\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 125, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}

\title[\hfilneg EJDE-2004/125\hfil n-dimensional pendulum-like equations]
{Existence of solutions to n-dimensional pendulum-like equations}

\author[P. Amster, P. De N\'apoli, M.C. Mariani\hfil EJDE-2004/125\hfilneg]
{Pablo Amster, Pablo L. De N\'apoli, Mar\'ia Cristina Mariani}  
% in alphabetical order

\address{Pablo Amster \hfill\break
Universidad de Buenos Aires\\
FCEyN - Departamento de Matem\'atica \\
Ciudad Universitaria, Pabell\'on I\\
(1428) Buenos Aires, Argentina\\
and Consejo Nacional de Investigaciones Cient\'\i ficas y T\'ecnicas (CONICET)}
\email{pamster@dm.uba.ar}


\address{Pablo L. De N\'apoli \hfill\break
Universidad de Buenos Aires\\
FCEyN - Departamento de Matem\'atica \\
Ciudad Universitaria, Pabell\'on I\\
(1428) Buenos Aires, Argentina}
\email{pdenapo@dm.uba.ar}

\address{Mar\'ia Cristina Mariani \hfill\break
Department of Mathematical Sciences\\
New Mexico State University\\
Las Cruces, NM 88003-0001, USA}
\email{mmariani@nmsu.edu}

\date{}
\thanks{Submitted June 3, 2004. Published October 20, 2004.}
\subjclass[2000]{35J25, 35J65}
\keywords{Pendulum-like equations; boundary value problems; \hfill\break\indent
topological methods}

 
\begin{abstract}
 We study the elliptic boundary-value problem
 \begin{gather*}
 \Delta u + g(x,u)  = p(x) \quad \hbox{in } \Omega \\
 u\big|_{\partial \Omega}  = \hbox{\rm constant}, \quad
 \int_{\partial\Omega} \frac {\partial u}{\partial \nu} = 0, 
 \end{gather*}
 where $g$ is $T$-periodic in $u$, 
 and $\Omega \subset \mathbb{R}^n$ is a bounded domain.
 We prove the existence of a solution under a condition on
 the average of the forcing term $p$.
 Also, we prove the existence of a compact interval 
 $I_p \subset \mathbb{R}$ such that the problem is solvable 
 for $\tilde p(x) = p(x) + c$ if and only if
 $c\in I_p$. 
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

Existence and multiplicity of periodic solutions to the
one-dimensional pendulum  like equation
\begin{gather} \label{pend}
u''+ g(t,u) = p(t) \\
\label{per}
u(0)-u(T) = u'(0)-u'(T) = 0
\end{gather}
where $g$ is $T$-periodic in $u$
have been studied by many authors;
see e.g. \cite{FM} and  for the history and
a survey of the problem see \cite{M1,M2}.
In this work, we consider a generalization of this problem
to higher dimensions. With this aim, note that
the boundary condition \eqref{per} can be written as
$$
u(0) = u(T) = c,\quad \int_0^T u '' = 0
$$
where $c$ is a non-fixed constant.
Thus, by the divergence Theorem, \eqref{pend}-\eqref{per}
can be generalized to a boundary-value problem for an
elliptic PDE in the following way:
\begin{equation}
\label{1} 
\begin{gathered}
\Delta u + g(x,u)  = p(x) \quad \hbox{in } \Omega \\
u\big|_{\partial \Omega}  = \hbox{\rm constant},\quad
\int_{\partial\Omega} \frac {\partial u}{\partial \nu} = 0, 
\end{gathered}
\end{equation}
where $\Omega \subset \mathbb{R}^n$ is a bounded $C^{1,1}$ domain.
We shall assume that $p\in L^2(\Omega)$, and that
$g \in L^\infty \big( \Omega\times \mathbb{R}\big)$ is $T$-periodic in $u$.
For simplicity we shall assume also that
$\frac{\partial g}{\partial u} \in L^\infty ( \Omega\times \mathbb{R})$.

