\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 209, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/209\hfil Range of semilinear operators] {Range of semilinear operators for systems at resonance} \author[P. Amster, M. P. Kuna \hfil EJDE-2012/209\hfilneg] {Pablo Amster, Mariel Paula Kuna} % in alphabetical order \address{Departamento de Matem\'atica, Facultad de Ciencias Exactas y Naturales\\ Universidad de Buenos Aires\\ Ciudad Universitaria, Pabell\'on I, (1428) Buenos Aires, Argentina \newline Consejo Nacional de Investigaciones Cient\'\i ficas y T\'ecnicas (CONICET), Argentina} \email[Pablo Amster] {pamster@dm.uba.ar} \email[Mariel Paula Kuna]{mpkuna@dm.uba.ar} \thanks{Submitted October 7, 2011. Published November 27, 2012.} \subjclass[2000]{34B15, 34L30} \keywords{Resonant systems; semilinear operators; critical point theory} \begin{abstract} For a vector function $u:\mathbb{R} \to \mathbb{R}^N $ we consider the system \begin{gather*} u''(t)+ \nabla G(u(t))= p(t)\\ u(t)=u(t+T), \end{gather*} where $G: \mathbb{R}^N \to \mathbb{R}$ is a $C^1$ function. We are interested in finding all possible $T$-periodic forcing terms $p(t)$ for which there is at least one solution. In other words, we examine the range of the semilinear operator $S:H^2_{\rm per}\to L^2([0,T],\mathbb{R}^N)$ given by $Su= u''+ \nabla G(u)$, where $$ H^2_{\rm per}= \{ u\in H^2([0,T], \mathbb{R}^N); u(0) - u(T) = u'(0)-u'(T)=0 \}. $$ Writing $p(t)= \overline{p} + \widetilde{p}(t)$, where $\overline{p}:=\frac 1T\int_0^Tp(t)\, dt$, we present several results concerning the topological structure of the set $$ \mathcal{I}(\widetilde{p})=\{ \overline{p} \in \mathbb{R}^N; \overline{p} + \widetilde{p}\in \operatorname{Im}(S)\}. $$ \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction}\label{intro} Let $G\in C^1(\mathbb{R}^N, \mathbb{R})$. A well known result establishes that if $\nabla G$ is bounded, then the Dirichlet problem \begin{gather} \label{ecua1} u'' + \nabla G(u) = p(t)\\ u(0)= u(T)= 0 \end{gather} has at least one solution for any $p\in L^2([0,T],\mathbb{R}^N)$; that is to say, the operator $S(u):= u''+\nabla G(u)$, regarded as a continuous function from $H^2\cap H_0^1([0,T],\mathbb{R}^N)$ to $L^2([0,T],\mathbb{R}^N)$, is surjective. This is due to the fact that the associated linear operator $Lu:=-u''$ is invertible; thus, a simple proof follows as a straightforward application of Schauder's fixed point theorem. The boundedness condition ensures that the nonlinearity does not interact with the spectrum of $L$. The situation is different at resonance, namely when the associated linear operator is non-invertible. In particular, if we consider the periodic problem for \eqref{ecua1}, then integrating we have $$ \frac 1T\int_0^T\nabla G(u(t))\, dt=\overline{p}. $$ Thus, the geometric version of the Hahn-Banach Theorem implies that a necessary condition for the existence of solutions is that $\overline{p} \in \operatorname{co}(\operatorname{Im}(\nabla G))$, where `co' stands for the convex hull. In particular, if we decompose $L^2([0,T],\mathbb{R}^N)$ as the orthogonal sum of $\mathbb{R}^N$ and the set $\widetilde{L}^2$ of zero-average functions; i.e., \begin{gather*} L^2([0,T],\mathbb{R}^N) = \mathbb{R}^N \oplus \widetilde{L}^2\\ p = \overline{p} + \widetilde{p}, \end{gather*} with $$ \widetilde{L}^2 := \{ v \in L^2([0,T], \mathbb{R}^N); \; \overline v=0\}, $$ then the range of $S$, now defined on $H^2_{\rm per}$, is contained in $\operatorname{co}(\operatorname{Im}(\nabla G))\oplus \widetilde{L}^2$. Thus, it is useful to study, for a given $\widetilde{p}\in \widetilde{L}^2$, the set $$\mathcal{I}(\widetilde{p}):= \{\overline{p} \in \mathbb{R}^N: \overline{p} + \widetilde{p}\in \operatorname{Im}(S)\}\subset \operatorname{co}(\operatorname{Im}(\nabla G)). $$ When $\nabla G$ is bounded it can be proven, generalizing the arguments given in \cite{HT} for a scalar equation, that $\mathcal{I}(\widetilde{p})$ is non-empty and connected; if $\nabla G$ is also periodic, then $\mathcal{I}(\widetilde{p})$ is compact (see e.g. \cite{tesis}). For example, a quite precise description of this set can be given when the radial limits $$ \lim_{s\to +\infty} \nabla G(sv):= \Gamma(v) $$ exist uniformly for $v\in S^{N-1}$, the unit sphere of $\mathbb{R}^N$. In this case, a well-known result by Nirenberg \cite{n} implies that all the interior points of the field $\Gamma:S^{N-1}\to \mathbb{R}^N$ (i. e. those points $\overline{p}$ such that the winding number of $\Gamma$ with respect to $\overline{p}$ is nonzero) is contained in $\mathcal{I}(\widetilde{p})$. If also $\operatorname{co}(\operatorname{Im}(\nabla G))\subset int(\Gamma)$, then the condition $deg(\Gamma,\overline{p})\ne 0$ is both necessary and sufficient, indeed: $$ \operatorname{Im}(S)=\operatorname{Int}(\Gamma)\oplus \widetilde{L}^2. $$ A different situation occurs when $\nabla G$ is unbounded; in particular, $\mathcal{I}(\widetilde{p})$ might be empty. The following result, adapted from the main theorem in \cite{ALP}, is useful to verify