\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 21, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/21\hfil A nonresonance condition at zero]
{A note on a nonresonance condition at zero \\
for first-order planar systems}

\author[M. Garrione \hfil EJDE-2015/21\hfilneg]
{Maurizio Garrione}

\address{Maurizio Garrione \newline
Dipartimento di Matematica ed Applicazioni,
Universit\`a di Milano-Bicocca,
Via Cozzi 55,  I-20125 Milano, Italy}
\email{maurizio.garrione@unimib.it}

\thanks{Submitted November 10, 2014. Published January 27, 2015.}
\subjclass[2000]{34C25, 37E45}
\keywords{Landesman-Lazer condition; rotation number;  multiplicity}

\begin{abstract}
 We introduce a Landesman-Lazer type nonresonance condition at zero
 for planar systems and discuss its rotational interpretation.
 We then show an application concerning multiplicity of $T$-periodic solutions
 to unforced Hamiltonian systems like
 $$
 Ju'=\nabla H(t, u), \quad \nabla H(t, 0) \equiv 0,
 $$
 for which the nonlinearity is resonant both at zero and at infinity,
 refining and complementing some recent results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

This note can be seen as a complement to some recent multiplicity results
for periodic solutions of first order Hamiltonian systems of ODEs
in the plane, wishing to fill in a little asymmetric gap in this kind of results.
Our starting point is embodied by planar systems at resonance having the form
\begin{equation}\label{eqintro}
Ju'=F(u)  + R(t, u), \quad t \in [0, T],
\end{equation}
where $J$ denotes the standard symplectic matrix. Here, resonance - always
considered with respect to $T$-periodic boundary conditions -
essentially arises when looking at a sublinear perturbation of a center-type
dynamics ($Ju'=F(u)$) for which the minimal period on the
orbits is \emph{constant and equal to a submultiple of $T$};
nevertheless, with reference to the beginning of Section \ref{sezdue}, the
stronger assumption
$$
F(u)=\nabla V(u) \quad \text{with }  V \in \mathcal{P}
$$
(roughly speaking, the principal part of the nonlinearity being a
positively homogeneous gradient)  is usually needed in order to
formulate plain and readable results.
Indeed, if $V \in \mathcal{P}$ the origin is an isochronous center for
$Ju'=\nabla V(u)$, and the common period of all the solutions
can be computed in an immediate way from the expression of $V$, namely
$$
\tau_V=\int_0^{2\pi} \frac{d\theta}{2V(\cos \theta, \sin \theta)}.
$$
This setting, introduced in the paper \cite{Fon04}, takes inspiration from
scalar second order asymmetric equations and represents a possible extension
of the concept of Dancer-Fu\v{c}ik spectrum to general planar systems.

We incidentally remark that similar considerations seem to be possible imposing
that $F$ is only positively $1$-homogeneous, without
being  a gradient, provided that there exists a closed orbit for $Ju'=F(u)$.
 However, apart from its possible interest in the context of
general homogeneous systems, this extension seems to be quite artificial,
so that we prefer to set our results in the mentioned framework.

It is well known (for the precise statements, see  \cite{Fon04}) that, under
a resonance assumption, the existence of a $T$-periodic
solution to \eqref{eqintro} is not ensured. In the last years, several conditions
were produced in order to overcome this problem and obtain
the solvability of \eqref{eqintro}. Among these, we have both Landesman-Lazer
\cite{FonGar11} and Ahmad-Lazer-Paul \cite{BosGar13}  type
ones, the former being particularly useful when employing topological methods,
the latter being more suitable to be exploited in a
variational setting. The common point between such conditions is that they are
formulated (up to some few exceptions, mainly for
Ahmad-Lazer-Paul type results) mostly at infinity, requiring a certain behavior
of the perturbation $R(t, u)$ when $| u |$ becomes
large. To the author's knowledge, much less has been said regarding existence
conditions at zero, namely when $| u |$ is small.

Wishing to deal with this situation, it is clear that, since we are working
with sublinear perturbations, $R(t, u)$ has to
satisfy $R(t, 0) \equiv 0$, so that mere existence results will be useless
in principle, as they could lead to find the trivial solution.
However, the Landesman-Lazer condition has been likewise shown to be particularly
effective \cite[Section 5]{BosGar11} in order to refine
results of \emph{multiplicity} of $T$-periodic solutions for unforced planar
Hamiltonian systems like
\begin{equation}\label{ilplanare}
Ju' = \nabla H(t, u), \quad \nabla H(t, 0) \equiv 0,
\end{equation}
when $\nabla H(t, u)$ has different asymptotic expansions with positively
 homogeneous principal term at zero and at infinity -
for some references about this kind of statements, we remind the reader,
e.g., to the bibliography in \cite{BosGar11}. In this setting, a
successful way of obtaining multiple solutions is represented by the
Poincar\'e-Birkhoff fixed point theorem, providing a number of fixed
points of the Poincar\'e map associated with \eqref{ilplanare} which grows
according to the gap between the integer parts of the number of
revolutions made by ``small'' and ``large'' solutions around the origin,
with precise nodal characterizations. As has been shown in
\cite{BosGar11}, assuming a Landesman-Lazer condition at infinity turns out
to be particularly useful when the principal term (at infinity)
is resonant, making it possible to give finer estimates of the rotation number
of large solutions, in principle ``dangerously close'' to an
integer.  One can then conclude the existence of multiple $T$-periodic solutions
in larger number than predicted by rougher estimates (see
\cite[Theorem 5.2]{BosGar11}).

The main goal of this paper is to extend this picture at zero, after having
introduced a suitable Landesman-Lazer type condition.
We observe that this way of proceeding in order to obtain multiplicity of
$T$-periodic solutions to systems like \eqref{ilplanare},
which works in the general positively homogeneous framework, turns to be
fruitful in dimension two but finds more difficulty to be applied
in higher dimension (even if a significant step in this direction seems
to have been achieved in the recent work \cite{FonUrePP}).
Alternatively, particularly in the asymptotically linear case
$$
\nabla H(t, u)=B_0(t) u + R_0(t, u)  \text{ at }0, \quad
 \nabla H(t, u)= B_\infty(t) u + R_\infty(t, u)   \text{ at } \infty,
$$
where $R_0(t, u), R_\infty(t, u)$ are negligible with respect to $| u |$,
Morse, Conley-Zehnder or Maslov index theory are quite
successful in any dimension, possibly considered jointly with Landesman-Lazer
or Ahmad-Lazer-Paul conditions (we refer, e.g., to
\cite{ChaLiuLiu97, LiuSuWan08} and the references therein; we mention
\cite{MarRebZan02} as a bridge between Maslov index type results and
Poincar\'e-Birkhoff type ones). However, one of their possible drawbacks is
 represented by the fact that the number of solutions found does
not change significantly according to the gap between the behaviors at zero and
infinity; moreover, in case the indexes at zero and infinity
are the same - for instance, if the principal parts coincide, they seldom allow
to infer existence.

It is looking at this second group of results that we find one of the few
contributions, to the best of our knowledge, about a possible
formulation of Landesman-Lazer type conditions at zero for second order problems
\cite{Han07} (to be compared with similar expressions at
infinity, e.g., \cite{Maw77, PerSch00}). The role of such conditions is
therein to allow a precise computation of the critical groups of the
action functional associated with the considered problem at the origin.
 However, also in this case the indexes at zero and infinity are
required to be different, otherwise existence is not guaranteed.

In this article, adopting a qualitative approach in the plane, we thus try to
 make the situation for first order planar systems a little bit
more symmetric, providing a nonresonance condition at zero which carries a
suitable rotational characterization. Consequently, we will be
able to complement some existing results of multiplicity of $T$-periodic solutions,
taking into account resonant principal parts at zero;
furthermore, we will be able to admit the same resonant expansion at zero and
at infinity.