This kind of problems have been considered for example in
\cite{BB}, where the authors study a model describing the
equilibrium of a plasma confined in a toroidal cavity.
Under appropriate conditions this model can be reduced to the
nonhomogeneous boundary-value problem
\begin{equation}
\begin{gathered}
\Delta u + h(x,u)  = 0 \quad \hbox{in } \Omega \\
u\big|_{\partial \Omega}  = \hbox{\rm constant},\quad
-\int_{\partial\Omega} \frac {\partial u}{\partial \nu} = I. 
\end{gathered} 
\end{equation}
The authors prove
the existence of at least one solution $u\in H^2$
of the problem for any $h$ satisfying the following assumptions:
\begin{itemize}

\item[(A1)] $h:\overline\Omega\times \mathbb{R} \to [0,+\infty)$ is continuous,
nondecreasing on $u$, with $h(x,u)=0$ for $u\le 0$.

\item[(A2)] $\lim_{u\to +\infty} \int_\Omega {h(x,u)}dx >I$.

\item[(A3)] $\lim_{u\to +\infty} \frac {h(x,u)}{u^r} =0$ for some
$r \in\mathbb{R}$ (with $r \le \frac{n}{n-2}$ when $n>2$).
\end{itemize}
On the other hand, for the particular case $h(x,u)= [u]_+^p$ and
$\Omega = B_1(0)$, Ortega
has proved in \cite{O2} that if $n >2$ and $p\ge \frac{n}{n-2}$ then
there exists a finite constant
$I_p$ such that the problem has no solutions for $I>I_p$.

In the second section we obtain a solution of \eqref{1} by
variational methods under a condition on the average of the
forcing term $p$.

In the third section we prove by topological methods
that for a given
$p$ there exists a nonempty closed and bounded interval $I_p$ such
that problem \eqref{1} is solvable for $\tilde p = p + c$ if and
only if $c\in I_p$.
A similar result for the one-dimensional case
has been proved by Castro \cite{C}, using variational methods,
and by Fournier and Mawhin \cite{FM}, using topological methods.


\section{Solutions by variational methods}

For fixed $x\in \Omega$, define $a_g(x)$ as the average of
$g$ with respect to $u$, namely:
$$
a_g(x) = \frac 1T \int_0^T g(x,u)du\,.
$$
For $\varphi \in L^1(\Omega)$ denote by
$\overline\varphi$ the average of $\varphi$, i.e.
$$
\overline\varphi = \frac 1{|\Omega|}\int_\Omega \varphi(x) dx.
$$

\begin{theorem} \label{teo1}
If
\begin{equation}
\label{av}
\overline p = \overline{a_g},
\end{equation}
then \eqref{1} admits at least one solution $u\in H^2(\Omega)$.
\end{theorem}

\begin{proof}
Let $\mathbb{R} + H_0^1(\Omega) = \{ u\in H^1(\Omega): u\big|_{\partial\Omega}
= \hbox{\rm constant} \}$,
and consider the functional $\mathcal{I}:\mathbb{R} + H_0^1(\Omega) \to \mathbb{R}$
given by
$$
\mathcal{I}(u) = \int_\Omega \Big(\frac{|\nabla u(x)|^2}2
- G(x,u(x)) + p(x)u(x)\Big) dx,
$$
where
$$
G(x,u) = \int_0^u g(x,s)ds.
$$
By standard results, $\mathcal{I}$ is weakly lower
semicontinuous in $\mathbb{R} + H_0^1(\Omega)$.
We remark that $u$ is a critical point of $\mathcal{I}$ if and only if
\begin{equation}
\label{*}
\int_\Omega (\nabla u.\nabla
\varphi - g(x,u)\varphi + p\varphi) dx = 0
\end{equation}
for any $\varphi\in \mathbb{R}+ H_0^1(\Omega)$.
In this case, if $c=u\big|_{\partial\Omega}$ then
$u$ is a weak solution of the problem
\begin{equation}
\label{**}
\Delta u + g(x,u) = p(x),\quad u\big|_{\partial\Omega} = c.
\end{equation}
It follows that $u\in H^2(\Omega)$. We claim that
$\int_{\partial\Omega} \frac {\partial u}{\partial \nu} =0$.
Indeed, taking $\varphi \equiv 1$ in \eqref{*} we obtain:
$$
\int_\Omega g(x,u)dx = \int_\Omega p(x)dx.
$$
Integrating (\ref{**}) over $\Omega$, we deduce that
$$
\int_{\partial\Omega} \frac {\partial u}{\partial \nu} =
\int_{\Omega} \Delta u = 0.
$$
Thus, any critical point of $\mathcal{I}$ is a weak solution 
of \eqref{1}.