that this is not the case if $G$ tends to $+\infty$ or to $-\infty$ as $|u|\to \infty$. More generally: \begin{theorem}\label{alp} Let $G\in C^1(\mathbb{R}^N,\mathbb{R})$, $\widetilde{p}\in \widetilde{L}^2$ and $\overline{p}\in \mathbb{R}^N$. If $$ \lim_{|u|\to \infty} G(u) - \overline{p}\cdot u =+\infty\quad \text{or}\quad \lim_{|u|\to \infty} G(u) - \overline{p}\cdot u = - \infty, $$ then $\overline{p}\in \mathcal{I}(\widetilde{p})$. \end{theorem} In particular, if $G(u)\to +\infty$ or $G(u)\to -\infty$ as $|u|\to \infty$, then $0\in \mathcal{I}(\widetilde{p})$. Furthermore, if $G$ is strongly convex, in the sense that $G(u) - c |u|^2$ is convex for some constant $c > 0$, then $\mathcal{I}(\widetilde{p}) = \mathbb{R}^N$ and hence $S$ is surjective. The same conclusion is obviously true when $G$ is strongly concave. \begin{remark} \label{rmk1.2} \rm When $N=1$, Theorem \ref{alp} generalizes the well-known Landesman-Lazer conditions. However, although \cite{n} can be regarded as an extension of these conditions, Theorem \ref{alp} does not necessarily generalize Nirenberg's result. \end{remark} This paper is organized as follows. In the next section, we prove a basic criterion which ensures that $\overline{p}\in \mathbb{R}^N$ belongs to $\mathcal{I}(\widetilde{p})$ for some given $\widetilde{p}$. In section \ref{bilin}, we give sufficient conditions for a point $\overline{p}_0\in \mathcal{I}(\widetilde{p})$ to be interior. In section \ref{periodic}, we extend a well known result by Castro \cite{AC} for the pendulum equation; more precisely, we prove that if $\nabla G$ is periodic then $\mathcal{I}$ regarded as a function from $\widetilde{L}$ to the set of compacts subsets of $\mathbb{R}^N$ (equipped with the Hausdorff metric) is continuous. Finally, in section \ref{convex}, we prove that if $G$ is strictly convex and satisfies some accurate growth assumptions, then $\mathcal{I}(\widetilde{p}) = \operatorname{Im}(\nabla G)$ for all $\widetilde{p}$. \section{A basic criterion for general $G$}\label{basic} \begin{proposition}\label{psi} Let $\overline{p}\in \mathbb{R}^N$ and define $\psi_{\overline{p}}: \mathbb{R}^N \to \mathbb{R}$ by $\psi_{\overline{p}}(u):=\overline{p}\cdot u-G(u)$. Assume that: \begin{enumerate} \item $\psi_{\overline{p}}$ is bounded from below, \item $\liminf_{|u| \to +\infty} \psi_{\overline{p}}(u) > \inf_{u\in \mathbb{R}^N}\psi_{\overline{p}}(u) + \frac{T}{8\pi ^2} \| \widetilde{p}\|^2_{L^2(0,T)}$. \end{enumerate} Then $\overline{p} \in \mathcal{I}(\widetilde{p})$. \end{proposition} \begin{proof} Consider the functional $J:H_{\rm per}^1:= \{ u\in H^1([0,T], \mathbb{R}^N): u(0) = u(T) \}\to \mathbb{R}$ given by $$ J(u):= \int^T_0 \frac{|u'(t)|^2}{2} + \psi_{\overline{p}} (u(t)) + \widetilde{p}(t) \cdot u(t)\, dt. $$ It is readily seen that $J$ is continuously Fr\'echet differentiable, and \begin{equation}\label{dif} DJ(u)(v)=\int^T_0 u'(t)\cdot v'(t)- \nabla G(u(t))\cdot v(t) + p(t)\cdot v(t) \, dt. \end{equation} Thus, if $u$ is a minimum of $J$, $u$ is a weak solution of \eqref{ecua1}, and by standard arguments we deduce that it is classical. Also, it is known that $J$ is weakly lower semicontinuous; thus, due to Theorem 1.1 of \cite{MW}, it suffices to prove that $J$ has a bounded minimizing sequence. Without loss of generality, we may suppose that $G(0)=0$. \noindent \emph{Claim 1: $-\infty < \inf J \leq T\inf\psi_{\overline{p}} \leq 0$.} Indeed, let us recall the well known Wirtinger inequality: \begin{equation}\label{wirti} \| u- \overline u \|^2_2 \leq \big( \frac{T}{2\pi} \big)^2 \|u' \|^2_2. \end{equation} From \eqref{wirti} and Cauchy-Schwarz inequality we deduce: $$ J(u) \geq \frac{1}{2} \|u'\|^2_2 - \| \widetilde{p} \|_2 \| u-\overline u \|_2 + \int_0^T \psi_{\overline{p}}(u(t))\,dt. $$ Thus, \begin{equation} J(u)\ge \label{desig} \frac{1}{2} \Big( \|u'\|_2 - \frac{T}{2\pi} \| \widetilde{p}\|_2 \Big)^2 - \frac{T^2}{8\pi^2} \| \widetilde{p} \|^2_2 + T\inf_{u\in \mathbb{R}^N} \psi_{\overline{p}} \end{equation} an the first inequality is proven. For the second inequality, it is sufficient to observe that $$ \inf_{u\in H^1_{\rm per}} J(u) \leq \inf_{u\in \mathbb{R}^N} J(u) \leq T\inf_{u\in \mathbb{R}^N} \psi_{\overline{p}} (u). $$ The third inequality is obvious since $\psi_{\overline{p}}(0)=0$. Next, consider a sequence $(u_{n})_{n\in \mathbb{N}}$ such that $\lim_{n \to \infty} J(u_{n})= \inf J$. \noindent \emph{Claim 2: The sequence $(u_{n})_{n\in \mathbb{N}}$ is bounded in $H^1_{\rm per}$.} From the previous claim, for any given $\epsilon >0$ there exists $n_0\in \mathbb{N}$ such that \begin{equation}\label{epsi} J(u_{n}) < T\inf \psi_{\overline{p}} + \epsilon,\quad\text{for all }n\geq n_0. \end{equation} Then \eqref{desig} yields $$ \Big( \|u'_{n}\|_2 - \frac{T}{2\pi} \| \widetilde{p}\|_2 \Big)^2 < \frac{T^2}{4\pi^2} \| \widetilde{p} \|^2_2 + 2 \epsilon $$ so $$ \|u'_{n}\|_2^2 < \frac{T}{\pi} \|\widetilde{p}\|_2\|u'_{n}\|_2 +2\epsilon.