The rotational characterization will be obtained by directly estimating the
 winding number of small solutions, while multiplicity of
solutions will follow from the Poincar\'e-Birkhoff fixed point theorem, having
the possibility of exploiting a finer estimate of the
rotations of ``small'' solutions.
Section \ref{sezdue} is completed by few applications to scalar second order
equations, while in Section \ref{seztre} we will highlight the
relationships between Landesman-Lazer type nonresonance conditions at zero
and at infinity, through a suitable change of variables.
Finally, we dedicate Section \ref{sezquattro} to a final discussion about
some complementary issues related to recent papers on planar
systems.

\section{A Landesman-Lazer condition at zero} \label{sezdue}

Let us denote by $J$ the standard $2 \times 2$ symplectic matrix, namely
$J=\begin{pmatrix}
0 & -1 \\
1 & 0 \\
\end{pmatrix}$,
and by $\mathcal{P}$ the set of the $C^1$-functions 
$V: \mathbb{R}^2 \to \mathbb{R}$, with locally Lipschitz continuous gradient, which
are positively homogeneous of degree $2$ and positive, i.e.,
$$
0 < V(\lambda u)=\lambda^2 V(u), \quad \lambda > 0, \;
 u \in \mathbb{R}^2 \setminus \{0\};
$$
if we replace \emph{positive} with \emph{nonnegative} in this definition, 
we will speak about the class $\mathcal{P}^*$, as in \cite{Gar12}. 
We remind that, given $V \in \mathcal{P}$, all the solutions to $Ju'=\nabla V(u)$ 
are periodic with the same minimal period $\tau_V > 0$; moreover, 
fixed one of them, denoted by $\varphi_V$, all the other ones can be 
written as $u(t)=C \varphi_V(t+\omega)$, for suitable $C \geq 0$, 
$\omega \in [0, \tau_V[\,$. We choose $\varphi_V$ in such a way 
that $V(\varphi_V(t)) \equiv 1/2$ (we recall that $V$ is preserved along 
the solutions).

We are interested in the planar $T$-periodic boundary value problem
\begin{equation}\label{problemaper}
\begin{gathered}
Ju' = \nabla V(u) + R(t, u) \\
u(0)=u(T),
\end{gathered}
\end{equation}
where $V \in \mathcal{P}$ and $R: [0, T] \times \mathbb{R}^2 \to \mathbb{R}^2$ 
is an $L^1$-Carath\'eodory function, i.e.,
\begin{itemize}
\item $t \mapsto R(t, u)$ is measurable, for every $u \in \mathbb{R}^2$;
\item $u \mapsto R(t, u)$ is continuous, for almost every $t \in [0, T]$;
\item for every compact subset $\mathcal{K} \subset \mathbb{R}^2$, 
there exists $h_{\mathcal{K}} \in L^1(0, T)$ such that
$| R(t, u) | \leq h_{\mathcal{K}}(t)$, for almost every
$t \in [0, T]$ and every $u \in \mathcal{K}$.
\end{itemize}
Wishing to study a resonant problem, we will assume that there exists 
a positive integer $N$ such that
\begin{equation}\label{risonanza}
\tau_V= \frac{T}{N},
\end{equation}
according to the setting introduced in \cite{Fon04}.
To guarantee the solvability of \eqref{problemaper} when 
$R: [0, T] \times \mathbb{R}^2 \to \mathbb{R}^2$ is sublinear at infinity -
as is usual in the spirit of resonance - and satisfies some mild control from 
below (see condition (LL+$_\infty$) below), a planar version
of the Landesman-Lazer condition was introduced in \cite{FonGar11} and 
extensively discussed in \cite{BosGar11, FonGar11}. Its explicit
expression reads as
\begin{equation}\label{LLi}
\int_0^T \liminf_{(\lambda, \omega) \to (+\infty, \theta)} 
\langle R(t, \lambda \varphi_V(t+\omega)) | \varphi_V(t+\omega) \rangle \,
dt > 0,
\end{equation}
for every $\theta \in [0, T]$.
The effect of \eqref{LLi} is to force the angular coordinate (normalized by $2\pi$) 
of ``large solutions'' to the differential equation
in \eqref{problemaper} not to display an integer gap between the values taken for
 $t=0$ and $t=T$ (see also \cite{BosGar11}). The
solvability follows then from a standard application of the Poincar\'e-Bohl 
theorem, providing a fixed point of the Poincar\'e map
associated with \eqref{problemaper}.

Our aim is now to introduce a counterpart of condition \eqref{LLi} at zero - namely, 
for $\lambda \to 0$, when considering perturbations
 $R(t, u)$ which are sublinear at zero. Precisely, we will always assume that,
 uniformly for almost every $t \in [0, T]$,
\begin{equation}\label{sottol}
\lim_{| u | \to 0} \frac{R(t, u)}{| u |} = 0.
\end{equation}
This immediately implies, in view of the continuity in the $u$-variable, that
$R(t, 0) = 0$ a.e., so that problem \eqref{problemaper} has the trivial solution 
$u(t) \equiv 0$.
As a consequence, in order to formulate a Landesman-Lazer condition at zero, 
the mere replacement of ``$+\infty$'' with ``$0$'' in
condition \eqref{LLi} would be useless, since the left-hand side in \eqref{LLi} 
would vanish. For this reason and inspired also by
\cite[Section 8]{FonGar11}, it seems quite natural to divide the quantity 
appearing in \eqref{LLi} by some power of $\lambda$, in order to
avoid that the integral vanishes.
Precisely, we give the following definition.

\begin{definition} \rm
Let $R(t, u)$ fulfill \eqref{sottol}. We will say that $R(t, u)$ 
\emph{satisfies (LL+$_0$) with respect to $V \in \mathcal{P}$}
if there exists $\alpha > 1$ such that the two following conditions are satisfied:
\begin{itemize}
\item
there exist a neighborhood $\mathcal{U}$ of the origin and $\eta^+_0 \in L^1(0, T)$ 
such that
\begin{equation}\label{controllofatou}
\frac{\langle R(t, u) | u \rangle}{| u |^{\alpha+1}} \geq \eta^+_0(t),
\end{equation}
for almost every $t \in [0, T]$ and every $u \in \mathcal{U}$;
\item
for every $\theta \in [0, T]$,
\begin{equation}\label{LL0}
\int_0^T \liminf_{(\lambda, \omega) \to (0^+, \theta)} 
\frac{\langle R(t, \lambda \varphi_V(t+\omega)) | \varphi_V(t+\omega) \rangle}
{\lambda^\alpha} \, dt > 0.
\end{equation}
\end{itemize}
\end{definition}

Conditions \eqref{controllofatou} and \eqref{LL0} have to be satisfied for 
the same value of $\alpha$; the fact that $\alpha > 1$ comes
from the sublinearity assumption \eqref{sottol}. Moreover,  we observe that \eqref{LL0} is weaker than a sign assumption like $\langle
R(t,u) | u \rangle \geq h(t)$ with $\int_0^T h(t) \, dt > 0$,
as is immediately seen.
We prove the following rotational interpretation of (LL+$_0$).

\begin{proposition}\label{rotaz}
Let $V \in \mathcal{P}$ satisfy \eqref{risonanza} and let $R(t, u)$ be an 
$L^1$-Carath\'eodory function fulfilling \eqref{sottol}. Then,
if $R(t, u)$ satisfies (LL+$_0$), there exists $0 < R_0 \ll 1$ such that 
every solution $u(t)$ to
$$
Ju'=\nabla V(u) + R(t, u),
$$
with $| u(0) | = R_0$, fulfills
$$
\operatorname{Rot} (u(t); [0, T]) > \frac{T}{\tau_V} = N.
$$
\end{proposition}