To prove the existence of critical points of $\mathcal{I}$,
let $\{u_n\} \subset
\mathbb{R}+ H_0^1(\Omega)$ be a minimizing sequence,
and let $c_n = u_n\big|_{\partial\Omega}$.
For any $u\in \mathbb{R}+ H_0^1(\Omega)$ it holds that
$$
\mathcal{I} (u + T) - \mathcal{I} (u) = T \int_\Omega p(x) dx
- \int_\Omega [G(x,u+T)-G(x,u)] dx.
$$
For fixed $x$, we have
$$
G(x,u(x)+T)-G(x,u(x)) = \int_{u(x)}^{u(x)+T} g(x,s)ds =
\int_{0}^{T} g(x,s)ds = Ta_g(x),
$$
and from \eqref{av} we deduce that
$\mathcal{I} (u + T) = \mathcal{I} (u)$.
Hence, we may assume that $c_n\in [0,T]$. By Poincar\'e's
inequality we have that
$$
\|u_n - c_n\|_{L^2} \le C \|\nabla u_n \|_{L^2},
$$
and then
$$
I(u_n) = \frac 12 \|\nabla u_n \|_{L^2}^2 +
\int_\Omega pu_n dx - \int_\Omega G(x,u_n)dx
\ge \frac 12 \|\nabla u_n \|_{L^2}^2 - r
\| \nabla u_n \|_{L^2} - s
$$
for some constants $r,s$. Thus, $\{u_n\}$ is bounded, and
by classical results $\mathcal{I}$ has a minimum on
$\mathbb{R}+ H_0^1(\Omega)$.
\end{proof}

\section{The maximal interval $I_p$}

Fix $p\in L^2(\Omega)$ such that $\overline p = \overline{a_g}$
and consider the problem
\begin{equation}\label{2} 
\begin{gathered}
\Delta u + g(x,u)  = p(x) + c \quad \hbox{in }\Omega \\
u\big|_{\partial \Omega}  = \hbox{\rm constant} \quad
\int_{\partial\Omega} \frac {\partial u}{\partial \nu} = 0 
\end{gathered} 
\end{equation}
with $c\in \mathbb{R}$.
It is easy to establish a necessary condition on $c$
for the solvability of \eqref{2}: indeed, if $u$ is a
solution of \eqref{2} then
$$
\frac 1{|\Omega|} \int_\Omega g(x,u(x)) dx = \overline p + c.$$
Thus, if we define $g_u(x) = g(x,u(x))$, we obtain:
$$
c = \overline {g_u} - \overline {a_g}.
$$
Furthermore, if
$$
g_+(x) = \sup_{0\le u\le T}g(x,u), \quad
g_-(x) = \inf_{0\le u\le T}g(x,u),
$$
it follows that $\overline {g_-}\le \overline {g_u}\le
\overline {g_+}$, and hence
$$
\overline {g_-}-\overline {a_g} \le c \le
\overline {g_+}-\overline {a_g}.$$
In particular,
$$
\inf_{[0,T]\times\mathbb{R}} {g} -\overline {a_g} \le c \le
\sup_{[0,T]\times\mathbb{R}}-\overline {a_g}.
$$
In the next theorem we obtain also a sufficient condition.
More precisely, if we define
$$
I_p = \{ c \in \mathbb{R}: \hbox{ \eqref{2} admits a solution in }
H^2(\Omega) \},$$
we shall prove that $I_p$ is a nonempty compact interval. From
Theorem \ref{teo1}, it follows that
$$
I_p = [\alpha_p, \beta_p],
$$
where
$$
\overline {g_-} - \overline{a_g} \le \alpha_p \le 0 \le \beta_p
\le \overline {g_+} - \overline{a_g}.
$$