$$ Hence, there exists $\tau > 0$, independent of $n$, such that $\|u'_{n} \|_2\leq \frac{T}{\pi}\| \widetilde{p}\|_2 + \tau$. As before, $$ J(u_{n}) \geq \frac{1}{2} \Big( \|u'_{n}\|_2 - \frac{T}{2\pi} \| \widetilde{p} \|_2 \Big)^2 - \frac{T^2}{8\pi^2} \| \widetilde{p} \|^2_2 + \int^T_0 \psi_{\overline{p}} (u_{n}(t))\, dt, $$ and from \eqref{epsi} we deduce that \begin{equation}\label{cota} \int^T_0 \psi_{\overline{p}} (u_{n}(t))\, dt \leq \frac{T^2}{8\pi^2} \| \widetilde{p} \|^2_2 + T\inf \psi_{\overline{p}} + \epsilon. \end{equation} Suppose that $\|u_n\|_{H^1}\to \infty$, then from the bound for $\|u_n'\|_2$ and the standard inequality $$ \|u_{n} - \overline u_{n} \|_{ \infty} \leq \frac{\sqrt{T}}{2} \| u'_{n} \|_2 $$ we deduce that $|\overline u_n|\to \infty$ and $|u_n(t)|\to \infty$ uniformly in $t$. Thus, for a given $\delta >0$ there exists $n_{1}\ge n_0$ such that $\psi_{\overline{p}} (u_{n}(t)) \geq \liminf_{ |u| \to \infty} \psi_{\overline{p}} (u) - \frac{\delta}{T}$ for all $n\geq n_{1}$. Hence $$ \int^T_0 \psi_{\overline{p}} (u_{n}(t))\, dt \geq T\liminf_{ |u| \to \infty} \psi_{\overline{p}} (u) - \delta \quad\text{for all } n\geq n_1. $$ Then, by \eqref{cota} \begin{equation} \label{desi} T\liminf_{ |u| \to \infty} \psi_{\overline{p}} (u) \leq T\inf \psi_{\overline{p}} + \frac{T^2}{8\pi^2} \| \widetilde{p} \|^2_2 + \epsilon + \delta, \end{equation} which contradicts hypothesis 2 when $\epsilon + \delta$ is small enough. So $(u_{n})_{n\in \mathbb{N}}$ is bounded in $H^1_{\rm per}$. \end{proof} \begin{remark} \label{rmk2.1} \rm In particular, if $$ \liminf_{|u| \to +\infty} \psi_{\overline{p}}(u)- \inf \psi_{\overline{p}}=r>0, $$ then $\overline{p} \oplus \tilde B_r(0)\subset \operatorname{Im}(S)$, where $\tilde B_r(0)\subset \widetilde{L}^2$ denotes the open ball of radius $r$ centered at $0$. \end{remark} \begin{example} \label{examp2.2} \rm Suppose that $$ \limsup_{|u| \to \infty} \frac{G(u)}{|u|}=-R<0. $$ Then $B_{R}(0) \subseteq \mathcal{I}(\widetilde{p})$ for any $\widetilde{p}$. Indeed, if $| \overline{p} | |u|(R - \epsilon - |\overline{p}|) =\epsilon |u|\to +\infty $$ as $|u|\to \infty$ and the result follows. This particular case is obviously covered by Theorem \ref{alp}; however, Proposition \ref{psi} is still applicable for example if $$ \limsup_{|u| \to \infty} \frac{G(u)}{|u|}\le 0\quad \text{and} \quad \limsup_{|u| \to \infty, u\in \mathcal C} \frac{G(u)}{|u|} = -R < 0 $$ with $$ \mathcal C:= \{u\in \mathbb{R}^N: u\cdot w > -c|u|\} $$ for some $w\in S^{n-1}$ and $c\in (0,1)$. In this case, $I(\widetilde{p})$ contains all the vectors $\overline{p}\in B_R(0)$ such that the angle between $\overline{p}$ and $-w$ is smaller than $\frac \pi 2 - \arccos(c)$. \end{example} \section{Interior points} \label{bilin} In this section we give sufficient conditions for a point $\overline{p}_0\in \mathcal{I}(\widetilde{p})$ to be interior. Roughly speaking, we shall prove that if the Hessian matrix of $G$ does not interact with the spectrum of the operator $Lu:= -u''$ then $\mathcal{I}(\widetilde{p})$ is a neighborhood of $\overline{p}_0$. More precisely: \begin{theorem}\label{prop2} Let us assume that $G\in C^2(\mathbb{R}^N, \mathbb{R})$ and let $\overline{p}_0\in \mathcal{I}(\widetilde{p})$ for some $\widetilde{p} \in \widetilde{L}^2$. Further, let $u_0$ be a solution of \eqref{ecua1} for $\overline{p}=\overline{p}_0$ and assume there exist symmetric matrices $A$, $B \in \mathbb{R}^{N\times N}$ such that $$ A\leq d^2G(u_0(t)) \leq B \quad t\in [0,T] $$ and $$ \Big(\frac{2\pi N_{k}}{T} \Big)^2 < \lambda_{k} \leq \mu_{k} < \Big( \frac{2\pi (N_{k}+1)}{T}\Big)^2 $$ for some integers $N_k\ge 0$, $k=1,\dots, N$, where $\lambda_{1} \leq \lambda_2 \leq \dots \leq \lambda_{N}$ and $\mu_{1} \leq \mu_2 \leq \dots \leq \mu_{N}$ are the respective eigenvalues of $A$ and $B$. Then there exists an open set $U\subset \mathbb{R}^N$ such that $\overline{p}_0 \in U\subseteq \mathcal{I}(\widetilde{p})$. \end{theorem} The proof relies in the following uniqueness result, which has been proven by Lazer in \cite{L3} using a lemma on bilinear forms. \begin{theorem}\label{formas} Let $Q$ be a real $N\times N$ symmetric matrix valued function with elements defined, continuous and $2\pi$-periodic on the real line. Suppose there exist real constant symmetric $A,B \in \mathbb{R}^{N\times N}$ such that \begin{equation}\label{condQ} A \leq Q(t) \leq B, \quad t\in [0,2\pi], \end{equation} and such that if $\lambda_{1}\leq \lambda_2 \leq \dots \leq \lambda_{N}$ and $\mu_{1} \leq \mu_2 \leq \dots \leq \mu_{N}$ denote the eigenvalues of $A$ and $B$ respectively and there exist integers $N_{k} \geq 0$, $k=1, \dots, N$, such that \begin{equation}\label{condauto} N_{k}^2 < \lambda_{k} \leq \mu_{k} < (N_{k}+1)^2. \end{equation} Then, there exist no non-trivial $2\pi-$periodic solution of the vector differential equation \begin{equation}\label{Lmono} w'' + Q(t)w = 0. \end{equation} \end{theorem} \begin{proof}[Proof of Theorem \ref{prop2}] Let us consider the operator \begin{gather*} F: H^2_{\rm per} \times \mathbb{R}^N \to L^2,\\ (u, \overline{p}) \mapsto u''+ \nabla