This is a multi-part message in MIME format. ---------------1406160953428 Content-Type: text/plain; name="14-48.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="14-48.comments" 24 pages, 4 figures ---------------1406160953428 Content-Type: text/plain; name="14-48.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="14-48.keywords" minimization, metric graphs, rearrangement, NLS ---------------1406160953428 Content-Type: application/x-tex; name="AdamiSerraTilli.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="AdamiSerraTilli.tex" \documentclass[a4paper,leqno,11pt]{article} \usepackage{amsmath,amssymb,amsthm,%showkeys, verbatim%,refcheck } \usepackage{tikz} \usetikzlibrary{decorations.pathreplacing} \newcommand\n{\noindent} \newcommand\be{\begin{equation}} \newcommand\ee{\end{equation}} \newcommand\R{{\mathbb R}} \newcommand\C{{\mathbb C}} \newcommand\implica{\quad\Rightarrow\quad} \newcommand\G{\mathcal G} \newcommand\Gbis{\mathcal H} \newcommand\lgr{{L^2(\G)}} \newcommand\lp{{L^p(\G)}} \newcommand\V{\textsc{v}} \newcommand\U{\textsc{u}} \newcommand\W{\textsc{w}} \newcommand\NN{{\mathbb N}} \newcommand\f{\frac} \newcommand\vv{\textsc{v}} \newcommand\ww{\textsc{w}} \newcommand\mis{\mathop{\rm meas}} \newcommand\eps{\varepsilon} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{problem}{Problem} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem*{remark*}{Remark} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \tikzstyle{nodo}=[circle,draw,fill,inner sep=0pt,minimum size=\widthof{k}] \tikzstyle{infinito}=[circle,inner sep=0pt,minimum size=0mm] \date{} \newcommand\iii{$\infty$} \newcommand\IIname{\textsc{l}} \newcommand\OOname{\textsc{r}} \newcommand\VVname{\textsc{v}} \newcommand\II[1]{\IIname_#1} \newcommand\OO[1]{\OOname_#1} \newcommand\N{\mathop{\mathcal N}} \newcommand\dx{{\,dx}} \newcommand\dt{{\,dt}} \newcommand\inti{{\int_I}} \title{NLS ground states on graphs} \author{Riccardo Adami\thanks{Author partially supported by the FIRB 2012 project ``Dispersive dynamics: Fourier Analysis and Variational Methods".}, Enrico Serra\thanks{Author partially supported by the PRIN 2012 project ``Aspetti variazionali e perturbativi nei problemi differenziali nonlineari".}, Paolo Tilli \\ \ \\{\small Dipartimento di Scienze Matematiche ``G.L. Lagrange'', Politecnico di Torino } \\ {\small Corso Duca degli Abruzzi, 24, 10129 Torino, Italy}} \begin{document} \maketitle \begin{abstract} We investigate the existence of ground states for the subcritical NLS energy on metric graphs. In particular, we find out a topological assumption that guarantees the nonexistence of ground states, and give an example in which the assumption is not fulfilled and ground states actually exist. In order to obtain the result, we introduce a new rearrangement technique, adapted to the graph where it applies. Owing to such a technique, the energy level of the rearranged function is improved by conveniently mixing the symmetric and monotone rearrangement procedures. \end{abstract} \noindent{\small AMS Subject Classification: 35R02, 35Q55, 81Q35, 49J40, 58E30.} \smallskip \noindent{\small Keywords: Minimization, metric graphs, rearrangement, nonlinear Schr\"odinger \\ \hbox{} \hskip 1.65cm Equation.} \section{Introduction} In this paper we investigate the existence of a ground state for the NLS energy functional \begin{equation} \label{NLSe} E (u,\G) = \frac 1 2 \| u' \|^2_{L^2 (\G)} - \frac 1 p \| u \|^p_{L^p (\G)} \end{equation} with the {\em mass constraint} \begin{equation} \label{mass} \| u \|^2_{L^2 (\G)} \ = \ \mu, \end{equation} where $\mu>0$ and $p\in (2,6)$ are given numbers and $\G$ is a \emph{connected metric graph}. Here we present a rather informal description of the problem and of the main results of the paper, whereas a precise setting and formal definitions are given in Section~\ref{sec:defi}, A metric graph $\G$ (\cite{berkolaiko, Friedlander,Kuchment}) is essentially a one-dimensional singular variety, made up of several, possibly unbounded intervals (the edges of the graph) some of whose endpoints are glued together according to the topology of the graph. The spaces $L^p(\G)$, $H^1(\G)$ etc. are defined in the natural way. All the functions we consider are real valued: this is not restrictive, because $E(|u|,\G)\leq