\begin{theorem} \label{teo2}
Assume that
$\overline p = \overline{a_g}$ and define
$$E = \{ u \in \mathbb{R}+ H^2 \cap H_0^1(\Omega):
\Delta u + g(x,u) = p + \overline {g_u} - \overline {a_g}\}.
$$
Then the set
$$
E_g := \{ \overline {g_u}: u\in E\}\subset \mathbb{R}
$$
is a nonempty compact interval.
Furthermore,
$E_g = \overline {a_g}+ I_p$.
\end{theorem}

For the proof of this theroem,  we  need
Lemmas \ref{lem3}, \ref{lem4}, \ref{lem5}, \ref{lem6}, \ref{lem7}
and Theorem \ref{teo8} below.

\begin{lemma}[Poincar\'e-Wirtinger inequality] \label{lem3}
There exists a constant $c\in\mathbb{R}$ such that
$$
\| u - \overline u\|_{L^2} \le c\| \nabla u\|_{L^2} 
$$
for all $u\in H^1(\Omega)$.
\end{lemma}

The proof of the above lemma can be found in  \cite{Ke}.


\begin{lemma} \label{lem4}
Assume that $\overline p = \overline{a_g}$.
Then for any $r\in \mathbb{R}$ the problem
\begin{gather*} 
\Delta u + g(x,u) = p + \overline {g_u} - \overline {a_g} \\
u\big|_{\partial \Omega}  = \hbox{\rm constant},\quad
\int_{\partial\Omega} \frac {\partial u}{\partial \nu} = 0 
\end{gather*}
admits at least one solution $u$ such that $\overline u = r$.
\end{lemma}

\begin{proof}
For $u\in H^1(\Omega)$ define $Tu= v$ as the unique solution
of the problem
\begin{equation} \label{3} 
\begin{gathered}
\Delta v  = p + \overline {g_u} - \overline {a_g} - g(x,u) \\
v\big|_{\partial \Omega}  = \hbox{\rm constant},\quad
\overline v = r. 
\end{gathered}
\end{equation}
Then $T:H^1(\Omega)\to H^1(\Omega)$ is
well defined and compact.
Indeed, if $u_0 $ is the unique element of $H^2\cap H_0^1(\Omega)$
such that
$$
\Delta u_0 = p + \overline {g_u} - \overline {a_g} - g(x,u),
$$
it is clear that $v = u_0 - \overline {u_0} + r$ is the unique
solution of \eqref{3}, and compactness follows immediately from
the compactness of the mapping $u\to u_0$.
Moreover, integrating the equation, it is immediate that
$$
\int_{\partial\Omega} \frac {\partial v}{\partial \nu} =
\int_{\Omega} \Delta v = 0.
$$
Then
$$
\int_\Omega\Delta v(v-r) + \int_\Omega |\nabla v|^2=
(v\big|_{\partial\Omega} -r)
\int_{\partial\Omega} \frac {\partial v}{\partial \nu} = 0,
$$
and we deduce that
$$
\|v-r\|_{H^1} \le c \|\Delta v\|_{L^2} \le C
$$
for some constant $C$. Thus, the proof follows from Schauder
Theorem.
\end{proof}

\begin{lemma} \label{lem5}
Let $p, E, E_g$ be as in Theorem \ref{teo2} and
$$E_T = \{ u\in E: u\big|_{\partial\Omega} \in [0,T]\}.$$
Then:
\begin{enumerate}
\item $E_T \subset \mathbb{R}+ H_0^1(\Omega)$ is compact.
\item $E_g = \{ \overline {g_u}: u\in E_T\}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $\{ u_n\} \subset E_T$ and
$c_n = u_n\big|_{\partial\Omega} \in [0,T]$.
>From standard elliptic
estimates it follows that
$\| u_n\|_{H^2} \le C$ for some constant $C$.
Taking a subsequence we may assume that
$u_{n} \to u$ in $\mathbb{R}+H_0^1(\Omega)$.
>From the equalities
$$
\Delta u_{n} = p +\overline {g_{u_{n}}} - \overline {a_g} -g(x,u_{n})
$$
it follows easily that $u\in E_T$, and (1) is proved.
Moreover, for any
$u\in E$ there exists $k\in\mathbb{Z}$ such that
$u_T: = u + kT \in E_T$. As $g_{u_T} = g_u$, the proof of (2) follows.
\end{proof}