In the statement, we have denoted by $\operatorname{Rot} (u(t); [0, T])$ 
the standard (clockwise) rotation number of the path
$t \mapsto u(t)$ around the origin, in the   interval $[0,T]$, namely
$$
\operatorname{Rot} (u(t); [0, T]) = \frac{1}{2\pi} \int_{0}^{T}  
\frac{\langle Ju'(t) | u(t) \rangle}{| u(t) |^2} \, dt .
$$
In the proof, we will also make use of the notion of modified rotation 
number (see \cite{BosGar11, Zan96}) associated with
$V \in \mathcal{P}$, given by
$$
\operatorname{Rot}_V \, (u(t); [0, T]) 
= \frac{1}{\tau_{V}} \int_{0}^{T}  \frac{\langle Ju'(t) | 
u(t) \rangle}{2V(u(t))} \, dt.
$$

\begin{proof}[Proof of Proposition \ref{rotaz}]
By contradiction, assume that the thesis is not true: then there exists 
a sequence of functions $u_n(t)$, with
$| u_n(0) | \to 0$, such that
$$
Ju_n'= \nabla V(u_n) + R(t, u_n),
$$
and
\begin{equation}\label{assurdo}
\operatorname{Rot} (u_n(t); [0, T]) \leq N.
\end{equation}
First, we observe that, by Gronwall's lemma (see \cite[Lemma 2.2]{FonGar11}), 
one has that $\| u_n \|_{L^\infty(0, T)} \to 0$.
Moreover, setting
$$
v_n(t)=\frac{u_n(t)}{\| u_n \|_{L^\infty(0, T)}},
$$
we have that $v_n(t)$ satisfies, for every $n$, the equation
\begin{equation}\label{normalizzata}
Jv_n'= \nabla V(v_n) + \frac{R(t, u_n)}{\| u_n \|_{L^\infty(0, T)}},
\end{equation}
so that, thanks to \eqref{sottol}, there exists $\zeta \in L^1(0, T)$ such 
that $| v_n'(t) | \leq \zeta(t)$.
Hence, using Ascoli's theorem we have that there exists a nonzero $v \in C([0, T])$ 
such that $v_n \to v$ uniformly; more precisely,
thanks to the Dunford-Pettis theorem, we can pass to the weak $L^1$-limit in 
\eqref{normalizzata} and infer that $v(t)$ satisfies
$Jv'=\nabla V(v)$, implying that $v(t)=R \varphi_V(t+\theta)$ for suitable 
constants $R > 0$ and $\theta \in [0, \tau_V[$. 
Now we estimate the modified rotation number for $u_n(t)$: recalling that 
it coincides with the usual one when taking integer values
(see, for instance, \cite[Proposition 2.1]{BosGar11}), and thanks to 
\eqref{assurdo}, we have
\begin{align*}
N & \geq \operatorname{Rot}_V (u_n(t); [0, T])\\
& = \frac{1}{\tau_V} \int_0^T  \frac{\langle \nabla V(u_n(t)) 
 + R(t, u_n(t)) | u_n(t) \rangle}{2V(u_n(t))} \, dt \\
&=  N +  \int_0^T  \frac{\langle R(t, u_n(t)) | u_n(t) \rangle}{2V(u_n(t))} \, dt.
 \end{align*}
Hence,
\begin{equation}\label{lacontradd}
\int_0^T  \frac{\langle R(t, u_n(t)) | u_n(t) \rangle}{2V(u_n(t))} \, dt \leq 0.
\end{equation}
We now aim at showing that \eqref{lacontradd} leads to a contradiction. 
To this end, we change variables setting
$u_n(t)= r_n(t) \varphi_V(t+\omega_n(t))$, with $r_n(t) \geq 0$ and 
$\omega_n(0) \in [0, \tau_V[\,$, and observe that 
$\| u_n \|_{L^\infty(0, T)}\to 0$ implies that $r_n \to 0^+$ uniformly.
 On the other hand, since $v_n \to v$ uniformly, we have that
\begin{equation}\label{convergenzaR}
\frac{r_n}{\| u_n \|_{L^\infty(0, T)}} \to R
\end{equation}
uniformly, and it can also be seen that
$\omega_n \to \theta$ uniformly (cf. \cite{FonGar11}).
Now, with reference to \eqref{lacontradd} and using the $2$-homogeneity of $V$, 
we have
$$
\int_0^T  \frac{\langle R(t, r_n(t)\varphi_V(t+\omega_n(t))) 
| \varphi_V(t+\omega_n(t)) \rangle}{r_n(t)} \, dt \leq 0
$$
for every $n$, and dividing by $\| u_n \|_{L^\infty(0, T)}^{\alpha-1}$ 
it follows that
$$
\int_0^T \frac{r_n(t)^{\alpha-1}}{\| u_n \|_{L^\infty(0, T)}^{\alpha-1}}
\frac{\langle R(t, r_n(t)\varphi_V(t+\omega_n(t))) | \varphi_V(t+\omega_n(t)) 
\rangle}{r_n(t)^\alpha} \, dt \leq 0.
$$
In view of \eqref{controllofatou}, we can thus use Fatou's lemma to infer that
$$
\int_0^T  \liminf_{n \to +\infty}
\frac{\langle R(t, r_n(t)\varphi_V(t+\omega_n(t))) 
| \varphi_V(t+\omega_n(t)) \rangle}{r_n(t)^\alpha} \, dt \leq 0,
$$
using standard properties of the inferior limit and recalling 
\eqref{convergenzaR}.
Consequently, observing that the inferior limit grows when computed on 
subsequences, we obtain
$$
\int_0^T  \liminf_{(\lambda, \omega) \to (0^+, \theta)}
\frac{\langle R(t, \lambda \varphi_V(t+\omega)) 
| \varphi_V(t+\omega) \rangle}{\lambda^\alpha} \, dt \leq 0,
$$
against the assumption.
\end{proof}

As already remarked, it is clear that, in view of this rotational interpretation, 
condition (LL+$_0$) ensures the existence of a
$T$-periodic solution via the Poincar\'e-Bohl theorem. However, 
without any other localization of the fixed point found, there could be the
risk of recovering the trivial solution.

As a counterpart of assumption (LL+$_0$) with reversed signs, we give 
the following definition.

\begin{definition} \rm
Let $R(t, u)$ fulfill \eqref{sottol}. We will say that $R(t, u)$ 
\emph{satisfies (LL-$_0$) with respect to $V \in \mathcal{P}$}
if there exists $\alpha > 1$ such that the two following conditions are satisfied:
\begin{itemize}
\item
there exist a neighborhood $\mathcal{U}$ of the origin and 
$\eta^-_0 \in L^1(0, T)$ such that
$$
\frac{\langle R(t, u) | u \rangle}{| u |^{\alpha+1}} \leq \eta^-_0(t),
$$
for almost every $t \in [0, T]$ and every $u \in \mathcal{U}$;
\item
for every $\theta \in [0, T]$,
$$
\int_0^T \limsup_{(\lambda, \omega) \to (0^+, \theta)}
\frac{\langle R(t, \lambda \varphi_V(t+\omega)) 
| \varphi_V(t+\omega) \rangle}{\lambda^\alpha} \, dt < 0.
$$
\end{itemize}
\end{definition}

The following statement can be proved analogously to Proposition \ref{rotaz}.