E(u,\G)$ and any ground state is in fact real valued, up to multiplication by a constant phase $e^{i\theta}$. When $\G=\R$ the minimization problem \begin{equation} \label{minE} \min E(u,\G),\quad u\in H^1(\G),\quad \| u\|_{L^2(\G)}^2=\mu \end{equation} is well understood and the minimizers, called \emph{solitons}, are known explicitly. The same is true when $\G=[0,+\infty)$ is a half-line (the minimizer being ``half a soliton'' of mass $2\mu$) and, to some extent, when $\G=[a,b]$ is a bounded interval. Much more interesting is the case when $\G$ is non-compact and has a nontrivial topology with multiple junctions, loops and so on (see Figure~\ref{Fig_gen}). The aim of this paper is that of studying existence and qualitative properties of solutions to \eqref{minE}, under quite general assumptions on $\G$ (for papers devoted to particular graphs, see \cite{acfn12a, acfn14,cacciapuoti}). Since when $\G$ is compact existence of minimizers for \eqref{minE} is immediate, we focus on graphs where at least one edge is unbounded (a half-line), so that the embeddings $H^1(\G)\hookrightarrow L^r(\G)$ are not compact and existence for \eqref{minE} is non-trivial. In fact, even though the infimum of $E(u,\G)$ is always trapped between two finite values (Theorem~\ref{teoremzero}), it turns out that the existence of minimizers heavily depends on the topology of $\G$: if, for instance, $\G$ consists of two half-lines with a ``double bridge" in between (Figure~\ref{Figbridge}.a) then \eqref{minE} has \emph{no solution}, while if $\G$ is a straight line with one pendant attached to it (Figure~\ref{Figbridge}.b) then minimizers \emph{do exist}. \begin{figure}[t] \begin{center} \begin{tikzpicture} [scale=1.3,style={circle, inner sep=0pt,minimum size=7mm}] % % doppio ponte con label (a) % \node at (-1.5,0) [infinito] (1) {$\infty$}; \node at (0,0)[nodo] (2) {}; \node at (1,0) [nodo] (3) {}; \node at (2.5,0) [infinito] (4) {$\infty$}; % \draw [-] (2) -- (1) ; \draw [-] (4) -- (3) ; \draw [-] (2) to [out=-40,in=-140] (3); \draw [-] (2) to [out=40,in=140] (3); % \draw (-1.5,0.8) node {(a)}; % % grafo col baffo con label (b) % \node at (4,0) [infinito] (L) {$\infty$}; \node at (5.5,0)[nodo] (O) {}; \node at (7,0) [infinito] (R) {$\infty$}; \node at (5.5,1) [nodo] (V) {}; % \draw [-] (L) -- (O) ; \draw [-] (O) -- (R) ; \draw [-] (O) -- (V) ; % \draw (4,0.8) node {(b)}; \end{tikzpicture} \end{center} \caption{(a) Two half-lines with a double-bridge in between. (b) A straight line with one pendant attached to it.} \label{Figbridge} \end{figure} Our results extend in two directions. First, we prove a nonexistence result (Theorems~\ref{teoremunoA} and \ref{teoremunoB}) for a broad family of non-compact graphs (see condition (H) in Section~\ref{sec:defi}): roughly speaking, if no cut-edge of $\G$ segregates all the half-lines of $\G$ in the same connected component, then \eqref{minE} has no solution, the only exceptions being certain graphs with a particular topology which we characterize completely (see Example~\ref{example1}). Moreover, for all these graphs %(which necessarily contain no less than two half-lines) the unattained infimum of $E(u,\G)$ coincides with the NLS energy of a soliton on the real line, for the same values of $\mu$ and $p$. The mentioned condition on $\G$, that prevents existence of minimizers in \eqref{minE}, is incompatible with the presence of a bounded pendant edge attached to $\G$. This motivates a case study when $\G$ is the simplest non-compact graph with one pendant, namely the graph in Figure~\ref{Figbridge}.b: we prove that for this particular $\G$ problem \eqref{minE} does have a solution (Theorem~\ref{teoremdue}) and we establish some qualitative properties of the minimizers (Theorem~\ref{teoremdueB}). In fact, in this case the energy level of the minimizers is \emph{strictly lower} than the energy level of a soliton on the real line, and any minimizer $u$ ---in order to reduce the energy level--- does exploit the topology of $\G$, in that $\sup u$ is attained at the tip of the pendant: indeed, the fact that concentrating mass on the pendant is energetically convenient prevents an a priori possible loss of compactness of a minimizing sequence along a half-line, and existence can be proved. In this respect, a key