To complete the proof of
Theorem \ref{teo2}, it suffices to show that
$I_p$ is connected. Indeed, it is clear that
$u$ is a solution of \eqref{2} if and only if
$u \in E$ with $c = \overline{g_u}-\overline{a_g}$,
and by continuity of the mapping $u\to \overline{g_u}$
it follows that $I_p$ is compact.

\begin{remark} \rm
>From Lemma \ref{lem4}, $E$ is infinite. In particular,
if $I_p = \{ 0\}$ then \eqref{1} admits a continuum of
solutions.
\end{remark}

To apply the method of upper and lower solutions
to our problem,
we shall first prove an associated maximum principle:

\begin{lemma} \label{lem6}
Let $\lambda > 0$ and assume that
$u\in H^{2}(\Omega)$ satisfies:
\begin{gather*}
\Delta u - \lambda u \ge 0,\\
u\big|_{\partial \Omega} = \hbox{\rm constant},\quad
\int_{\partial\Omega} \frac{\partial u}{\partial \nu} \le 0.
\end{gather*}
Then $u\le 0$.
\end{lemma}

\begin{proof}
If $u\big|_{\partial \Omega} = c\le 0$ the result
follows by the
classical maximum principle. If $c>0$, let
$
\Omega^+ = \{ x\in \Omega: u(x) > 0\}$ and
$u^+(x) = \max\{ u(x), 0\}$.
Then
$$
0 \le \int_{\Omega}\lambda u.u^+ 
\le \int_{\Omega} \Delta u.u^+ 
=-\int_{\Omega^+} |\nabla u|^2 + c\int_{\partial\Omega}
\frac{\partial u}{\partial \nu} < 0\,,
$$
a contradiction.
\end{proof}

\begin{lemma} \label{lem7}
Let $\theta \in L^2(\Omega)$ and $\lambda>0$.
Then the problem
\begin{gather*}
\Delta u - \lambda u = \theta \quad \hbox{in }\Omega \\
u\big|_{\partial \Omega} = \quad \hbox{\rm constant}, \quad
\int_{\partial\Omega} \frac {\partial u}{\partial \nu} = 0 
\end{gather*}
admits a unique solution $u_\theta\in H^{2}(\Omega)$.
Furthermore, the mapping $\theta\to u_\theta$ is continuous.
\end{lemma}

\begin{proof}
Let $\mathcal{J}:\mathbb{R}+H_0^1(\Omega)\to \mathbb{R}$ be the functional
$$
\mathcal{J}(u) = \int_\Omega \frac {|\nabla u|^2}2 +
\frac {\lambda u^2}2 + \theta u.
$$
It is immediate that $\mathcal{J}$ is weakly lower semicontinuous
and coercive, then it has a minimum $u$.
Furthermore, $u\in H^{2}(\Omega)$ and
$\int_{\partial\Omega} \frac {\partial u}{\partial \nu} = 0$.
Integrating the equation, we also obtain that
$-\lambda \overline u = \overline \theta$.

By standard elliptic estimates and Lemma \ref{lem3}, there exists
a constant $c$ such that
$$
\|w-\overline w\|_{H^{2}} \le c \| \Delta w - \lambda w\|_{L^2}
$$
for any $w\in H^{2}\cap (\mathbb{R} + H_0^{1})$ such that
$\int_{\partial\Omega} \frac {\partial w}{\partial \nu} =0$;
thus, uniqueness follows.
Finally, if $\theta_1$, $\theta_2 \in L^2(\Omega)$
then
$$
\| u_{\theta_1}-u_{\theta_2}\|_{H^{2}}
\le |\Omega|.|\overline \theta_1-\overline\theta_2| +
c \|\theta_1-\theta_2\|_{L^{2}},
$$
and the proof is complete.
\end{proof}