G(u)-\widetilde{p} -\overline{p}, \end{gather*} then clearly $F(u_0, \overline{p}_0)=0$. On the other hand, $F$ is Fr\'echet differentiable, and its differential with respect to $u$ at $(u_0,\overline{p}_0)$ is computed by \begin{align*} D_{u}F(u_0, \overline{p}_0)(\varphi) &=\lim_{t\to 0} \frac{F(u_0+t\varphi, \overline{p}_0)-F(u_0, \overline{p}_0)}{t} \\ &=\lim_{t\to 0} \frac{t\varphi '' + \nabla G(u_0+t\varphi) - \nabla G(u_0)}{t} \\ &=\varphi''+\lim_{t\to 0} \frac{\nabla G(u_0+t\varphi)- \nabla G(u_0)}{t}\\ &=\varphi''+d^2G(u_0)\varphi \end{align*} Taking $Q(t)=d^2G(u_0(t))$ in Theorem \ref{formas} we deduce that $D_{u}F(u_0, \overline{p}_0): H^2_{\rm per} \to L^2$ is a monomorphism; furthermore, from the Fredholm Alternative (see e. g. \cite{C}) we conclude that $D_{u}F(u_0, \overline{p}_0)$ is an isomorphism. By the Implicit Function Theorem (see \cite[Theorem 1.6]{T}), there exists an open neighborhood $U$ of $\overline{p}_0$ and a unique function $u: U\to H^2_{\rm per}$ such that $$ F(u(\overline{p}), \overline{p})=0,\quad \text{for all }\overline{p} \in U. $$ Thus $U\subset \mathcal{I}(\widetilde{p})$ and the proof is complete. \end{proof} \begin{remark} \label{rmk3.3} \rm A simple computation shows that a similar result is obtained when $d^2G(u_0(t))$ lies at the left of the first eigenvalue. Indeed, it suffices to assume: \begin{enumerate} \item $d^2G(u_0(t)) \leq 0$ for all $t$. \item There exists $A\subset [0,T]$ with $meas(A) >0$ such that $d^2G(u_0(t)) < 0$ for $t\in A$. \end{enumerate} As before, it suffices to prove that $L\varphi:= \varphi'' + d^2G(u_0)\varphi$ is a monomorphism. Suppose that $L\varphi=0$, then $$ 0=-\int^T_0L\varphi(t)\cdot \varphi(t)\, dt = \int^T_0| \varphi'(t)|^2\, dt - \int^T_0 d^2G(u_0(t))\varphi(t)\cdot\varphi(t)\, dt, $$ Then $$ \int^T_0| \varphi'(t)|^2\, dt = \int^T_0 d^2G(u_0(t))\varphi(t)\cdot\varphi(t)\, dt \le \int_{A} d^2G(u_0(t))\varphi(t)\cdot\varphi(t)\, dt, $$ and we conclude that $\varphi \equiv 0$. \end{remark} The following corollary is immediate. \begin{corollary} \label{coro3.1} Let $\widetilde{p}\in \widetilde{L}^2$ and assume that $$ d^2G(u) < 0\, \text{ for all } \, u\in \mathbb{R}^N $$ or that $$ A\leq d^2G(u)\leq B\, \text{ for all } \, u\in \mathbb{R}^N $$ with $A$ and $B$ as in Theorem \ref{prop2}. Then $\mathcal{I}(\widetilde{p})$ is open. \end{corollary} \section{Continuity of the function $\mathcal{I}$} \label{periodic} In this section we assume that $\nabla G$ is periodic and give a characterization of the set $\mathcal{I}(\widetilde{p})$ which, in particular, will allow us to prove the continuity of the function $\mathcal{I}:\widetilde{L}\to \mathcal K(\mathbb{R}^N)$, where $\mathcal K(\mathbb{R}^N)$ denotes the set of compacts subsets of $\mathbb{R}^N$ equipped with the Hausdorff metric. In fact, we prove a little more. \begin{theorem}\label{castro} Assume that $G\in C^2(\mathbb{R}^N,\mathbb{R})$ satisfies: \begin{enumerate} \item $\nabla G$ is periodic, that is: for every $j=1, \dots, N$ there exists $T_{j}>0$ such that $\nabla G(u+T_{j}e_{j})=\nabla G(u)$. \item There exists a discrete set $S\subset \mathbb{R}^N$ such that \begin{equation} \label{hipoG} \left( \nabla G(u)- \nabla G(v) \right)\cdot ( u-v ) < \Big(\frac T{2\pi}\Big)^2\|u-v\|_2^2\quad \text{for } u, v \in \mathbb{R}^N\backslash S. \end{equation} \end{enumerate} Then for every $\widetilde{p}\in \widetilde{L}^2$ there exists a periodic function $F_{\widetilde{p}}\in C(\mathbb{R}^N,\mathbb{R}^N)$ such that $\mathcal{I}(\widetilde{p})=\operatorname{Im}(F_{\widetilde{p}})$. Furthermore, if $\widetilde{p}_n\to \widetilde{p}$ weakly in $\widetilde{L}^2$, then $\mathcal{I}(\widetilde{p}_n)\to \mathcal{I}(\widetilde{p})$ for the Hausdorff metric. \end{theorem} \begin{remark} \label{rmk4.2} \rm In particular, it follows that $\mathcal{I}(\widetilde{p})$ is compact and arcwise connected. This has been proven also by topological methods in \cite{tesis}. As mentioned in the introduction, we also know that $\mathcal{I}(\widetilde{p})\subset \operatorname{co}(\operatorname{Im}(\nabla G))$. \end{remark} For convenience, let us consider the decomposition $H^1_{\rm per} = \mathbb{R}^N \oplus \tilde H^1_{\rm per}$, where $\tilde H^1_{\rm per}= H^1_{\rm per}\cap \widetilde{L}^2$, and denote the functional defined in section \ref{basic} by $J_p:H^1_{\rm per}\to \mathbb{R}$. The proof of Theorem \ref{castro} shall be based on a series of lemmas. From now on, we shall assume that all the preceding assumptions on $G$ are satisfied. \begin{lemma}\label{lema1} For each $x\in \mathbb{R}^N$ and $p\in L^2([0,T],\mathbb{R}^N)$, there exists a unique $\phi(x, p)\in \tilde H^1_{\rm per}$ such that \begin{equation} \label{condj} DJ_p(x+\phi(x,p))(v)=0 \quad \text{for all } v\in \tilde H^1_{\rm per}. \end{equation} Moreover, the function $\phi(\cdot, p): \mathbb{R}^N \to \tilde H^1_{\rm per}$ is continuous. \end{lemma} \begin{proof} Let us first prove the uniqueness of $\phi(x,p)$. Suppose $u_{1}$, $u_2 \in \tilde H^1_{\rm per}$ are such that $$ DJ_p(x+u_{1})(v)=0=DJ_p(x+u_2)(v) \quad \text{for all }v\in \tilde H^1_{\rm per}. $$ Taking $v=u_1-u_2$, using \eqref{dif} it follows that \begin{equation} \int^T_0 |(u_{1}-u_2)'|^2 \, dt=\int_0^T (\nabla G(x+u_{1}) -\nabla G(x+u_2))\cdot(u_{1}-u_2)\, dt. \end{equation} This fact, \eqref{wirti} and \eqref{hipoG} imply that $u_{1}=u_2$. Next we prove the existence of $\phi(x,p)$. Let $I_x:\tilde H^1_{\rm per}\to \mathbb{R}$ be the functional defined by $I_x(v)=J_p(x+v)$, then \begin{equation} \label{coerc} \begin{split} I_x(v) &= \frac{1}{2} \|v'\|_2^2 + \int^T_0 p(t)\cdot (x+v(t))-G(x+v(t)) \, dt \\ & \ge \frac{1}{2} \|v'\|_2^2 - \|\widetilde{p}\|_2\|v\|_2 + T(\overline{p}\cdot x - \|\nabla G\|_\infty) \end{split} \end{equation} It follows that $I_x$ is coercive and hence it achieves an absolute minimum, which satisfies \eqref{condj}. Finally, let $x_n\to x$ and suppose that $\phi(x_n,p)\not\to \phi(x,p)$. From \eqref{coerc}, the sequence $\left(\phi(x_n,p)\right)_n$ is bounded in $\tilde H^1_{\rm per}$. Taking a subsequence, if necessary, we may assume that it converges weakly to some $w\in H^1_{\rm per}$, uniformly and $\|\phi(x_n,p)-\phi(x,p)\|_{H^1}\ge \epsilon >0$ for all $n$. Passing to the limit in the equalities $$ DJ_p(x_n+\phi(x_n,p))(v) = 0 \quad \text{for all } v\in \tilde H^1_{\rm per}$$ we deduce that $DJ_p(x+w)(v) = 0$ for all $v\in \tilde H^1_{\rm per}$ and hence $w = \phi(x,p)$. Moreover, as $$ J_p(x_n + \phi(x_n,p))\le J_p(x_n + \phi(x,p)) \quad \text{and} \quad J_p(x + \phi(x,p))\le J_p(x + \phi(x_n,p)) $$ for all $n$, we deduce that $$ \limsup_{n\to \infty} \int_0^T |\phi(x_n,p)'|^2\, dt \le \int_0^T |\phi(x,p)'|^2 \, dt \le \liminf_{n\to \infty} \int_0^T |\phi(x_n,p)'|^2\, dt $$ and hence $\|\phi(x_n,p)'\|_2\to \|\phi(x,p)'\|_2$. Thus, $$ \|\phi(x_n,p)'-\phi(x,p)'\|_2^2 = \|\phi(x_n,p)'\|_2^2 + \|\phi(x,p)'\|_2^2 - 2\int_0^T \phi(x_n,p)'\cdot \phi(x,p)'\, dt \to 0 $$ as $n\to \infty$, which contradicts the fact that $\phi(x_n,p)\not\to \phi(x,p)$. \end{proof} \begin{lemma}\label{lema2} The function $\phi(\cdot,p)$ depends only on $\widetilde{p}$. \end{lemma} \begin{proof} Let $c \in \mathbb{R}^N$, then \begin{align*} DJ_{p+c}(x+\phi(x,p))(v) &= \int^T_0\phi(x,p)'\cdot v'-\nabla G(x+\phi(x,p))\cdot v+(p+c)\cdot v \, dt\\ &=\int^T_0\phi(x,p)'\cdot v'-\nabla G(x+\phi(x,p))\cdot v+p\cdot v\, dt=0 \end{align*} for all $v\in \tilde H^1_{\rm per}$. From uniqueness, we deduce that $\phi(\cdot, p)=\phi(\cdot,p+c)$. \end{proof} Let us denote by $\tilde J_p: \mathbb{R}^N \to \mathbb{R}$ the function defined by $$ \tilde J_p(x)= J_p(x+\phi(x,p)). $$ It is readily seen that $\tilde J_p\in C^1(\mathbb{R}^N,\mathbb{R})$ and \begin{equation}\label{condtilde} \nabla \tilde J_p(x)\cdot y = DJ_p(x+\phi(x,p))(y + v) \quad \text{for all } y\in \mathbb{R}^N,\, v\in \tilde H^1_{\rm per}. \end{equation} The following lemma will allow us to reduce the problem of finding a critical point in $H_{\rm per}^1$ to a finite-dimensional problem. \begin{lemma} \label{lema3} The element $x+v \in \mathbb{R}^N \oplus \tilde H^1_{\rm per}$ is a critical point of $J_p$ if and only if $v=\phi(x,p)$ and $x$ is a critical point of $\tilde J_p$. \end{lemma} \begin{proof} By Lemma \ref{lema1}, if $x+v$ is a critical point of $J_p$, then $v=\phi(x,p)$. From \eqref{condtilde}, $\nabla \tilde J_p(x)\cdot y=0$ for every $y\in \mathbb{R}^N$ and hence $x$ is a critical point of $\tilde J_p$. Conversely, suppose $v=\phi(x,p)$ and $\nabla \tilde J_p(x)=0$. For $u\in H^1_{\rm per}$, let us write $u= \overline u + \tilde u$ with $\overline u \in \mathbb{R}^N$ and $\tilde u \in \tilde H^1_{\rm per}$. Then $$ DJ_p(x+v)(u) = DJ_p(x+\phi(x,p))(\overline u + \tilde u) = \nabla \tilde J_p(x)\cdot \overline u = 0,$$ so $x+v$ is a critical point of $J_p$. \end{proof} \begin{lemma} The function $\phi(\cdot,p)$ is periodic. \end{lemma} \begin{proof} Let $x\in \mathbb{R}^N$. From the periodicity of $\nabla G$ we deduce that $$ DJ_p(x+T_{j}e_{j}+\phi(x,p))(v) = DJ_p(x+\phi(x,p))(v)=0 $$ for all $v\in \tilde H^1_{\rm per}$. By Lemma \ref{lema1}, $\phi(x+T_{j}e_{j}, p)=\phi(x,p)$. \end{proof} The following proposition will provide the proof of Theorem \ref{castro}. \begin{proposition} \label{prop4.1} Let $\widetilde{p}\in \widetilde{L}^2$ and define the function $F_{\widetilde{p}}: \mathbb{R}^N \to \mathbb{R}^N$ by $$ F_{\widetilde{p}}(x)=\int^{2\pi}_0 \nabla G(x+\phi(x,\widetilde{p}(t)))\, dt. $$ Then $F_{\widetilde{p}}$ is continuous and $\mathcal{I}(\widetilde{p})= \operatorname{Im}(F_{\widetilde{p}})$. Moreover, if $\widetilde{p}^n \in \widetilde{L}^2$ converges weakly to some $\widetilde{p}\in \widetilde{L}^2$, then $\mathcal{I}(\widetilde{p}^n)$ converges to $\mathcal{I}(\widetilde{p})$ for the Hausdorff topology. \end{proposition} \begin{proof} The continuity of $F_{\widetilde{p}}$ is clear from the continuity of $\phi(\cdot, \widetilde{p})$ and the embedding $\tilde H^1_{\rm per}\hookrightarrow C([0,T],\mathbb{R}^N)$. Let us prove that $\mathcal{I}(\widetilde{p})= \operatorname{Im}( F_{\widetilde{p}})$. According to \eqref{condtilde}, Lemma \ref{lema2} and Lemma \ref{lema3}, problem \eqref{ecua1} has a weak solution if and only if there exists $x\in \mathbb{R}^N$ such that $D J_p(x+\phi(x,\widetilde{p}))(y)=0$ for every $y\in \mathbb{R}^N$. From \eqref{dif}, this is equivalent to $$ 0= \int^T_0 -\nabla G(x+ \phi(x,\widetilde{p}))\cdot y+ p\cdot y\, dt = y\cdot \int^T_0 -\nabla G(x+\phi(x,\widetilde{p}))+ \overline{p} \, dt $$ for all $y\in \mathbb{R}^N$. Thus, the problem has a solution if and only if $$ \overline{p} = \int_0^T \nabla G(x+ \phi(x,\widetilde{p}))\, dt, $$ for some $x\in \mathbb{R}^N$; that is, $\overline{p} \in \operatorname{Im}(F_{\widetilde{p}})$. Finally, suppose that $\widetilde{p} ^n\to \widetilde{p}$ weakly in $\widetilde{L}^2$ and denote $J_{n}:= J_{\widetilde{p}^n}$, $J:=J_{\widetilde{p}}$, $\phi_{n}(\cdot):=\phi(\cdot, \widetilde{p}^n)$, $\phi(\cdot):= \phi(\cdot, \widetilde{p})$, $F_n:=F_{\widetilde{p}^n}$ and $F:=F_{\widetilde{p}}$. We claim that $F_{n} \to F$ pointwise. Indeed, for fixed $x\in \mathbb{R}^N$ proceeding as in the proof of Lemma \ref{lema1} it is easy to see that if $n\to \infty$, then $\phi_{n}(x) \to \phi(x)$. As $\nabla G$ is continuous, we deduce from the Lebesgue's dominated convergence theorem that $F_{n}(x)\to F(x)$. To prove that $\mathcal{I}(\widetilde{p}^n) \to \mathcal{I}(\widetilde{p})$ as ${n\to \infty}$ for the Hausdorff topology, we need to see that: \begin{itemize} \item[(i)] $\sup_{\overline q^n \in \mathcal{I}(\widetilde{p}^n)} \operatorname{dist}(\overline q^n, \mathcal{I}(\widetilde{p})) \to 0$, \item[(ii)] $\sup_{\overline q \in \mathcal{I}(\widetilde{p})} \operatorname{dist}(\overline q, \mathcal{I}(\widetilde{p}^n)) \to 0$. \end{itemize} For (i), denote $S_{n}=\sup_{\overline q^n \in \mathcal{I}(\widetilde{p}^n)}\operatorname{dist}(\overline q^n, \mathcal{I}(\widetilde{p}))$ and let $\overline{p}^n \in \mathcal{I}(\widetilde{p}^n)$ be chosen in such a way that $\operatorname{dist}(\overline{p}^n, \mathcal{I}(\widetilde{p})) \geq S_{n} - \frac{1}{n}$. We shall prove that $\operatorname{dist}(\overline{p}^n, \mathcal{I}(\widetilde{p}))\to 0$. By contradiction, suppose there exists a subsequence, still denoted $\{ \overline{p}^n \}$, such that \begin{equation} \label{ce} \operatorname{dist}(\overline{p}^n, \mathcal{I}(\widetilde{p}))\geq \epsilon >0. \end{equation} Moreover, we know that $\mathcal{I}(\widetilde{p})\subset \operatorname{co}(\operatorname{Im}(\nabla G))$; in particular, taking a convergent subsequence if necessary we may suppose that $\overline{p}^n\to \overline{p}$ for some $\overline{p}\in \mathbb{R}^N$. For each $n$, let $u_{n} \in H^1_{\rm per}$ be a solution of the problem for $\overline{p}^n$. From the periodicity of $\nabla G$, we may assume that the sequence $\{ \overline u_{n} \}$ is bounded in $\mathbb{R}^N$. Thus, $\{u_{n}\}$ is bounded in $H^1_{\rm per}$ and \begin{equation} \label{uk} \int^T_0 u'_{n}\cdot v' - \nabla G(u_{n})\cdot v +(\widetilde{p}^n +\overline{p}^n)\cdot v \,dt = 0 \end{equation} for all $v\in H^1_{\rm per}$. Taking again a subsequence, we may assume that $u_{n}\to u_0$ weakly in $H^1_{\rm per}$ and hence $$ \int^T_0 u'_0\cdot v' - \nabla G(u_0)\cdot v + (\widetilde{p}+\overline{p})\cdot v \, dt = 0 $$ for all $v\in H^1_{\rm per}$. Then $u_0$ is a weak solution of \eqref{ecua1} with $p=\widetilde{p}+\overline{p}$ and $\overline{p} \in \mathcal{I}(\widetilde{p})$, which contradicts \eqref{ce}. Thus $\operatorname{dist}(\overline{p}^n, \mathcal{I}(\widetilde{p}))\to 0$ and consequently $S_{n} \to 0$. Next we prove (ii). Denote now $S_{n} =\sup_{\overline q \in \mathcal{I}(\widetilde{p})} \operatorname{dist}(\overline q, \mathcal{I}(\widetilde{p}^n))$ and take $\overline q^n\in \mathcal{I}(\widetilde{p})$ such that $\operatorname{dist}(\overline q^n, \mathcal{I}(\widetilde{p}^n)) \geq S_{n} -\frac{1}{n}$. As before, suppose there exists a subsequence, still denoted $\{ \overline q^n\}$, such that \begin{equation} \label{enejota} \operatorname{dist}(\overline q^n, \mathcal{I}(\widetilde{p}^n)) \geq \epsilon >0. \end{equation} Passing to a subsequence if necessary, there exist $\overline q \in \mathcal{I}(\widetilde{p})=\operatorname{Im}(F)$ and $n_{1}\in \mathbb{N}$ such that $\operatorname{dist}(\overline q^n ,\overline q)< \frac{\epsilon}{2}$ for all $n\geq n_{1}$. Fix $x_0\in \mathbb{R}^N$ such that $F(x_0)=\overline q$ and let $\overline{p}^n=F_{n}(x_0) \in \mathcal{I}(\widetilde{p}^n)$. As $F_{n}(x_0) \to F(x_0)$, there exists $n_2 \in \mathbb{N}$ such that $\operatorname{dist}(\overline{p}^n, \overline q)< \frac{\epsilon}{2}$ for all $n\geq n_2$. Take $n_0= \max \{n_{1}, n_2 \}$ and hence \[ \operatorname{dist}(\overline q^n, \mathcal{I}(\widetilde{p}^n) \leq\operatorname{dist}(\overline q^n, \overline{p}^n) \leq\operatorname{dist}(\overline q^n, \overline q) +\operatorname{dist}(\overline q, \overline{p}^n)< \epsilon \] for $n\ge n_0$. This contradicts \eqref{enejota}, so we conclude that $S_{n} \to 0$. \end{proof} \section{Characterization of $\mathcal{I}$ for convex $G$} \label{convex} In