role is played by a new rearrangement technique, introduced in the proof of Lemma~\ref{lemmaH}. The interesting feature is that a \emph{hybrid} rearrangement of a function $u\in H^1(\G)$ is needed, adapted to the topology of $\G$: \emph{high} values of $u$ are rearranged \emph{increasingly} on the pendant, while \emph{small} values of $u$ are rearranged \emph{symmetrically} on the straight line of $\G$. It should be pointed out, however, that the presence of a bounded pendant attached to $\G$ is not enough, alone, to guarantee solutions to \eqref{minE}. The problem of characterizing all non-compact $\G$ such that \eqref{minE} has a solution is certainly a challenging one, since the topology of $\G$ alone is not enough to answer this question and, in general, also the \emph{metric} properties of $\G$ (i.e. the lengths of it edges) play a relevant role. This issue will be discussed in more detail in a forthcoming paper. \bigskip Among the physical motivations for this problem (see \cite{noja} and references therein), nowadays the most topical is probably given by the Bose-Einstein condensation (see \cite{stringari}). It is widely known that, under a critical temperature, a boson gas undergoes a phase transition that leads a large number $N$ of particles into the same quantum state, represented by the {\em wave function} that minimizes the {\em Gross-Pitaevskii energy} functional \be \label{gp} E_{GP} (\varphi, \Omega) \ = \ \| \varphi' \|^2_{L^2 (\Omega)} + 8 \pi \alpha \| \varphi \|^4_{L^4 (\Omega)} \ee under the the {\em normalization condition} $ \| \varphi \|_{L^2 (\Omega)}^2 \ = \ N. $ The real number $\alpha$ is the scattering length associated with the two-body interaction between the particles in the gas. The functional in \eqref{NLSe} corresponds to the case of a negative scattering length, that is realized, for instance, by an attractive two-body interaction. Besides, in \eqref{NLSe} we consider a general subcritical nonlinearity power. \noindent In \eqref{gp}, the domain $\Omega$ corresponds to the shape of the (magnetic and/or optical) trap where the gas has been confined in order to induce the phase transition. Present technology allows various shapes for traps, like discs, cigars, and so on. Recently, the possibility of building ramified traps (\cite{kevrekidis,vidal}) has been envisaged theoretically, even though, at least to our knowledge, they have not been experimentally realized so far. \noindent To give a mathematical description of such an experimental setting, one should choose a spatial domain $\Omega$ that reproduces the shape of the trap and then minimize the energy in \eqref{gp}. One would expect that, for branched traps, the domain $\Omega$ may be replaced by a suitable graph $\G$. The possible ground state then provides the state of the condensate in the trap $\Omega$, while the absence of a ground state would in principle signal an instable character of the system. For instance, in the situation depicted by Theorems \ref{teoremzero}, \ref{teoremunoA}, the system would run away along an infinite edge, mimicking the shape of a soliton. For further results on nonlinear evolution on graphs, see e.g. \cite{ali, bona, smi,matrasulov}. The paper is organized as follows: Section \ref{sec:defi} contains a precise setting of the problem and the statements of the main results. In Sections \ref{sec:prelim} and \ref{sec:aux} we discuss some preliminary facts and techniques (in particular, rearrangements on graphs) and some auxiliary statements that may have some interest in themselves. Finally, Sections \ref{sec:proofs1} and \ref{sec:baffo} contain the proofs of the results stated in Section~\ref{sec:defi}. \emph{Remark on Figures:} some figures have been included to better describe the topology of certain metric graphs. In these pictures, vertices ``at infinity'' are denoted by the symbol $\infty$, while ordinary vertices (i.e. junctions of two or more edges) are denoted by a bullet. \section{Setting, notation and main results} \label{sec:defi} Although we shall not need deep results from graph theory, the notion of graph is central to this paper: we refer the reader to \cite{balakrishnan,bollobas} for a modern account on the subject. Throughout the paper a graph is always meant as a (connected) \emph{multigraph}, that is, we allow for multiple edges joining the same pair of vertices. Self-loops (i.e. edges starting and ending at the same vertex) are also allowed. More precisely, the central objects of the paper are \emph{metric graphs} (see \cite{Kuchment, Friedlander}), i.e. (connected) graphs $\G=(V,E)$ where each edge $e\in E$ is associated with either a closed bounded interval $I_e=[0,\ell_e]$ of length $\ell_e>0$, or a closed half-line $I_e=[0,+\infty)$, letting $\ell_e=+\infty$ in this case. Two edges $e,f\in E$ joining the same pair of vertices, if present, are \emph{distinct objects} in all respects: in particular, the corresponding intervals $I_e$ and $I_f$ need not have the same length, and must be considered \emph{distinct} even in the case where $\ell_e=\ell_f$. For every $e\in E$ joining two vertices $\vv_1,\vv_2\in V$, a coordinate $x_e$ is chosen along $I_e$, in such a way that $\vv_1$ corresponds to $x_e=0$ and $\vv_2$ to $x_e=\ell_e$, or \emph{viceversa}: if $\ell_e=+\infty$, however, we always assume that the half-line $I_e$ is attached to the remaining part of the graph at $x_e=0$, and the vertex of the graph corresponding to $x_e=+\infty$ is called a \emph{vertex at infinity}. The subset of $V$ consisting of all vertices at infinity will be denoted by $V_\infty$. With this respect, we shall always assume that \begin{equation} \label{H0} \text{all vertices at infinity of $\G$ (if any) have degree one} \end{equation} where, as usual, the degree %$\deg(\vv)$ of a vertex $\vv\in V$ is the number of edges incident at $\vv$ (counting twice any self-loop at $\vv$, of course). %we do not %consider in and out degrees separately, as the orientation of $I_e$ %induced by the coordinate %$x_e$ is irrelevant to our purpose. Finally, the cardinalities of $E$ and $V$ are assumed to be finite. An example of a typical metric graph $\G$ is given in Figure~\ref{Fig_gen}. \medskip \noindent A connected metric graph $\G$ has the natural structure of a locally compact metric space, the metric being given by the shortest distance measured along the edges of the graph. Observe that \begin{equation} \label{comp} \text{$\G$ is compact $\iff$ no edge of $\G$ is a half-line $\iff$ $V_\infty=\emptyset$.} \end{equation} With some abuse of notation, we often identify an edge $e$ with the corresponding interval $I_e$: thus, topologically, the \emph{metric space} $\G$ is the \emph{disjoint} union $\bigsqcup I_e$ of its edges, with some of their endpoints \emph{glued together} into a single point (corresponding to a vertex $\vv\in V\setminus V_\infty$), according to the topology of the \emph{graph} $\G$ (using the same symbol $\G$ for both the metric graph and the induced metric space should cause no confusion). We point out that any vertex at infinity $\vv\in V_\infty$ is of course a vertex of the graph $\G$, but is \emph{not} a point of the metric space $\G$ (this is consistent with \eqref{H0}). \begin{figure}[t] \begin{center} \begin{tikzpicture} [scale=1.3,style={circle, inner sep=0pt,minimum size=7mm}] % \node at (0,0) [nodo] (1) {}; \node at (-1.5,0) [infinito] (2){$\infty$}; \node at (1,0) [nodo] (3) {}; \node at (0,1) [nodo] (4) {}; \node at (-1.5,1) [infinito] (5) {$\infty$}; \node at (2,0) [nodo] (6) {}; \node at (3,0) [nodo] (7) {}; \node at (2,1) [nodo] (8) {}; \node at (3,1) [nodo] (9) {}; \node at (4.5,0) [infinito] (10) {$\infty$}; \node at (5.5,0) [infinito] (11) {$\infty$}; \node at (4.5,1) [infinito] (12) {$\infty$}; % \draw [-] (1) -- (2) ; \draw [-] (1) -- (3); \draw [-] (1) -- (4); \draw [-] (3) -- (4); \draw [-] (5) -- (4); \draw [-] (3) -- (6); \draw [-] (6) -- (7); \draw [-] (6) to [out=-40,in=-140] (7); \draw [-] (3) to [out=10,in=-35] (1.4,0.7); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw [-] (1.4,0.7) to [out=145,in=100] (3); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw [-] (6) to [out=40,in=140] (7); \draw [-] (6) -- (8); \draw [-] (6) to [out=130,in=-130] (8); \draw [-] (7) -- (8); \draw [-] (8) -- (9); \draw [-] (7) -- (9); \draw [-] (9) -- (12); \draw [-] (7) -- (10); \draw [-] (7) to [out=40,in=140] (11); % \end{tikzpicture} \end{center} \caption{A metric graph with $5$ half-lines and $13$ bounded edges, one of which forms a self-loop.