T}({P}_{})$ on ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$ through the ‘intermediate’ point ${\mathrm{\Phi}}_{0}^{T}(R)$ on ${\mathrm{\Gamma}}^{{\ell}_{1}}$. Observe also that ${\mathrm{\Phi}}_{0}^{T}({P}_{+})$ and ${\mathrm{\Phi}}_{0}^{T}({P}_{})$ are antipodal. In fact, ${P}_{}$ and ${P}_{+}$ are antipodal and ${\mathrm{\Phi}}_{0}^{T}$ is an odd map (indeed, $(x(t),y(t))$ is a solution of (2.7) if and only if $(x(t),y(t))$ is a solution of the same equation). Now, if the trivial situation $\{{\mathrm{\Phi}}_{0}^{T}({P}_{}),{\mathrm{\Phi}}_{0}^{T}({P}_{+})\}=\{{P}_{},{P}_{+}\}$ occurs, we have ${\mathrm{\Phi}}_{0}^{T}(\mathcal{L})\cap {\mathcal{L}}^{\prime}=\{{P}_{},{P}_{+}\}\ne \mathrm{\varnothing}$. Otherwise, we find that ${\mathrm{\Phi}}_{0}^{T}({P}_{})$ and ${\mathrm{\Phi}}_{0}^{T}({P}_{+})$ belong to the two different components of $\mathcal{E}\setminus {\mathcal{L}}^{\prime}$. Since ${\mathrm{\Phi}}_{0}^{T}({P}_{})$ and ${\mathrm{\Phi}}_{0}^{T}({P}_{+})$ are the end points of the arc ${\mathrm{\Phi}}_{0}^{T}(\mathcal{L})\subseteq \mathcal{E}$, by an elementary connectivity argument, we conclude that ${\mathrm{\Phi}}_{0}^{T}(\mathcal{L})\cap {\mathcal{L}}^{\prime}\ne \mathrm{\varnothing}$. This proves the first assertion of the theorem.
For the search of heteroclinic solutions from ${P}_{+}$ to ${P}_{}$, we adopt a similar procedure, choosing as a target the set ℒ itself. The fact that, for every $T>0$, there is always at least one intersection point between ${\mathrm{\Phi}}_{0}^{T}(\mathcal{L})$ and ℒ follows by the same argument developed for homoclinic solutions. Namely, if the trivial situation $\{{\mathrm{\Phi}}_{0}^{T}({P}_{}),{\mathrm{\Phi}}_{0}^{T}({P}_{+})\}=\{{P}_{},{P}_{+}\}$ occurs, we have ${\mathrm{\Phi}}_{0}^{T}(\mathcal{L})\cap \mathcal{L}=\{{P}_{},{P}_{+}\}\ne \mathrm{\varnothing}$. Otherwise, we find that ${\mathrm{\Phi}}_{0}^{T}({P}_{})$ and ${\mathrm{\Phi}}_{0}^{T}({P}_{+})$ belong to the two different components of $\mathcal{E}\setminus \mathcal{L}$. Therefore, it follows that ${\mathrm{\Phi}}_{0}^{T}(\mathcal{L})\cap \mathcal{L}\ne \mathrm{\varnothing}$.
We study now the problem of multiplicity of solutions. As before, we consider at first the case of solutions homoclinic to ${P}_{+}$.
We introduce in the inner region ℰ of a phaseplane for (2.7) a system of modified polar coordinates $(\theta ,E)$ with center in the origin. In this system, every point is determined by its angular coordinate θ and its energy E defined in (2.8). Since the energy level lines in ℰ are strictly starshaped with respect to the origin, we obtain a coordinate system equivalent to the polar one.
Observe that the set $\mathcal{L}\cup {\mathcal{L}}^{\prime}$ intersects every halfline from the origin exactly in one point. Accordingly, we can parameterize those points using the angular coordinate. With this convention, the initial points in the arc ℒ are parameterized as $\gamma (s)$ for $s\in [\pi ,2\pi ]$ (s is the angle) with $\gamma (\pi )={P}_{}$, $\gamma (2\pi )={P}_{+}$ and $\gamma (3\pi /2)=R$. As a consequence, $E(\gamma (s))$ is strictly decreasing for $s\in [\pi ,3\pi /2]$ and strictly increasing in $[3\pi /2,2\pi ]$. The target set ${\mathcal{L}}^{\prime}$ (the symmetric of ℒ with respect to the xaxis) is parameterized with the angle $s\in [0,\pi ]$.
Hence, the natural parametrization of ℒ in the $(\theta ,E)$plane is given by $\gamma (s)=(s,E(s))$ for $s=\theta \in [\pi ,2\pi ]$.
(recall that $\gamma (s)=(s,E(s))$ for $s=\theta \in [\pi ,2\pi ]$).
(this follows from the fact that the orbit ${\mathrm{\Gamma}}^{{\ell}^{\ast}}$ has a period ${\tau}_{{\ell}^{\ast}}$ and is symmetric with respect to the xaxis).