this section, we shall assume that $G$ is a strictly convex function, namely $$ G(sx+(1-s)y)< sG(x)+(1-s)G(y)\quad \text{for all } s\in (0,1),\; x, y\in \mathbb{R}^N. $$ Our main result reads as follows. \begin{theorem} \label{prop1} Assume that: \begin{enumerate} \item There exist $\alpha < \left(\frac T{2\pi}\right)^2$ and $\beta\in \mathbb{R}$ such that \begin{equation} \label{k} G(u) \leq \frac{\alpha}{2} |u|^2+ \beta \quad \text{for all $u\in \mathbb{R}^N$} \end{equation} \item For every $a\in \mathbb{R}^N$ there exists $r_0>0$ such that \begin{equation} \label{partial} \frac{\partial G}{\partial w}(rw+x)\geq \frac{\partial G}{\partial w}(a) \end{equation} for all $r\geq r_0$, $w\in S^{n-1}$ and $|x|\leq C$, where $C=C(a,\widetilde{p})$ is the constant defined below in \eqref{const}. \end{enumerate} Then $\mathcal{I}(\widetilde{p})=\operatorname{Im}(\nabla G)$. \end{theorem} \begin{proof} Firstly, we shall prove the inclusion $\operatorname{Im}(\nabla G) \subseteq \mathcal{I}(\widetilde{p})$. For simplicity, from the rescaling $v(t)=u(\frac T{2\pi} t)$ we may assume that $T=2\pi$. Let $K: \widetilde{L}^2\to H^2\cap \widetilde{L}^2$ be the inverse of the operator $Lu:=u''$, namely $Kh= u$, where $u$ is the unique solution of the problem \begin{gather*} u''=h\\ u(0)=u(2\pi), \quad u'(0)=u'(2\pi)\\ \overline u=0. \end{gather*} \noindent \emph{Claim 1: $\int^{2\pi}_0Kh(t)\cdot h(t)\, dt + \int^{2\pi}_0 | h(t) |^2\, dt \geq 0$.} Indeed, from \eqref{wirti} it is seen that $$ \int_0^{2\pi} |(Kh)'(t)|^2 \, dt = -\int_0^{2\pi} Kh(t)\cdot h(t)\, dt \le \|(Kh)'\|_2\|h\|_2, $$ which implies that $\|(Kh)'\|_2\le \|h\|_2$, and hence $$ -\int_0^{2\pi} Kh(t)\cdot h(t)\, dt = \int_0^{2\pi} |(Kh)'(t)|^2 \, dt \le \|h\|^2_2. $$ For $\overline{p} \in \operatorname{Im}(\nabla G)$, fix $a \in \mathbb{R}^N$ such that $\nabla G(a)= \overline{p}$, and define the functions $$ F(t,u):=G(u)-p(t)\cdot u; $$ and, for given $\epsilon >0$, $$ F_{\epsilon}(t,u):= G(u) -p(t)\cdot u+ \frac{\epsilon}{2} | u|^2 $$ where $p(t)=\widetilde{p}(t) + \overline{p}$. Next, consider the Fenchel transform $F^{*}_{\epsilon}$ of the function $F_{\epsilon}$ defined as \begin{equation} \label{lege} F^{*}_{\epsilon}(t, v) = \max_{w\in \mathbb{R}^N} \left( v\cdot w- F_{\epsilon}(t,w) \right) . \end{equation} Observe that $F^{*}_{\epsilon}$ is well defined, since $F_{\epsilon}$ is strongly concave; hence a unique global maximum $w$ is achieved and satisfies the following properties: \begin{enumerate} \item $v=\nabla F_{\epsilon}(t,w)$, \item $w=\nabla F^{*}_{\epsilon}(t,v)$, \item $v\cdot w-F_{\epsilon}(t,w)= F^{*}_{\epsilon}$. \end{enumerate} Properties $1$ and $2$ are known as \emph{Fenchel duality} (see \cite{MW}). Define the functional $I_\epsilon:\widetilde{L}^2\to \mathbb{R}$ given by $$ I_{\epsilon} (v)=\int^{2\pi}_0 \frac 12 Kv(t)\cdot v(t) + F^{*}_{\epsilon}(t,v(t))\, dt. $$ From \eqref{lege} and \eqref{k}, \begin{align*} F^{*}_{\epsilon}(t,v)\geq |v |^2 - F_{\epsilon}(t,v) &= |v |^2+ p\cdot v - \frac{\epsilon}{2} |v |^2 - G(v) \\ & \geq |v |^2 + p\cdot v - \frac{\epsilon + \alpha}{2} |v |^2 - \beta \end{align*} and using Claim 1, Cauchy-Schwarz Inequality and the fact that $v\in \widetilde{L}^2$ we deduce: $$ I_{\epsilon}(v) \geq -\frac 12 \int^{2\pi}_0 |v(t) |^2\, dt + \int^{2\pi}_0 |v(t) |^2 + \widetilde{p}(t)\cdot v(t) -\frac{\epsilon + \alpha}{2} |v(t) |^2 - \beta\, dt; $$ that is, \begin{equation} \label{coercive} I_{\epsilon}(v) \geq \frac{1-\alpha - \epsilon}{2} \| v \|^2_2 - \|\widetilde{p}\|_2\|v\|_2 - 2\pi \beta. \end{equation} Thus $I_{\epsilon}$ is coercive for $\epsilon < 1-\alpha$ and hence it achieves a minimum $u_{\epsilon}$. As $K$ is self-adjoint, it is easy to verify that $$ \int^{2\pi}_0 [K u_{\epsilon}(t) + \nabla F^{*}_{\epsilon}(t, u_{\epsilon}(t))] \cdot \varphi(t)\, dt =0, \;\; \text{for all }\varphi \in \widetilde{L}^2. $$ Then $K(u_{\epsilon}) + \nabla F^{*}_{\epsilon}(s, u_{\epsilon}) =A \in \mathbb{R}^N$. Let $v_{\epsilon}=\nabla F^{*}_{\epsilon}(s, u_{\epsilon})= A-K(u_{\epsilon})$, then by the Fenchel duality $u_{\epsilon}= \nabla F_{\epsilon}(s, v_{\epsilon})$. In other words, $u_{\epsilon}= \nabla G(v_{\epsilon})- p(t) + \epsilon v_{\epsilon}$. On the other hand, $v''_{\epsilon}=(-K(u_{\epsilon}))''=-u_{\epsilon}$; hence, $v_{\epsilon}$ satisfies \begin{equation} \label{vepsilon} \begin{gathered} v''_{\epsilon}+ \nabla G(v_{\epsilon})+ \epsilon v_{\epsilon} = p(t)\\ v_{\epsilon}(0)=v_{\epsilon}(2\pi), \; \; v_{\epsilon}'(0)=v_{\epsilon}'(2\pi). \end{gathered} \end{equation} Moreover, if $F^*$ denotes the Legendre transform of $F$ defined by $$ F^*(t,v)= \sup_{w\in \mathbb{R}^N} \left( v\cdot w- F_{\epsilon}(t,w) \right) $$ then it is obvious that $F^{*}_{\epsilon} \leq F^{*}$. As $u_\epsilon$ is the minimum, it follows that \begin{equation} \label{uepsilon} I_{\epsilon}(u_{\epsilon}) \leq I_{\epsilon} (-\widetilde{p}) = \int^{2\pi}_0 \frac{1}{2} K\widetilde{p}(t) \cdot \widetilde{p}(t) + F^{*}(t, -\widetilde{p}(t))\, dt. \end{equation} For fixed $t$, let $\Psi (y):= -\widetilde{p}\cdot y-F(t,y)= \overline{p}\cdot y-G(y)$, then $$ \nabla \Psi(y)= - \widetilde{p} - \nabla F(t,y)= \overline{p} -\nabla G(y). $$ Thus, $a$ is a critical point of $\Psi$ and, as $\Psi$ is strictly concave, we conclude that $a$ is the absolute maximum. Then $$ -\widetilde{p}\cdot a-F(t,a)= \max_{w\in \mathbb{R}^N} \left( -\widetilde{p}(t)\cdot w-F(t,w) \right)= F^{*}(t, -\widetilde{p}(t)). $$ Hence, from \eqref{uepsilon} and the fact that $\widetilde{p}\in \widetilde{L}^2$ we obtain: $$ I_{\epsilon}(u_{\epsilon}) \leq \int^{2\pi}_0 \frac{1}{2} K\widetilde{p}(t) \cdot \widetilde{p}(t) -F(t,a)\, dt = 2\pi \left(a\cdot \nabla G(a) - G(a)\right) - \frac 12 \|(K\widetilde{p})'\|_2^2. $$ Fixing $c < (1-\alpha)/2$, we conclude from \eqref{coercive} that if $\epsilon$ is small enough then $$ c \|u_{\epsilon} \|^2_2 - \|\widetilde{p}\|_2\|u_{\epsilon}\|_2 \le 2\pi \left(a\cdot \nabla G(a) - G(a) + \beta \right) - \frac 12 \|(K\widetilde{p})'\|_2^2. $$ As $v''_{\epsilon}=-u_{\epsilon}$, it follows that $\tilde v_{\epsilon}$ is bounded for the $H^2$ norm; in particular, \begin{equation} \|v_{\epsilon} \|_{\infty}\le C \label{const} \end{equation} for some constant $C$, depending only on ${\widetilde{p}}$ and $a$. Let us prove now that $\overline v_{\epsilon}$ is bounded. By direct integration of \eqref{vepsilon} we obtain: \begin{equation} \label{inte} \frac{1}{2\pi} \int^{2\pi}_0 \nabla G(v_{\epsilon}(t))dt + \epsilon \overline v_{\epsilon} = \overline{p}. \end{equation} Writing $\overline v_{\epsilon}= rw$, where $r= | \overline v_{\epsilon} |$ and $| w| =1$, and multiplying \eqref{inte} by $w$, we obtain $$ \epsilon r + \frac{1}{2\pi} \int_0^{2\pi} \frac{\partial G}{\partial w}(rw + \tilde v_{\epsilon}(t))\, dt = \overline{p}\cdot w = \nabla G(a)\cdot w= \frac{\partial G}{\partial w}(a). $$ As $|\tilde v_{\epsilon}(t)|\le C$, for $r\ge r_0$ inequality \eqref{partial} yields: $$ 0=\epsilon r +\frac{1}{2} \int^{2\pi}_0 \Big( \frac{\partial G}{\partial w}(rw + \tilde v_{\epsilon}(t)) -\frac{\partial G}{\partial w}(a) \Big)dt \geq \epsilon r, $$ a contradiction. So, $| \overline v_{\epsilon} |\le r_0$ and $v_{\epsilon}$ is bounded for the $H^2$ norm. From the compact embedding $H^2([0, 2\pi ], \mathbb{R}^N)\hookrightarrow C^1([0, 2\pi ], \mathbb{R}^N)$, there exists a sequence $\{v_{\epsilon_{n}} \}_{n\in \mathbb{N}}$ that converges in $C^1([0, 2\pi ], \mathbb{R}^N)$ to some function $v$. From \eqref{vepsilon}, $$ \int^{2\pi}_0 \Big( v''_{\epsilon_{n}}(t)+ \nabla G(v_{\epsilon_{n}}(t))+\epsilon_{n}v_{\epsilon_{n}}(t) \Big) \cdot \varphi(t)\,dt = \int^{2\pi}_0 p(t)\cdot \varphi(t)\, dt \quad \forall\, \varphi \in \widetilde{L}^2. $$ Integrating by parts and passing to the limit, we obtain: $$ -\int^{2\pi}_0v'(t)\cdot \varphi'(t)\, dt + \int^{2\pi}_0 \nabla G(v(t))\cdot\varphi(t)\, dt = \int^{2\pi}_0 p(t)\cdot \varphi(t)\,dt. $$ Then $v$ is a solution of \eqref{ecua1}. Finally, let us prove that $\mathcal{I}(\widetilde{p})\subseteq \operatorname{Im}(\nabla G)$. As previously mentioned, we know that $\mathcal{I}(\widetilde{p})\subseteq \operatorname{co} (\operatorname{Im}(\nabla G))$, so it remains to see that $\operatorname{Im}(\nabla G)$ is convex. \noindent \emph{Claim 2}: If $F\in C^1(\mathbb{R}^N,\mathbb{R})$ is strictly convex, then $$ 0\in \operatorname{Im}(F)\quad \iff \lim_{|x | \to +\infty} F(x)= +\infty. $$ The sufficiency is obvious. In order to prove the necessity, assume that $\nabla F(x_0)=0$ for some $x_0\in \mathbb{R}^N$ and for each $w\in S^{n-1}$ define $\Phi_{w} (t):= \frac{\partial F}{\partial w}(x_0+tw)$. From the convexity of $F$ we deduce that $\Phi_{w}$ is strictly increasing. Furthermore, the function $\Phi:S^{n-1}\times [0,+\infty)\to \mathbb{R}$ given by $\Phi(w,t):=\Phi_{w}(t)$ is continuous and $\Phi(w,1)>0$ for all $w\in S^{n-1}$. Hence, there exists a constant $c>0$, such that $\Phi_{w}(1) \geq c>0$ for all $w\in S^{n-1}$. As $\Phi_{w}$ is strictly increasing, we conclude that $\Phi_{w}(t) > c$ for all $t >1$. Thus, $$ F(x_0+Rw)-F(x_0+w)=R\nabla F(x_0+ \xi w)\cdot w = R\frac{\partial F}{\partial w}(x_0+\xi w)\geq cR. $$ We conclude that $F(x_0+Rw)\geq F(x_0+w)+cR$ and the claim is proved. Next, let us consider $y_{1}$, $y_2\in Im (\nabla G)$ and $y=a_{1}y_{1}+a_2y_2$, with $a_{1}+a_2=1$ and $a_{1}$, $a_2\geq 0$. Define $$ F(x)=G(x)-y\cdot x= a_{1} \left( G(x)-y_{1}\cdot x \right) + a_2( G(x)-y_2\cdot x). $$ As $G(x)-y_{1}\cdot x$ and $G(x)-y_2\cdot x$ are strictly convex, it follows from Claim 2 that both functions tend to $+\infty$ as $|x|\to \infty$, and hence \begin{equation} \label{lim} \lim_{|x | \to +\infty} F(x)= +\infty. \end{equation} Using Claim 2 again, \eqref{lim} implies that $0\in \operatorname{Im}(\nabla F)=\operatorname{Im}( \nabla G-y)$, then $y \in \operatorname{Im}(\nabla G)$ and so completes the proof. \end{proof} \subsection*{Acknowledgments} This research was partially supported by the projects PIP 11220090100637 from CONICET, and UBACyT 20020090100067. 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