}\label{Fig_gen} \end{figure} With $\G$ as above, a function $u:\G\to\R$ can be regarded as a bunch of functions $(u_e)_{e\in E}$, where $u_e:I_e\to\R$ is the restriction of $u$ to the edge $I_e$. Endowing each edge $I_e$ with Lebesgue measure, one can define $L^p$ spaces over $\G$ in the natural way, with norm \[ \| u\|_{L^p(\G)}^p=\sum_{e\in E} \| u_e\|_{L^p(I_e)}^p,\quad u=(u_e). \] Similarly, the Sobolev space $H^1(\G)$ is defined as the set of those functions $u:\G\to\R$ such that \begin{equation} \label{eq:18} \text{$u=(u_e)$ is continuous on $\G$, and $u_e\in H^1(I_e)$ for every edge $e\in E$,} \end{equation} with the natural norm \[ \Vert u\Vert_{H^1(\G)}^2=\int_{\G} \bigl(|u'(x)|^2+|u(x)|^2\bigr)\,dx= \sum_{e\in E} \int_{I_e} \bigl(|u_e'(x_e)|^2 +|u_e(x_e)|^2\bigr)\,dx_e. \] Note that $H^1(\G)$ can be identified with a closed subspace (determined by the continuity of $u$ at the vertices of $\G$) of the Cartesian product $\bigoplus_{e} H^1(I_e)$. In terms of the coordinate system $\{x_e\}$, continuity on $\G$ means that, whenever two edges $e,f$ meet at a vertex $\vv$ of $\G$, the corresponding branches of $u$ satisfy a no-jump condition of the kind $u_e(0)=u_f(0)$ (or $u_e(\ell_e)=u_f(\ell_f)$, or $u_e(\ell_e)=u_f(0)$ etc., depending on the orientation of $I_e,I_f$ induced by the coordinates $x_e,x_f$). Notice that, according to \eqref{H0}, vertices at infinity are never involved in these continuity conditions: on the other hand, if $u\in H^1(\G)$, then automatically \begin{equation} \label{zeroallinfinito} I_e=[0,+\infty)\implica \lim_{x_e\to +\infty} u_e(x_e)=0, \end{equation} because in particular $u_e\in H^1(I_e)$. \medskip \noindent Within this framework, we are now in a position to state our main results. Fix $\G$ as above, and numbers $\mu,p$ satisfying \begin{equation} \label{mup} \mu>0\quad\text{and}\quad 2
0$ by \eqref{mup}, and
$\phi_1(x)=C_p \mathop{\rm sech}(c_p x)^{\alpha/\beta}$ with $C_p,c_p>0$.
\end{remark}
%
The following is a general result for non-compact graphs.
%
\begin{theorem}\label{teoremzero}
If $\G$ contains at least one half-line, then
\begin{equation}
\label{eq:14bi}
\inf_{u\in H^1_\mu(\G)} E(u,\G)\leq
\min_{\phi\in H^1_\mu(\R)} E(\phi,\R)=
E(\phi_\mu,\R)
\end{equation}
and, in the other direction,
\begin{equation}
\label{eq:14tri}
\inf_{u\in H^1_\mu(\G)} E(u,\G)\geq
\min_{\phi\in H^1_\mu(\R^+)} E(\phi,\R^+)=
\frac 1 2 E(\phi_{2\mu},\R).
\end{equation}
\end{theorem}
In order to investigate
whether the infimum in \eqref{eq:14bi} is attained or not, the following
structure assumption on the graph $\G$ will play a crucial role:
\begin{itemize}
\item[(H)] After removal of any edge
$e\in E$, \emph{every} connected component of the \emph{graph} $(V,E\setminus\{e\})$ contains
at least one vertex $\vv\in V_\infty$.
\end{itemize}
Some remarks are in order. Firstly, (H) entails that $\G$ has \emph{at least} one
vertex at infinity $\vv_1\in V_\infty$, whence $\G$ is not compact
by \eqref{comp}. Secondly, the condition on
$e$ is relevant only when $e$ is a \emph{cut-edge} for $\G$ (i.e. when the
removal of $e$ disconnects $\G$), because when $(V,E\setminus\{e\})$
is connected the presence of $\vv_1\in V_\infty$ makes the condition trivial. On the other hand,
the edge $e$ (half-line) that has $\vv_1$ as vertex at infinity is necessarily
a cut-edge, since by \eqref{H0} its removal leaves vertex $\vv_1$ \emph{isolated}
in the graph $(V,E\setminus\{e\})$: therefore, the other connected component
necessarily contains a vertex at infinity $\vv_2\not=\vv_1$. Hence we see that,
in particular,
\begin{equation}
\label{almenodue}
\text{(H)}\quad\Rightarrow\quad
\text{$\G$ has at least two vertices at infinity.}
\end{equation}
Roughly speaking, assumption (H) says that there is always a vertex at infinity
on \emph{both sides} of any cut-edge.
Any cut-edge $e$ that violates (H),
would therefore
leave \emph{all} the vertices at
infinity on the same connected component
thus forming a sort of ``bottleneck'',
as regards the location of $V_\infty$ relative to $e$.
Thus,
in a sense, we may consider (H) as a no-bottleneck condition on $\G$.
Finally,
the fact that
(H) concerns cut-edges only makes it easy to test algorithmically, when the topology
of $\G$ is intricate: for instance, one can easily check that the graph in Figure~\ref{Fig_gen}
satisfies (H).