Then, the open interval $]2T/{\tau}_{{\ell}^{\ast}},2T/{\tau}_{{\ell}_{1}}[$ contains at least one odd integer.
By definition, for any nonnegative integer i, the set ${\mathcal{L}}_{\mathrm{left}}^{\prime}(2i\pi ,0)$ is a simple arc connecting the points $(2i\pi ,{\ell}^{\ast})$ and $(2i\pi +(\pi /2),{\ell}_{1})$ in the rectangle $[2i\pi ,2i\pi +(\pi /2)]\times [{\ell}_{1},{\ell}^{\ast}]$ and, similarly, ${\mathcal{L}}_{\mathrm{right}}^{\prime}(2i\pi ,0)$ is a simple arc connecting the points $(2i\pi +(\pi /2),{\ell}_{1})$ and $(2i\pi +\pi ,{\ell}^{\ast})$ in the rectangle $[2i\pi +(\pi /2),2i\pi +\pi ]\times [{\ell}_{1},{\ell}^{\ast}]$. On the other hand, the set $\mathrm{\Phi}({\mathcal{L}}_{\mathrm{left}})$ is a simple arc connecting the points $(\theta (T,\pi ),{\ell}^{\ast})$ and $(\theta (T,3\pi /2),{\ell}_{1})$ in $\mathcal{E}(\mathrm{\infty},2\pi ]$ and, similarly, $\mathrm{\Phi}({\mathcal{L}}_{\mathrm{right}})$ is a simple arc connecting the points $(\theta (T,3\pi /2),{\ell}_{1})$ and $(\theta (T,2\pi ),{\ell}^{\ast})=(\pi +\theta (T,\pi ),{\ell}^{\ast})$ in the same strip.
for $2k=j1$ an even integer. Hence the points $(\theta (T,3\pi /2),{\ell}_{1})$ and $(\theta (T,\pi ),{\ell}^{\ast})$ are separated by the arc ${\mathcal{L}}_{\mathrm{left}}^{\prime}(2k\pi ,0)$, while the points $(\theta (T,3\pi /2),{\ell}_{1})$ and $(\theta (T,2\pi ),{\ell}^{\ast})$ are separated by ${\mathcal{L}}_{\mathrm{left}}^{\prime}(2k\pi ,0)$ and ${\mathcal{L}}_{\mathrm{right}}^{\prime}(2k\pi ,0)$. As a consequence, $\mathrm{\Phi}(\mathcal{L})\cap \mathrm{\Xi}$ contains at least three points, precisely the nonempty intersections of $\mathrm{\Phi}({\mathcal{L}}_{\mathrm{left}})\cap ({\mathcal{L}}_{\mathrm{left}}^{\prime}(2k\pi ,0))$, $\mathrm{\Phi}({\mathcal{L}}_{\mathrm{right}})\cap ({\mathcal{L}}_{\mathrm{left}}^{\prime}(2k\pi ,0))$ and $\mathrm{\Phi}({\mathcal{L}}_{\mathrm{right}})\cap ({\mathcal{L}}_{\mathrm{right}}^{\prime}(2k\pi ,0))$.
for $2k={\mathbf{m}}_{o}(T)1$ an even integer. Hence the points $(\theta (T,3\pi /2),{\ell}_{1})$ and $(\theta (T,\pi ),{\ell}^{\ast})$ are separated by the arcs ${\mathcal{L}}_{\mathrm{left}}^{\prime}(2k\pi ,0)$ and ${\mathcal{L}}_{\mathrm{right}}^{\prime}(2k\pi ,0)$. The points $(\theta (T,3\pi /2),{\ell}_{1})$ and $(\theta (T,2\pi ),{\ell}^{\ast})$ are separated by ${\mathcal{L}}_{\mathrm{left}}^{\prime}(2k\pi ,0)$, ${\mathcal{L}}_{\mathrm{right}}^{\prime}(2k\pi ,0)$ and ${\mathcal{L}}_{\mathrm{left}}^{\prime}((2k1)\pi ,0)$. As a consequence, $\mathrm{\Phi}(\mathcal{L})\cap \mathrm{\Xi}$ contains at least five points: two points coming from the intersections of $\mathrm{\Phi}({\mathcal{L}}_{\mathrm{left}})$ with the target set Ξ and three from the intersections of $\mathrm{\Phi}({\mathcal{L}}_{\mathrm{right}})$.
Proceeding by induction, with the same argument, the following result is obtained.
Claim 1 $\mathrm{\Phi}(\mathcal{L})\cap \mathrm{\Xi}$ has at least $2{n}^{\mathrm{\#}}+1$ solutions, where ${n}^{\mathrm{\#}}$ is the number of integers less than or equal to ${\mathbf{m}}_{o}(T)$ which are contained in the open interval $]2T/{\tau}_{{\ell}^{\ast}},2T/{\tau}_{{\ell}_{1}}[$.