Now we have the following result.

\begin{theorem} \label{teo8}
If $\varphi \in L^{2}(\Omega)$ and
there exist $\alpha, \beta\in H^{2}(\Omega)$ with
$\alpha\le \beta$ such that
\begin{gather*}
\Delta \beta + g(\cdot,\beta)\le \varphi(x)\le \Delta \alpha + g(\cdot,\alpha),\\
\beta\big|_{\partial \Omega} = \quad \hbox{\rm constant}, \quad
\alpha\big|_{\partial \Omega} = \quad \hbox{\rm constant},\\
\int_{\partial\Omega} \frac {\partial \beta}{\partial \nu} \ge 0
\ge
\int_{\partial\Omega} \frac {\partial \alpha}{\partial \nu},
\end{gather*}
then the problem
\begin{gather*}
\Delta u + g(x,u) = \varphi(x) \\
u\big|_{\partial \Omega}  = \quad \hbox{\rm constant},\quad
\int_{\partial\Omega} \frac {\partial u}{\partial \nu} = 0 
\end{gather*} 
admits at least one solution $u\in H^{2}(\Omega)$
such that $\alpha\le u\le \beta$.
\end{theorem}

\begin{proof}
Let $\lambda\ge R$, where
$R = \|\frac{\partial g}{\partial u}\|_{L^\infty}$.
For fixed $v \in L^2(\Omega)$
define $Tv = u$ as the unique solution of the
problem
\begin{gather*}
\Delta u - \lambda u  = \varphi - g(x,v) - \lambda v
\quad \hbox{in }\Omega \\
u\big|_{\partial \Omega}  = \quad \hbox{\rm constant}, \quad
\int_{\partial\Omega} \frac {\partial u}{\partial \nu} =0. 
\end{gather*} 
By the lemmas above, the mapping $T:L^2(\Omega)\to
L^2(\Omega)$ is well defined and compact. Moreover
for $\alpha\le v\le \beta$, we have
$$
 \Delta u - \lambda u = \varphi - g(x,v) - \lambda v
\ge \varphi - g(x,\beta) - \lambda \beta
\ge \Delta \beta - \lambda \beta.
$$
Hence,
$$
\Delta (u-\beta) - \lambda (u-\beta) \ge 0
$$
and
$$
(u-\beta)\big|_{\partial \Omega} = \quad \hbox{\rm constant},\quad 
\int_{\partial\Omega} \frac {\partial (u- \beta)}{\partial \nu}
\le 0.
$$
 From Lemma \ref{lem6}, we deduce that $u\le \beta$. In the same way,
we obtain that $u\ge \alpha$
and the result follows by Schauder Theorem.
\end{proof}

\begin{proof}[Proof of Theorem \ref{teo2}]
Let $P\in H^2(\Omega)$ be any solution of the problem
\begin{gather*}
\Delta P = p - \overline{a_g} \\
P\big|_{\partial \Omega}  = \quad \hbox{\rm constant},\quad
\int_{\partial\Omega} \frac {\partial P}{\partial \nu} =0. 
\end{gather*}
Taking $v = u - P$, problem \eqref{2} is equivalent to the problem
\begin{gather*}
\Delta v + \tilde g (x, v)  = c + \overline{a_g} \\
P\big|_{\partial \Omega} = \quad \hbox{\rm constant},\quad
\int_{\partial\Omega} \frac {\partial P}{\partial \nu} =0\,, 
\end{gather*}
where $\tilde g(x,v) := g(x,v+P(x))$ is
continuous and $T$-periodic in $v$.
Thus, we may assume without loss of generality that
$p$ is continuous.
Let $c_1, c_2 \in I_p$, $c_1 < c_2$, and take
$u_1, u_2 \in E$ such that
$\overline{g_{u_i}} = c_i - \overline{a_g}$.
As $u_i \in C(\overline \Omega)$,
adding $kT$ for some integer $k$ if necessary, we may
suppose that $u_1 \le u_2$.
For $c\in [c_1,c_2]$ we have that
$$
\Delta u_1 + g(x,u_1) = p + c_1 - \overline {a_g} \le
p + c - \overline {a_g} \le p + c_2 - \overline {a_g} =
\Delta u_2 + g(x,u_2).
$$
 From the previous theorem, there exists $u\in E$ such that
$\overline{g_{u}} = c-\overline{a_g} $. The proof is complete.
\end{proof} 