Under assumption (H), the inequality in \eqref{eq:14bi} is in fact an equality:
\begin{theorem} \label{teoremunoA}
If $\G$ satisfies (H), then
\begin{equation}
\label{infuguale}
\inf_{u\in H^1_\mu(\G)} E(u,\G)=
\min_{\phi\in H^1_\mu(\R)} E(\phi,\R)=
E (\phi_\mu, \R).
\end{equation}
\end{theorem}
%
%
\begin{remark*}
Hypothesis (H) and hence Theorem \ref{teoremunoA} apply to
examples of graphs
previously treated in the literature: star-graphs with
unbounded edges (\cite{acfn12a}) and general multiple bridges
(\cite{acfn14}). Furthermore,
they apply to any
semi-eulerian graph
with two vertices at infinity, as well as
to more complicated networks like the
one represented in Figure \ref{Fig_gen}.
\end{remark*}
It is easy to construct examples of graphs $\G$ satisfying (H),
for which the infimum in \eqref{infuguale} is achieved.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
[scale=1.3,style={circle,
inner sep=0pt,minimum size=7mm}]
%
\node at (0,3) [nodo] [label=below:$x_1$] (OO) {};
\node at (-1,3) [infinito] (RR) {\iii};
\node at (1,3) [infinito] (LL) {\iii};
%
\draw [-] (LL) -- (OO) ;
\draw [-] (RR) -- (OO) ;
\draw (-1,4) node {(a)};
%
%
\node at (0,0) [nodo] [label=below:$x_1$] (O) {};
\node at (-1,0) [infinito] (R) {\iii};
\node at (1,0) [infinito] (L) {\iii};
%
\draw [-] (L) -- (O) ;
\draw [-] (R) -- (O) ;
\draw [-] (O) to [out=40,in=0] (0,1);
\draw [-] (O) to [out=140,in=180] (0,1);
%
\draw (-1,1.5) node {(b)};
%
\end{tikzpicture}\quad\quad
\begin{tikzpicture}
[scale=1.3,style={circle,
inner sep=0pt,minimum size=7mm}]
%
\node at (0,0) [nodo] [label=below:$x_2$] (O) {};
\node at (1,0) [infinito] (L) {\iii};
\node at (-1,0) [infinito] (R) {\iii};
\node at (0,1) [nodo] [label=right:$\quad x_1$] (1) {};
%
\draw [-] (L) -- (O) ;
\draw [-] (R) -- (O) ;
\draw [-] (1) to [out=40,in=0] (0,2);
\draw [-] (1) to [out=140,in=180] (0,2);
\draw [-] (O) to [out=40,in=-40] (1);
\draw [-] (O) to [out=140,in=-140] (1);
\draw (-1,1.5) node {(c)};
%
\end{tikzpicture}\quad\quad
\begin{tikzpicture}
[scale=1.3,style={circle,
inner sep=0pt,minimum size=7mm}]
%
\node at (0,0) [nodo] [label=below:$x_n$] (O) {};
\node at (1,0) [infinito] (L) {\iii};
\node at (-1,0) [infinito] (R) {\iii};
\node at (0,1) [nodo] [label=right:$\quad x_{n-1}$] (1) {};
\node at (0,3) [nodo] [label=right:$\quad x_{1}$] (3) {};
\node at (0,2) [nodo] [label=right:$\quad x_{2}$] (2) {};
\draw (0.5,1.6) node {$\vdots$};
%
\draw [-] (L) -- (O) ;
\draw [-] (R) -- (O) ;
\draw [-] (3) to [out=40,in=0] (0,4);
\draw [-] (3) to [out=140,in=180] (0,4);
\draw [-] (O) to [out=40,in=-40] (1);
\draw [-] (O) to [out=140,in=-140] (1);
\draw [-] (2) to [out=40,in=-40] (3);
\draw [-] (2) to [out=140,in=-140] (3);
%\draw [dashed] (1) to [out=40,in=-40] (2);
%\draw [dashed] (1) to [out=140,in=-140] (2);
\draw [dashed] (1) to [out=40,in=-90] (0.267,1.5);
\draw [dashed] (1) to [out=140,in=-90] (-0.267,1.5);
\draw [dashed] (2) to [out=-40,in=90] (0.267,1.5);
\draw [dashed] (2) to [out=-140,in=90] (-0.267,1.5);
%
%\filldraw [fill=white,draw=white] (-0.5,1.3) rectangle (0.5,1.7);
\draw (-1,1.5) node {(d)};
\end{tikzpicture}
\end{center}
\caption{Graphs described in Example \ref{example1}, for which \eqref{eq:14bi} is an equality.}
\label{figtorre}
\end{figure}
\begin{example}\label{example1} \emph{(a)}
If $\G$ is isometric to $\R$ (see Remark~\ref{remclassic} and Figure~\ref{figtorre}.a), then
the soliton $\phi_\mu$ can be seen as an element of $H^1_\mu(\G)$,
and by \eqref{ensol} the infimum in \eqref{infuguale} is achieved.