A completely symmetric argument leads to the same multiplicity result for solutions which are homoclinic to ${P}_{}$.
where $E(\theta )$ is the function defined in (2.15).
As before, we assume the validity of condition (2.18). Then the open interval $]2T/{\tau}_{{\ell}^{\ast}},2T/{\tau}_{{\ell}_{1}}[$ contains at least one even integer.
For any positive integer j with $2T/{\tau}_{{\ell}^{\ast}}<j\le {\mathbf{m}}_{e}(T)$, we have that (2.20) holds too. At this point, we have simply to repeat (with obvious changes due to the fact that the target set is ${\mathrm{\Xi}}^{\prime}$ which is a shift of Ξ by $(2pi,0)$) the argument presented above for the case of homoclinic connections and obtain the following conclusion.
Claim 2 $\mathrm{\Phi}(\mathcal{L})\cap {\mathrm{\Xi}}^{\prime}$ has at least $2{n}_{\mathrm{\#}}+1$ solutions, where ${n}_{\mathrm{\#}}$ is the number of integers less than or equal to ${\mathbf{m}}_{e}(T)$ which are contained in the open interval $]2T/{\tau}_{{\ell}^{\ast}},2T/{\tau}_{{\ell}_{1}}[$.
A completely symmetric argument leads to a multiplicity result for heteroclinic solutions from ${P}_{}$ to ${P}_{+}$.
This concludes the proof of the theorem. □
3 Remarks and related results
 1.
A problem which naturally arises from the analysis that we have performed concerns what happens if, for a stepwise weight function satisfying (2.5), we suppose that $0<{\mu}_{1}<{\mu}_{0}$. Repeating the preliminary phaseplane analysis of Section 2.2, one can easily check that for any gap $T={t}_{+}{t}_{}>0$, there always exist a solution homoclinic to ${P}_{+}$ and another homoclinic to ${P}_{}$, as well as a heteroclinic from ${P}_{+}$ to ${P}_{}$ and another one from ${P}_{}$ to ${P}_{+}$. However, in general, if $0<{\mu}_{1}<{\mu}_{0}$, one cannot obtain multiplicity results like those achieved in Section 2 without some extra assumptions on $h(x)$. This is the reason for which, in the study of equation (2.1), we have considered only the case $0<{\mu}_{0}<{\mu}_{1}$. For different examples on related equations in which a weight coefficient can be above or below its limits at infinity, see, for instance, [4–6].
 2.
 3.Due to the special form of the weight coefficient, it is standard to verify (via a simple rescaling procedure) that (1.1) is equivalent to an equation of the form${x}^{\u2033}+b(t)xah(x)=0,$(3.1)
 4.
The approach used in the proofs, based on the properties of the Poincaré map, guarantees that our results are stable with respect to small perturbations. More precisely, fixed a suitable length $T={t}_{+}{t}_{}$ for the time interval, Theorems 2.1 and 2.2 provide a lower bound for the number of solutions. We can state that the same lower bound persists for a small perturbation of the coefficient in the ${L}^{1}$norm on $[{t}_{},{t}_{+}]$. Therefore, the assumption that the weight $a(t)$ in (1.1) or $b(t)$ in (3.1) are stepwise functions can be slightly relaxed, so we can ‘smooth’ them.
 5.With reference to equations (1.1) or (3.1) with stepwise coefficients, we observe that our approach can be adapted to boundary value problems on a compact interval $[0,T]$ like, e.g., the Dirichlet (twopoint) or the Neumann problem. In these cases, we have to find solutions connecting given lines which depart from the origin. For the sake of conciseness, we cannot describe the most general situation, but we just outline a possible application for the Neumann problem$\{\begin{array}{c}{x}^{\u2033}+2kx2g(t){x}^{3}=0,\hfill \\ {x}^{\prime}(0)=0,\phantom{\rule{2em}{0ex}}{x}^{\prime}(T)=0,\hfill \end{array}$(3.2)
 6.
As a final remark, we mention the fact that combining our technique with Ważewski’s method [31], following the approach developed by Conley in [18], one can deal with some more general classes of weight functions. For example, one could tackle with these techniques the cases of asymptotically constant or asymptotically periodic coefficients. These extensions, however, need a more delicate analysis and they are beyond the goals of the present paper.
We hope that the abundance of multiplicity results found in the present work (in the special case of stepwise coefficients) may suggest possible directions for extending Theorem 2.1 and Theorem 2.2 to more general weight functions. This will be our goal for a future investigation of the problem.
Declarations
Acknowledgements
The authors are deeply indebted with the referee for the careful checking of the manuscript and for his/her remarks, including a correction to an erroneous argument in the previous version of the proof. This research was partially supported by the project PRIN2009 Equazioni Differenziali Ordinarie e Applicazioni.
Authors’ Affiliations
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