\begin{proposition}\label{rotaz2}
Let $V \in \mathcal{P}$ satisfy \eqref{risonanza} and let $R(t, u)$ 
be an $L^1$-Carath\'eodory function fulfilling \eqref{sottol}. 
Then, if $R(t, u)$ satisfies (LL-$_0$), there exists $0 < R_0 \ll 1$ such 
that every solution $u(t)$ to
$$
Ju'=\nabla V(u) + R(t, u),
$$
with $| u(0) | = R_0$, is such that
$$
\operatorname{Rot} (u(t); [0, T]) < \frac{T}{\tau_V} = N.
$$
\end{proposition}

With these preliminaries, we can give a slightly more symmetric version of the
 multiplicity results in \cite{BosGar11}
for unforced planar Hamiltonian systems with positively homogeneous principal 
part, by possibly considering a resonant situation at zero.
Once taken into account Propositions \ref{rotaz} and \ref{rotaz2}, the proof 
is the same as in \cite{BosGar11}, so that we will only briefly
sketch the reasoning leading to the conclusion. For the sake of clarity, 
we recall the version of the Landesman-Lazer conditions at
infinity which will be used, together with
the needed controls in order for the integrals to make sense:
\smallskip

\noindent (LL+$_\infty$) for every $\theta \in [0, T]$,
$$
\int_0^T \liminf_{(\lambda, \omega) \to (+\infty, \theta)} 
\langle R(t, \lambda \varphi_V(t+\omega)) | \varphi_V(t+\omega) \rangle \,
dt > 0,
$$
assuming that there exists $\eta^+_\infty \in L^1(0, T)$ such that for almost
 every $t \in [0, T]$, for every $u \in \mathbb{R}^2$ with
$| u | \leq 1$ and every $\lambda  > 1$,
$$
\langle R(t, \lambda u)  | u \rangle \geq \eta^+_\infty(t);
$$
\smallskip

\noindent (LL-$_\infty$)
for every $\theta \in [0, T]$,
$$
\int_0^T \limsup_{(\lambda, \omega) \to (+\infty, \theta)}
\langle R(t, \lambda \varphi_V(t+\omega)) | \varphi_V(t+\omega) \rangle \, dt < 0,
$$
assuming that there exists $\eta^- _\infty\in L^1(0, T)$ such that for almost
 every $t \in [0, T]$,
for every $u \in \mathbb{R}^2$ with $| u | \leq 1$ and every $\lambda  > 1$,
$$
\langle R(t, \lambda u)  | u \rangle \leq \eta^-_\infty(t).
$$

We first give the statement for a common expansion at zero and infinity.

\begin{proposition}\label{molteplicitacomm}
Let $H: [0, T] \times \mathbb{R}^2 \to \mathbb{R}$ be differentiable in the 
second variable, with $\nabla H(t, u)$ an $L^1$-Carath\'eodory function, and 
assume that the uniqueness for the solutions to the Cauchy problems associated
with
\begin{equation}\label{ilproblema}
Ju'=\nabla H(t, u)
\end{equation}
is guaranteed. Moreover, let $\nabla H(t, 0) \equiv 0$ and assume that there 
exist a positive integer $k$ and
$V \in \mathcal{P}$ in such a way that
$$
\frac{T}{\tau_{V}} = k,
$$
and, setting
$$
R(t, u) = \nabla H(t, u) - \nabla V(u),
$$
it holds
$$
\lim_{| u | \to 0}\frac{R(t, u)}{| u |} = 0, \quad 
\lim_{| u | \to +\infty}\frac{R(t, u)}{| u |} = 0.
$$
Finally, suppose that
$$
R(t, u) \text{ satisfies  (LL+}_0\text{) and (LL-}_\infty) \text{ with respect to } V.
$$
Then, there exist at least two nontrivial $T$-periodic solutions 
$u_{1}(t), u_{2}(t)$ to (\ref{ilproblema}) such that
$$
\operatorname{Rot} \, (u_{1}(t); [0, T]) = \operatorname{Rot} \, (u_{2}(t); [0, T]) = k.
$$
\end{proposition}

\begin{proof}
It suffices to observe that, due to the Landesman-Lazer assumptions 
(cf. Proposition \ref{rotaz} and \cite[Proposition 4.1]{BosGar11}),
there exist $0 < r < R$ such that
$$
| u(0) | = r \quad \Rightarrow \quad \operatorname{Rot}  (u(t); [0, T]) >  k
$$
and
$$
| u(0) | = R \quad \Rightarrow \quad \operatorname{Rot}  (u(t); [0, T]) <  k.
$$
The conclusion follows then from the Poincar\'e-Birkhoff fixed point theorem.
\end{proof}

Of course, in presence of different asymptotic expansions at zero and infinity, 
a larger gap between the minimal periods associated with
the principal terms corresponds to a higher number of solutions. We briefly 
summarize this into the following proposition.

\begin{proposition}\label{molteplicitagap}
Let $H: [0, T] \times \mathbb{R}^2 \to \mathbb{R}$ be differentiable in the 
second variable, with $\nabla H(t, u)$ an $L^1$-Carath\'eodory
function, and assume that the uniqueness for the solutions to the Cauchy problems 
associated with \eqref{ilproblema} is guaranteed. Moreover, let
 $\nabla H(t, 0) \equiv 0$ and assume that there exist $V_0$, 
$V_\infty \in \mathcal{P}$, with
\begin{gather}\label{sviluppo0}
\nabla H(t,u) = \nabla V_0(u) + o(| u |), \quad | u | \to 0, \\
\label{sviluppoI}
\nabla H(t,u) = \nabla V_\infty(u) + o(| u |), \quad | u | \to \infty,
\end{gather}
such that it holds
$$
\frac{T}{\tau_{V_0}} = k_0, \quad  \frac{T}{\tau_{V_\infty}} = k_\infty,
$$
for some integers $k_0 \geq k_\infty$.
Then, setting
\begin{equation}\label{resto0}
R_0(t, u) = \nabla H(t, u) - \nabla V_0(u) \quad \text{and} \quad
 R_\infty(t, u) = \nabla H(t, u) - \nabla V_\infty(u),
\end{equation}
if
$$
R_0(t, u) \text{ satisfies  (LL+}_0\text{) with respect to } V_0,
$$
and
$$
R_\infty(t, u) \text{ satisfies  (LL-}_\infty\text{) with respect to } V_\infty,
$$
for every integer $k \in [k_\infty,k_0]$ there exist two $T$-periodic 
solutions $u_{1, k}(t), u_{2, k}(t)$ to (\ref{ilproblema}) such that
$$
\operatorname{Rot} (u_{1, k}(t); [0, T])
 = \operatorname{Rot} (u_{2, k}(t); [0, T]) = k.
$$
\end{proposition}

Obviously, one can obtain the same result if $R_0(t, u)$ satisfies
 (LL-$_0$) and $R_\infty(t, u)$ satisfies (LL+$_\infty$),
if $k_0 \leq k_\infty$.

\begin{remark} \rm
The choice of $T$-periodic boundary conditions is not essential in our results.
In the paper \cite{BosGarPP}, resonant problems with general homogeneous boundary 
conditions have been considered, and corresponding
Landesman-Lazer type assumptions at infinity, carrying a suitable rotational 
characterization, were introduced. For example, in the case of a
Sturm-Liouville boundary value problem like
\begin{gather*}
Ju'= \nabla V(u) + R(t, u)  \\
u(0) \in l_S, \quad u(T) \in l_A,
\end{gather*}
where $l_S$, $l_A$ are two lines through the origin, it was observed that 
only two different values of $\theta$ have to be
 considered when formulating the Landesman-Lazer conditions, namely the values 
$\theta_1, \theta_2 \in [0, \tau_V[\,$ such that
$\varphi_V(\theta_1), \varphi_V(\theta_2) \in l_S$.
In a similar way, one can give analogous Landesman-Lazer conditions at zero, 
by considering (LL+$_0$) or (LL-$_0$) only for
$\theta=\theta_1, \theta=\theta_2$ (we remind the reader to 
\cite[Remark 2.2]{BosGarPP}); we omit the details for briefness. In this way,
one can easily derive multiplicity of solutions satisfying prescribed homogeneous 
boundary conditions when both the principal terms at zero
and infinity are resonant, via a shooting argument. Right because of this 
technique of proof, we notice that in this case the nonlinearity
is not required to be of gradient type (as instead is needed for the 
Poincar\'e-Birkhoff theorem).
\end{remark}