\medskip
\noindent\emph{(b)} The symmetry of the soliton $\phi_\mu\in H^1(\R)$ can be exploited to
construct other examples. Given $a_1>0$, let $\G$ be
the quotient space $\R/\{\pm a_1\}$, obtained by gluing together the two points $a_1$ and
$-a_1$ into a unique point $x_1$. As a metric graph, $\G$ is depicted in
Figure~\ref{figtorre}.b, the length of the loop being $2a_1$. Since $\phi_\mu(a_1)=\phi_\mu(-a_1)$,
$\phi_\mu$ can be seen as an element of $H^1_\mu(\G)$, letting $x=0$ correspond to the north pole
of the loop in Figure~\ref{figtorre}.b. As before,
by \eqref{ensol} the infimum in \eqref{infuguale} is achieved.
\medskip
\noindent\emph{(c)} More generally, for $n\geq 2$
fix $a_n>\ldots> a_1>0$, and let $\G$ be obtained from $\R$ by gluing together
each pair of points $\pm a_1,\ldots,\pm a_n$, the corresponding new points
being denoted $\{x_j\}$. As a metric graph, $\G$ is as
in Figure~\ref{figtorre}.d (the length of the loop at the top being $2a_1$,
while the pairs of parallel edges have lengths $a_j-a_{j-1}$, $2\leq j\leq n$).
Since $\phi_\mu(a_i)=\phi_\mu(-a_i)$,
reasoning as in (b)
we see that the infimum in \eqref{infuguale} is attained.$\quad\square$
\end{example}
%
In fact, the graphs of the previous example are the \emph{only ones} for which
the infimum is attained.
%
%
\begin{theorem} \label{teoremunoB}
If $\G$ satisfies (H) then \eqref{infuguale} holds
true,
but the infimum is never achieved unless $\G$
%(after removal of
%possible spurious vertices
%of degree two, see Remark~\ref{remdeg2})
is isometric to one of the graphs discussed in Example~\ref{example1}.
\end{theorem}
%
%
Thus, with the only exception of the graphs of Example~\ref{example1},
assumption (H) rules out the existence of minimizers. Among metric graphs with
at least two half-lines, the simplest one that violates (H) is the graph in
Figure~\ref{Figbridge}.b, made up of two half-lines and one bounded edge (of arbitrary length)
joined at their initial point.
For this graph, we have the following result.
\begin{theorem} \label{teoremdue}
Let $\G$ consist of
two half-lines and one bounded edge (of arbitrary length $\ell>0$)
joined at their initial point. Then
\begin{equation}
\label{infminore}
\inf_{u\in H^1_\mu(\G)} E(u,\G)<
\min_{\phi\in H^1_\mu(\R)} E(\phi,\R)=E(\phi_\mu,\R)
\end{equation}
and the infimum is achieved.
\end{theorem}
As mentioned in the introduction, any minimizer exploits
the peculiar topology of this graph, and tends to concentrate on
the pendant. This is described in the following theorem.
\begin{theorem} \label{teoremdueB}
Let $\G$ be as in Theorem~\ref{teoremdue}, and let
$u\in H^1_\mu(\G)$ be any minimizer that achieves the infimum in \eqref{infminore}.
Then, up to replacing $u$ with $-u$, we have $u>0$ and
\begin{itemize}
\item[(i)] $u$ is strictly monotone along the pendant, with a
maximum at the tip.
\item[(ii)] If $u_1,u_2$ denote the restrictions of $u$ to the two half-lines,
with coordinates $x\geq 0$ starting both ways at the triple junction, then
\[
u_1(x)=u_2(x)=\phi_{\mu^*}(x+y)\quad\forall x\geq 0,
\]
for suitable $y>0$ and $\mu^*>\mu$ that depend on the mass $\mu$ and the length of
the pendant $\ell$. In particular, the restriction of $u$
to the straight line
is symmetric and radially decreasing, with a corner point at the origin.
\item[(iii)] For fixed $\mu$, the infimum in \eqref{infminore} is a strictly decreasing function
of $\ell$.
\end{itemize}
\end{theorem}
From (i) and (ii) it follows that the minimum of $u$ along the pendant coincides with
its maximum on the straight line (in other words, $u$ tends to concentrate on the pendant). Observe that,
on each half-line, $u$ coincides with a suitable
\emph{portion} of the soliton $\phi_{\mu^*}$.