\begin{remark}\rm
Using fixed point methods, Lemma \ref{lem7} can be
generalized; thus, it is easy to see
that Theorem \ref{teo2} is still valid for the
more general problem
\begin{gather*}
\Delta u + \langle b(x),\nabla u \rangle + g(x,u)  = p(x) \quad
 \hbox{in } \Omega \\
u\big|_{\partial \Omega}  = \hbox{\rm constant},\quad
\int_{\partial\Omega} \frac {\partial u}{\partial \nu} = 0\,, 
\end{gather*}
where $b$ is a $C^1$-field such that $\hbox{div }b =0$.
However, for $b\neq 0$ the problem is no longer variational, and
then the claim of Theorem \ref{teo1} is not necessarily true.
Indeed, in the particular case $n=1$, it is well known that
for the pendulum equation
$$
u'' + a u' + b\; \sin u =  f (t),
$$
where $a$ is a positive constant,
there exists a family of $T$-periodic
functions $f$ such that $\int_0^T f = 0$
for which the equation has no periodic solutions 
(see \cite{A,O,OST}).
\end{remark}

\begin{remark} \rm
As in \cite{FM}, it can be proved that
for any $c$ in the interior of
$I_p$ there exist at least two solutions of \eqref{2}
which are essentially different 
(i.e. not differing by a multiple of $T$).
\end{remark}

\subsection*{Acknowledgement}
The authors would like to thank the anonymous referee
for his/her careful reading of the original manuscript 
and the fruitful remarks.


\begin{thebibliography}{00}

\bibitem{A}
Alonso J.; \emph{Nonexistence of periodic solutions for a damped
pendulum equation}.
 Diff. and Integral Equations, 10 (1997), 1141-1148.

\bibitem{BB}
Berestycki B., Brezis H.; \emph{On a
free boundary problem arising in plasma physics}.
Nonlinear Analysis 4, 3 (1980), pp 415-436.


\bibitem{C}
Castro A.;
\emph{Periodic solutions of the forced pendulum equation}.
Differential equations (Proc. Eighth Fall Conf., Oklahoma State Univ.,
Stillwater, Okla., 1979), pp. 149--160, Academic Press, New
York-London-Toronto, Ont., 1980.


\bibitem{FM}
Fournier G., Mawhin J.; \emph{On periodic solutions of forced
pendulum-like equations}.
Journal of Differential Equations 60 (1985) 381-395.

\bibitem{Ke}
Kesavan, S.; \emph{Topics in Functional Analysis
and Applications}. John Wiley \& Sons, Inc., New York, NY, 1989.

\bibitem{M1}
Mawhin J.; \emph{Periodic oscillations of forced pendulum-like
equations}, Lecture Notes in Math, 964, Springer, Berlin 1982,
458-476.

\bibitem{M2} 
Mawhin J.; \emph{Seventy-five years of global analysis around the
forced pendulum equation}, Proc. Equadiff 9, Brno 1997.

\bibitem{O}
Ortega R.; \emph{A counterexample
for the damped pendulum equation}. Bull. de la
Classe des Sciences, Ac. Roy. Belgique, LXXIII (1987), 405-409.

\bibitem {O2}
Ortega R.; \emph{Nonexistence of radial solutions
of two elliptic boundary value problems}.
Proc. of the Royal Society of Edinburgh 114A (1990) 27-31.

\bibitem{OST}
Ortega R., Serra E., Tarallo M.;
\emph{Non-continuation of the periodic oscillations
of a forced pendulum in the presence of friction}.
Proc. of Am. Math. Soc. Vol. 128, 9 (2000), 2659-2665.


\end{thebibliography}


\end{document}