We conclude this section commenting briefly  on condition \eqref{LL0}, 
with particular attention to the second order case.
As a first remark, we observe that \eqref{LL0} can be satisfied, 
for instance, by functions with indefinite weight like
$$
R(t, u) = a(t) | u |^\beta u, \quad u \approx 0,
$$
where $a(t)$ changes sign at least once in $[0, T]$ and $\beta > 0$; 
once the form of $\varphi_V(t)$ is known, it is not difficult to assume
 conditions on $a(t)$ to the purpose (to make a trivial example, 
$\int_0^T a(t) \, dt > 0$ is sufficient if $\varphi_V(t)$ describes a
circumference in the plane).

Dealing, in particular, with the second-order asymmetric equation
$$
x'' + \mu x^+ - \nu x^- + r(t, x)=0, \quad  \mu, \nu > 0,
$$
where $r: [0, T] \times \mathbb{R} \to \mathbb{R}$ satisfies the
 sublinearity assumption
$\lim_{| x | \to 0} r(t, x)/x = 0$,
condition \eqref{LL0} reads as
\begin{equation}\label{prima}
\int_0^T \liminf_{(\lambda, \omega) \to (0^+, \theta)} 
\frac{r(t, \lambda \phi(t+\omega)) \phi(t+\omega)}{\lambda^\alpha} \, dt > 0,
\end{equation}
where
$$
\phi(t)= \begin{cases}
 \frac{1}{\sqrt{\mu}} \sin (\sqrt{\mu} t) & \text{if }    
t \in [0, \frac{\pi}{\sqrt{\mu}}]  \\[4pt]
 \frac{1}{\sqrt{\nu}} \sin \big(\sqrt{\nu}\Big(\frac{\pi}{\sqrt{\mu}}- t\Big)\big) 
& \text{if }   t \in [\frac{\pi}{\sqrt{\mu}}, \frac{T}{N}],
\end{cases}
$$
and
\begin{equation}\label{iltau}
\varphi_V(t)=(\phi(t), \phi'(t)),  \quad 
\tau_V = \frac{\pi}{\sqrt{\mu}} + \frac{\pi}{\sqrt{\nu}} = \frac{T}{N}.
\end{equation}
Exploiting the properties of inferior limits, we can easily obtain 
that \eqref{prima} is equivalent to
\begin{align*}
&\int_{\{\phi(\cdot+\theta) > 0\}}
\Big(\liminf_{x \to 0^+} \frac{r(t, x)}{| x |^\alpha}\Big) 
 |\phi(t+\theta)|^{\alpha+1} \, dt \\
&- \int_{\{\phi(\cdot+\theta) < 0\}} \Big(\limsup_{x \to 0^-} 
\frac{r(t, x)}{| x |^\alpha}\Big) |\phi(t+\theta)|^{\alpha+1} \, dt > 0,
\end{align*}
recovering the same condition used in \cite{Han07}.
Incidentally, we notice that, with reference to the form of the usual conditions 
at infinity, ``$0^+$'' and ``$0^-$'' replace ``$+\infty$''
and ``$-\infty$'', respectively.

As an example, if $r(t, x) \approx x^{2j+1}$ near $0$, with $j$ a positive integer, 
the condition is always satisfied for $\alpha=2j+1$,
since both the summands in the above inequality are positive. 
This is indeed intuitive, because the contribution given by $r(t, x)$ to the
angular speed of the solutions in the phase plane is here sign-defined 
(in clockwise sense).
More in general, a similar situation occurs if
$r_x(t, 0) > 0$, which guarantees the validity of a similar sign condition.
Of course, on the other hand, condition \eqref{LL0} cannot be satisfied 
by a nonlinearity which goes to zero faster than any power of $x$,
like, for instance, $r(t, x) \approx \exp(-1/| x |)$.

If a weight is present in front of the power, the situation can be of course 
less easy. For instance, for the equation
$$
x'' + \mu x^+ - \nu x^- + a(t) x^{2j+1} = 0,
$$
where $j$ is a positive integer and \eqref{iltau} is satisfied,
setting $\alpha=2j+1$ condition \eqref{LL0} reads as
\begin{align*}
&\int_{\{\phi(\cdot+\theta) > 0\}} a(t) | \phi(t+\theta)|^{2j+2} \, dt 
+\int_{\{\phi(\cdot+\theta) < 0\}} a(t) | \phi(t+\theta)|^{2j+2} \, dt\\
& = \int_0^T a(t) | \phi(t+\theta) |^{2j+2} \, dt > 0,
\end{align*}
for every $\theta > 0$.
We notice that such a condition can be fulfilled by a sign-indefinite weight 
$a(t)$ (up to suitably balancing the positive and the negative
contribution in the integral) and, in the context of indefinite nonlinearities, 
it was used, e.g., in \cite{Teh07}, to guarantee the
existence of an entire solution to a Schr\"odinger equation at resonance.

\section{A change of variables: analogies between zero and infinity}\label{seztre}

We are now going to introduce a change of variables highlighting the relationships 
between the Landesman-Lazer conditions (LL+$_0$) and
(LL+$_\infty$) exploited in the previous section.
In order to make the situation more symmetric and the computations homogeneous,
we adopt slightly different notation and formulations than in 
Section \ref{sezdue}, not affecting, however, the validity of the previous
results.
We thus focus on the following two conditions:
\smallskip

\noindent (LL+$_0$)$_\alpha$
there exists $\alpha > 1$ such that, for every $\theta \in [0, T]$,
\begin{equation}\label{centrale0}
\int_0^T \liminf_{(\mu, \omega) \to (0^+, \theta)} \frac{\langle R(t, \mu \varphi_V(t+\omega)) | \varphi_V(t+\omega) \rangle}
{\mu^\alpha} \, dt > 0,
\end{equation}
provided that there exist a neighborhood $\mathcal{U}$ of the origin and $\eta^+_0 \in L^1(0, T)$ such that
\begin{equation}\label{controllofatou0}
\frac{\langle R(t, u) | u \rangle}{| u |^{\alpha+1}} \geq \eta^+_0(t),
\end{equation}
for almost every $t \in [0, T]$ and every $u \in \mathcal{U}$;
\smallskip

\noindent (LL+$_\infty$)$_k$ there exists $k \geq 0$ such that, for every $\theta \in [0, T]$,
\begin{equation}\label{centrale1}
\int_0^T \liminf_{(\lambda, \omega) \to (+\infty, \theta)}
\lambda^k \langle R(t, \lambda \varphi_V(t+\omega)) | \varphi_V(t+\omega) 
\rangle \, dt > 0,
\end{equation}
provided that there exists $\eta^+_\infty\in L^1(0, T)$ such that for almost 
every $t \in [0, T]$,
for every $u \in \mathbb{R}^2$ with $| u | \leq 1$ and every $\lambda  > 1$,
 it holds
\begin{equation}\label{controllofatou1}
\lambda^k \langle R(t, \lambda u)  | u \rangle \geq \eta^+_\infty(t).
\end{equation}

We remark that, for $k=0$, condition (LL+$_\infty$)$_k$ turns into the 
usual Landesman-Lazer condition introduced in
\cite{FonGar11}; observe also that the rotational effect recalled in the 
previous section is not affected by the value of $k$.

To analyze the link between these two conditions, let us first consider the system
\begin{equation}\label{partenza}
Ju'=\nabla V(u) + R(t, u),
\end{equation}
where we assume that $R(t, 0) \equiv 0$, as previously. By the uniqueness 
of the solutions to the associated Cauchy problems, no nontrivial
solutions to \eqref{partenza} can reach the origin in a finite time.
Thus, for $\beta > 1$  (and $u \in \mathbb{R}^2 \setminus \{0\}$), 
we can perform the change of variables
\begin{equation}\label{cambiovar}
w=\frac{u}{V(u)^{\beta/2}}
\end{equation}
(observe that the denominator only vanishes in $u=0$). 
Notice that \eqref{cambiovar} leaves the set $\{u \mid V(u) = 1\}$
invariant and does not rescale $1$-homogeneously
when applied to $\lambda u$.

Denoting by $\odot$ one between $0^+$ and $+\infty$, and by $\odot^{-1}$ 
its reciprocal in extended sense,
i.e. $1/0^+=+\infty$ and $1/+\infty=0^+$, in view of the fact that $V(u)$ 
is positively $2$-homogeneous and $\beta > 1$ we
first see that
\begin{equation}\label{relazioneodot}
| w | \to \odot \Longleftrightarrow | u | \to \odot^{-1}.
\end{equation}
Moreover, we immediately derive the relations
$$
V(w)=\frac{1}{V(u)^{\beta -1}}, \quad
V(u)=\frac{1}{V(w)^{\frac{1}{\beta-1}}}, \quad
u= \frac{w}{V(w)^{\frac{\beta}{2(\beta-1)}}}\, .
$$
Recalling now that, if $u(t)$ is a nontrivial solution to \eqref{partenza}, then
$w(t)$ given by \eqref{cambiovar} is everywhere well defined and differentiable,
and using that
$$
\nabla (V(u)^{\beta/2}) = \frac{\beta}{2} V(u)^{\frac{\beta-2}{2}} \nabla V(u),
$$
we can compute the differential system corresponding to \eqref{partenza} 
in the new variable $w$, obtaining
$$
Jw' = \nabla V(w) + \tilde{R}(t, w) - \frac{\beta}{2}
V(w)^{-1} \langle J\nabla V(w) | \tilde{R}(t, w)\rangle Jw,
$$
where we have set
$$
\tilde {R}(t, w)=R\Big(t, \frac{w}{V(w)^{\frac{\beta}{2(\beta-1)}}}\Big)
V(w)^{\frac{\beta}{2(\beta-1)}}=\frac{1}{V(u)^{\beta/2}} R(t, u).
$$
We thus notice that the new system has the form
\begin{equation}\label{arrivo}
Jw'= \nabla V(w) + \tilde{R}(t, w) + S(t ,w) Jw,
\end{equation}
where the last term, being parallel to $Jw$, only influences the radial 
component of the solutions.

We now aim at showing that, for suitable choices of $k$ and $\alpha$,
there is a complete equivalence between condition (LL+$_0$)$_\alpha$ 
for system \eqref{partenza} (resp. \eqref{arrivo}) and
condition (LL+$_\infty$)$_k$ for system \eqref{arrivo} (resp. \eqref{partenza}),
based on the fact that such conditions only act on the angular component of 
the solutions.

Keeping the same notation as before for $\odot$, we preliminarily observe that
$$
\frac{\tilde{R}(t, w)}{| w |}
= \frac{R(t, u)}{| w | V(u)^{\beta/2}} = \frac{R(t, u)}{| u |},
$$
so that, in view of \eqref{relazioneodot},
$$
\lim_{| u | \to \odot} \frac{R(t, u)}{| u |} = 0 \quad \Longleftrightarrow \quad
\lim_{| w | \to \odot^{-1}} \frac{\tilde{R}(t, w)}{| w |} = 0.
$$
We now check the equivalence between \eqref{centrale0} and \eqref{centrale1}.

Assume first that (LL+$_\infty$)$_k$ is satisfied for \eqref{arrivo}, namely 
there exists $k \geq 0$ such that, for every $\theta \in [0, T]$,
$\tilde{R}(t, w)$ satisfies
\begin{equation}\label{arrinf}
\int_0^T \liminf_{(\lambda, \omega) \to (+\infty, \theta)} 
\lambda^k \langle \tilde{R}(t, \lambda \varphi_V(t+\omega)) |
\varphi_V(t+\omega) \rangle\, dt > 0
\end{equation}
(since the term coming from $S(t, w)Jw$ clearly vanishes).
Writing the explicit expression of $\tilde{R}(t, w)$, using the positive 
$2$-homogeneity of $V(u)$ and the fact that $V(\varphi_V(t))
\equiv 1/2$, we have that \eqref{arrinf} is
equivalent to
$$
\int_0^T \liminf_{(\lambda, \omega) \to (+\infty, \theta)}
\lambda^k \Big\langle R\Big(t, \frac{\lambda 
\varphi_V(t+\omega)}{\lambda^{\frac{2\beta}{2(\beta-1)}}
(1/2)^{\frac{\beta}{2(\beta-1)}}}\Big) \lambda^{\frac{\beta}{\beta-1}} 
(1/2)^{\frac{\beta}{2(\beta-1)}} \Big| \varphi_V(t+\omega)
\Big\rangle \, dt > 0,
$$
so that
$$
\int_0^T \liminf_{(\lambda, \omega) \to (+\infty, \theta)} 
\lambda^{\frac{\beta}{\beta-1}+k}
\Big\langle R\Big(t, \big(\frac{2^{\beta/2}}{\lambda}\big)^{\frac{1}{\beta -1}} 
\varphi_V(t+\omega)\Big) \Big|
\varphi_V(t+\omega) \Big\rangle \, dt > 0.
$$
Hence, setting
$$
\mu=\Big(\frac{2^{\beta/2}}{\lambda}\Big)^{\frac{1}{\beta-1}},
$$
we obtain
$$
\int_0^T \liminf_{(\mu, \omega) \to (0^+, \theta)}
\frac{\langle R\left(t, \mu \varphi_V(t+\omega)\right)| 
\varphi_V(t+\omega)\rangle}{\mu^{\beta+k\beta-k}} \, dt > 0,
$$
after having observed that we can omit multiplicative positive constants 
when considering the sign of this quantity.
Then, given $\alpha > 1$, condition (LL+$_\infty$)$_k$ for system 
\eqref{arrivo} and condition (LL+$_0$)$_\alpha$ for system
\eqref{partenza}
are equivalent through the choice
\begin{equation}\label{sceltabeta1}
\beta = \frac{\alpha + k}{1+k} > 1.
\end{equation}
Conversely, let us assume condition (LL+$_0$)$_\alpha$ for system 
\eqref{arrivo}, namely there exists $\alpha > 1$ such that, for every
$\theta \in [0, T]$, it holds
$$
\int_0^T \liminf_{(\mu, \omega) \to (0^+, \theta)} 
\frac{\langle \tilde{R}(t, \mu \varphi_V(t+\omega)) 
| \varphi_V(t+\omega) \rangle} {\mu^\alpha} \, dt > 0.
$$
Performing the same computations as before, we have that this implies
$$
\int_0^T \liminf_{(\mu, \omega) \to (0^+, \theta)} 
\mu^{\frac{\beta}{\beta-1}-\alpha}
\Big\langle R\Big(t, \big(\frac{2^{\beta/2}}{\mu}\big)^{\frac{1}{\beta-1}} 
\varphi_V(t+\omega)\Big) \Big|
\varphi_V(t+\omega) \Big\rangle \, dt > 0,
$$
and with the position
$$
\lambda=\Big(\frac{2^{\beta/2}}{\mu}\Big)^{\frac{1}{\beta-1}}
$$
we obtain, omitting again superflous multiplicative constants,
$$
\int_0^T \liminf_{(\lambda, \omega) \to (+\infty, \theta)}
\frac{\langle R\left(t, \lambda \varphi_V(t+\omega)\right)| 
\varphi_V(t+\omega)\rangle}{\lambda^{\beta+\alpha-\alpha \beta}} \, dt > 0.
$$
Then, given $k > 0$, condition (LL+$_0$)$_\alpha$ for system \eqref{arrivo}
 and condition (LL+$_\infty$)$_k$ for system \eqref{partenza}
are equivalent through the choice
\begin{equation}\label{sceltabeta2}
\beta= \frac{\alpha+k}{\alpha -1} > 1.
\end{equation}
We finally have to check that the controls from below make possible 
to apply Fatou's lemma after the change of variables. However, this is
quite immediate: assume for instance (LL+$_\infty$)$_k$ for system 
\eqref{partenza}, with the aim of proving Proposition \ref{rotaz} for
system \eqref{arrivo}. After computations similar to the
ones appearing in the proof of Proposition \ref{rotaz}, we obtain the inequality
$$
\int_0^T \frac{r_n(t)^{\alpha-1}}{\| u_n \|_{L^\infty(0, T)}^{\alpha-1}}
\frac{\langle \tilde{R}(t, r_n(t)\varphi_V(t+\omega_n(t))) 
| \varphi_V(t+\omega_n(t)) \rangle}{r_n(t)^\alpha} \, dt \leq 0,
$$
where $r_n \to 0^+$ uniformly and $\omega_n \to \theta$ uniformly.
In view of the previous computations,
this gives
$$
\int_0^T
\frac{r_n(t)^{\alpha-1}}{\| u_n \|_{L^\infty(0, T)}^{\alpha-1}} \rho_n(t)^{\alpha \beta - \alpha - \beta} \langle R\left(t, \rho_n(t)
\varphi_V(t+\omega)\right)|
\varphi_V(t+\omega)\rangle \, dt \leq 0,
$$
where
$$
\rho_n(t) \approx \frac{1}{r_n(t)^{\frac{1}{\beta-1}}}
$$
(up to multiplicative constants), so that
$\rho_n \to +\infty$ uniformly. Using \eqref{sceltabeta2}, this is equivalent to
$$
\int_0^T \frac{r_n(t)^{\alpha-1}}{\| u_n \|_{L^\infty(0, T)}^{\alpha-1}} 
\rho_n(t)^{k} \langle R\left(t, \rho_n(t)
\varphi_V(t+\omega)\right)|
\varphi_V(t+\omega)\rangle \, dt > 0.
$$
The possibility of applying Fatou's lemma now follows in a standard way, 
since the ratio under the integral sign converges to $R$ in
view of \eqref{convergenzaR}, and the reminder is bounded from below in 
view of \eqref{controllofatou1}.
This also shows that the use of \eqref{cambiovar} may be an alternative 
strategy to show the validity of the rotational interpretation of
(LL+$_0$)$_\alpha$, but to make a better comparison with the existing 
literature we have preferred to show explicitly the computations in
the proof of Proposition \ref{rotaz}.

\begin{remark} \rm
It is clear that all the discussion could be carried out, in the same way,
 for conditions (LL-$_0$)$_\alpha$ and
(LL-$_\infty$)$_k$. We omit any other detail for briefness.
\end{remark}

\begin{remark} \rm
The change of variables \eqref{cambiovar} suggests that analogous versions 
of recent results of existence and
multiplicity of harmonic and subharmonic solutions formulated at infinity, 
possibly with different growth than asymptotically positively
homogeneous (like, for instance, \cite{Bos12}), could be easily stated at 
zero only at the expense of suitably translating the assumptions
on the nonlinearity.
\end{remark}


\section{Final remarks}\label{sezquattro}

In this last section we collect some final observations, possibly related 
to little improvements of known results through the use of
condition  (LL+$_0$).
\smallskip

\noindent 1. \emph{Existence of subharmonics.}
 In the spirit of \cite{BosGar11}, we can deal with the existence of
subharmonic solutions for (\ref{ilproblema}), assuming that $H(t, u)$ is 
defined on $\mathbb{R} \times \mathbb{R}^2$ and $T$-periodic in the
first variable. As usual in this setting, we say that an $mT$-periodic 
function $u(t)$ solving \eqref{ilproblema} is a \emph{subharmonic of
order $m$} - where $m$ is an integer greater than or equal to $2$ - if 
$mT$ is its least period in the class of the integer multiples of
$T$, i.e., $u(t)$ is not $lT$-periodic for any integer $l < m$.
Notice that, if $\operatorname{Rot} \, (u(t); [0, mT]) = k$,
then $u(t)$ is a subharmonic of order $m$ whenever $m$ and $k$ are relatively 
prime integers, namely their greatest common divisor is~$1$.

Proceeding very similarly as in \cite{BosGar11}, to which we refer for 
further details, we can prove, for instance, the validity of the
following.

\begin{proposition}\label{subarmon}
Let $H: [0, T] \times \mathbb{R}^2 \to \mathbb{R}$ be differentiable in the 
second variable, with $\nabla H(t, u)$ an
$L^1$-Carath\'eodory function, and assume that the uniqueness for the 
solutions of the Cauchy problems associated with (\ref{ilproblema}) is
guaranteed. Moreover, assume \eqref{sviluppo0} and \eqref{sviluppoI}, 
and suppose that
$$
\frac{T}{\tau_{V_\infty}} = \frac{T}{\tau_{V_0}}.
$$
Using the same notation as in \eqref{resto0} for $R_0(t, u)$ and 
$R_\infty(t, u)$, assume finally that
\begin{gather*}
R_0(t, u) \text{ satisfies  (LL-}_0\text{) with respect to } V_0,\\
R_\infty(t, u) \text{ satisfies  (LL+}_\infty\text{) with respect to } V_\infty.
\end{gather*}
Then, for every positive integer $r$, there exists an integer $m^*(r)$ such that, 
for every $m \geq m^*(r)$, there exist $2r$
subharmonics of order $m$ solving (\ref{ilproblema}).
\end{proposition}

Of course, a symmetric statement can be obtained exchanging the considered
 Landesman-Lazer conditions at zero and infinity.

\begin{proof}
The proof can be obtained similarly as in \cite[Theorem 5.5]{BosGar11}, 
observing that, by virtue of the hypotheses, there is a gap in
the rotation numbers at zero and infinity:
\begin{gather*}
\operatorname{Rot}_0 (u(t); [0, T]) < \frac{T}{\tau_{V_0}} 
\quad \text{ (at zero)}, \\
\operatorname{Rot}_\infty (u(t); [0, T]) > \frac{T}{\tau_{V_\infty}} 
\quad \text{ (at infinity)}.
\end{gather*}
One can now find subharmonics of order $m$ making $k$ turns around the origin 
whenever there exist two integers $k$ and $m$,
relatively prime, such that
$$
\operatorname{Rot}_0 \, (u(t); [0, T]) < \frac{k}{m} 
< \operatorname{Rot}_\infty\, (u(t); [0, T]).
$$
We refer to \cite[Theorem 5.5]{BosGar11} for further details.
\end{proof}

\noindent 2. \emph{Comparison with different unperturbed systems.} 
Condition (LL+$_0$) can also be formulated in the framework of more general 
positively homogeneous comparison systems at zero and
infinity, like
$$
Ju'=\gamma_0(t) \nabla V_0(u), \quad Ju'=\gamma_\infty(t) \nabla V_\infty(u),
$$
where $\gamma_0, \gamma_\infty$ are positive functions with 
$\frac{1}{T} \int_0^T \gamma_0(s) \, ds = 1$,
$\frac{1}{T} \int_0^T \gamma_\infty(s) \, ds = 1$.
For the first system, in particular, \eqref{LL0} takes then the form
$$
\int_0^T \liminf_{(\lambda, \omega) \to (0^+, \theta)}
\frac{\langle R(t, \lambda \varphi_V(\Gamma_0(t)+\omega)) 
| \varphi_V(\Gamma_0(t)+\omega) \rangle}{\lambda^\alpha} \, dt > 0,
$$
where $\Gamma_0(t)=\int_0^t \gamma_0(s) \, ds$. The influence of such 
an assumption on the rotation number of small solutions can be
 shown similarly as in the case when $\gamma_0 \equiv 1$.

On the other hand, one could also consider nonlinearities whose principal 
term belongs to the class $\mathcal{P}^*$,
namely it vanishes on some lines (see \cite{Gar12}). In this case, 
fixed $V \in \mathcal{P}^*$, the usual Landesman-Lazer condition at
infinity is given by:
\begin{itemize}
\item for every $\xi  \in \mathbb{S}^1$ satisfying $V(\xi)=0$,
$$
\int_0^T \liminf_{(\lambda, \eta) \to (+\infty, \xi)} 
\langle R(t, \lambda \eta) | \eta \rangle \, dt > 0
$$
(actually, it could also be formulated in a slightly weaker form, with 
$F(t, u)=\nabla V(u) + R(t, u)$ replacing $R(t, u)$,
as seen in \cite{Gar12}; we omit to write the analogous of 
\eqref{controllofatou} and further details for the sake of briefness).
\end{itemize}
The counterpart of such an assumption at zero would be the following:
\begin{itemize}
\item there exists $\alpha > 1$ such that, for every $\xi  \in \mathbb{S}^1$ 
satisfying $V(\xi)=0$, it holds
$$
\int_0^T \liminf_{(\lambda, \eta) \to (0^+, \xi)} 
\frac{\langle R(t, \lambda \eta) | \eta \rangle}{\lambda^\alpha} \, dt > 0.
$$
\end{itemize}
Notice that, for the way it is used in the proof, in condition \eqref{LL0} 
it is not necessary that $\alpha$ is independent of
$\theta$ (in this case, the controls from below and above required to 
apply Fatou's lemma have to change according to the value of
$\theta$), even if in the applications this is usually fulfilled. 
Just to give an example, in the framework of the class $\mathcal{P}^*$,
let $T=2\pi$ and consider a Hamiltonian $V \in \mathcal{P}^*$ which 
vanishes in the points $(0, 1)$ and $(1, 0)$. If, for instance,
$$
R(t, x, y)= (x^2 y + | \sin t | \sqrt{x^4+y^4}, -x^3+y^3),
$$
one has
\begin{equation}\label{conticino}
\langle R(t, \lambda x, \lambda y) | (x, y) \rangle 
=  \lambda^2 | \sin t | x \sqrt{x^4+y^4} +  \lambda^3 y^4.
\end{equation}
Now, if $\xi=(1, 0)$, it is sufficient to choose $\alpha=2$ to get the 
sign assumption
(since the first summand in \eqref{conticino} has positive integral), while for
$\xi=(0, 1)$ a convenient choice is given by $\alpha=3$ 
(since the first summand in \eqref{conticino} vanishes).
\smallskip



\noindent 3. \emph{Comparison with Ahmad-Lazer-Paul type conditions.} 
In the spirit of \cite{BosGar13, FonGar11b}, we briefly compare \eqref{LL0} 
with Ahmad-Lazer-Paul type conditions at zero,
 in the case when $R(t, u)=\nabla Q(t, u)$  (so that the problem can be dealt 
with in a variational framework).
Our considerations will consist purely in a matter of computation, comparing 
the explicit expression of the two conditions,
without any other restriction imposed instead by the specific problem considered 
(referring, in particular, to the way the range of
exponents to be taken into account has to be chosen).

We thus consider an Ahmad-Lazer-Paul type condition like
\begin{equation}\label{ALPgen}
\liminf_{\substack{\| u \| \to 0 \\ u \in H_0^0}} \frac{\int_0^T Q(t, u) \, dt}
{\| u \|^{2\beta}} = M > 0,
\end{equation}
(see, for instance, \cite{Han07}), for $\beta > 0$, where 
$M \in \mathbb{R} \cup \{+\infty\}$ and $H^0_0$ is the nullity space
associated with the principal part of the nonlinearity at zero. 
This condition may be fulfilled, for instance, if $Q(t, u)$ satisfies a
suitable pointwise control at zero, as in \cite{Han07}.

We now proceed in comparing conditions \eqref{LL0} and \eqref{ALPgen}.
First, using the same argument in \cite[Lemma 4.2]{BosGar13}, 
it is possible to show that
\eqref{LL0} implies the existence of $\lambda_0 > 0$, 
$\theta_1, \ldots, \theta_j \in [0, T]$,
$\delta_1, \ldots, \delta_j$ $ > 0$ and $h_1, \ldots, h_j \in L^1(0, T)$ 
($j \in \mathbb{N}$), with $\int_0^T h_i(t) \, dt > 0$ for every
$i=1, \ldots, j$, such that
$[0, T] \subset \cup_{i=1}^j [\theta_i - \delta_i, \theta_i + \delta_i]$, 
and, for every $i =1,\ldots,j$ and almost every $t \in [0,T]$,
$$
\frac{\langle \nabla Q(t, \lambda \varphi_V(t+\omega)) 
| \varphi_V(t+\omega)  \rangle}
{\lambda^\alpha} \geq h_i(t), \quad \text{ if }\,
| \omega - \theta_i | \leq \delta_i, \, 0 < \lambda \leq \lambda_0.
$$
For almost every $t \in [0, T]$ and 
every $0< \lambda \leq \lambda_0$, $\theta \in [0, T]$,
we now have
\begin{align*}
\frac{Q(t,\lambda \varphi_V(t+\theta))}{\lambda^{2\beta}} 
& = \frac{1}{\lambda^{2\beta}} \int_0^1 \frac{d}{d\kappa}Q(t,\kappa\lambda \varphi_V(t+\theta))\, d\kappa \\
& = \frac{1}{\lambda^{2\beta}} \int_0^\lambda s^\alpha
\frac{\langle \nabla Q(t,s\varphi_V(t+\theta)) | \varphi_V(t+\theta) \rangle}{s^{\alpha}} \, ds \\
& \geq  \frac{\lambda^{\alpha+1-2\beta}}{\alpha+1} h_i(t),
\end{align*}
being the index $i$ such that
$\theta \in [\theta_i - \delta_i, \theta_i + \delta_i]$.
Integrating on $[0,T]$ we now obtain
$$
\int_0^T \frac{Q(t,\lambda \varphi_V(t+\theta))}{\lambda^{2\beta}} \, dt \geq
\frac{\lambda^{\alpha+1-2\beta}}{\alpha+1} \int_0^T h_i(t) \, dt,
$$
so that \eqref{ALPgen} is satisfied (recall that $\int_0^T h_i  > 0$) 
if \eqref{LL0} holds with $\alpha \leq 2\beta-1$.
As can be expected, the situation concerning the range of the exponents
 appears quite different with respect to what happens
 with the usual Landesman-Lazer condition at infinity 
(see \cite[Remark 4.3]{BosGar13} and \cite{FonGar11b}).

\subsection*{Acknowledgements}
 The author wishes to thank Alessandro Fonda for his useful comments 
on reading the draft of the paper.

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\end{document}