\section{Some preliminary results}\label{sec:prelim}
The \emph{decreasing rearrangement} $u^*$ of a function $u\in H^1(\G)$,
where $\G$ is a metric graph, was first used in \cite{Friedlander} where
it is proved that, as in the classical case where $\G$ is an interval (see \cite{Kawohl}),
this kind of rearrangement does not increase the Dirichlet integral (see also \cite{acfn12b}).
Besides the increasing
rearrangement $u^*$, we shall also need the \emph{symmetric rearrangement}
$\widehat{u}$, whose basic properties we now recall.
Given $u\in H^1(\G)$, assume for simplicity that
\begin{equation}
\label{eq:90}
m:=\inf_{\G} u\geq 0,\qquad M:=\sup_{\G} u>0
\end{equation}
and, as in \cite{Friedlander}, let $\rho(t)$ denote the distribution
function of $u$:
\[
\rho(t)=\sum_{e\in E}\mis\bigl(\{x_e\in I_e\,:\,\,
u_e(x_e)>t\}\bigr),\quad t\geq 0,
\]
where the $u_e$'s are
the branches of $u$ as in \eqref{eq:18}. Set
\begin{equation}
\label{defomega}
\omega:=\sum_{e\in E} \mis(I_e),\quad
I^*:=[0,\omega),\quad
\widehat{I}:=(-\omega/2,\omega/2)
\end{equation}
where $\omega\in [0,\infty]$ is the total length of $\G$. As usual,
one can define
the following rearrangements of $u$:
\begin{enumerate}
\item [(i)]the \emph{decreasing} rearrangement $u^*:I^*\to \R$ as the function
\begin{equation}
\label{defu*dec}
u^*(x):=\inf\{t\geq 0\,:\,\, \rho(t)\leq x\},\quad x\in I^*;
\end{equation}
\item [(ii)] the \emph{symmetric decreasing} rearrangement $\widehat{u}:\widehat{I}\to \R$ as the function
\begin{equation*}
%\label{defu*sim}
\widehat{u}(x):=\inf\{t\geq 0\,:\,\, \rho(t)\leq 2|x|\},\quad x\in \widehat{I}.
\end{equation*}
\end{enumerate}
Since $u$, $u^*$ and $\widehat u$ are equimeasurable, one has
\begin{equation}
\label{normelr}
\int_{I^*} |u^*(x)|^r\,dx=
\int_{\widehat{I}} |\widehat{u}(x)|^r\,dx=
\int_{\G} |u(x)|^r\,dx
\quad\forall r>0
\end{equation}
and
\begin{equation*}
%\label{eq:infsup}
\inf_{I^*}u^*=
\inf_{\widehat{I}}\widehat{u}=\inf_{\G}u=m,\quad
\sup_{I^*}u^*=
\sup_{\widehat{I}}\widehat{u}=\sup_{\G}u=M.
\end{equation*}
As in the classical case
where $\G$ is an interval (see \cite{Kawohl}),
when $\G$ is a
\emph{connected} metric graph
it turns out (see \cite{Friedlander}) that
$u^*\in H^1(I^*)$ and
$\widehat{u}\in H^1(\widehat{I})$ respectively (connectedness of $\G$ is not essential, as long
as the image of $u$ is connected).
However,
while the passage from $u$ to $u^*$ never increases the
Dirichlet integral (\cite{Friedlander}), this is not always true
for $\widehat{u}$, a sufficient condition being that the number of preimages
\begin{equation*}
%\label{eq:defN}
N(t):=\#\{x\in \G\,:\,\, u(x)=t\},
\quad t\in (m,M)
\end{equation*}
is \emph{at least two} (see
Remark~2.7 in \cite{Kawohl}).
More precisely, we have
\begin{proposition}
\label{proprear}
Let $\G$ be a connected metric graph, and let $u\in H^1(\G)$ satisfy \eqref{eq:90}.
Then
\begin{equation}
\label{eq:ens}
\int_{I^*} |(u^*)'|^2\,dx\leq\int_{\G} |u'|^2\,dx,
\end{equation}
with strict inequality unless $N(t)=1$ for a.e. $t\in (m,M)$. Finally,
\begin{equation}
\label{eq:ens2}
N(t)\geq 2\quad\text{for a.e. $t\in (m,M)$}
\quad\Rightarrow\quad\int_{\widehat{I}} |(\widehat{u})'|^2\,dx\leq\int_{\G} |u'|^2\,dx,
\end{equation}
where equality implies that $N(t)=2$ for a.e. $t\in (m,M)$.
\end{proposition}
The part concerning $u^*$ can be found in \cite{Friedlander},
while the corresponding statements for $\widehat u$ can be proved in exactly
the same way.
\begin{remark}\label{remomega}
If $\G$ is non-compact, i.e. if $\G$ contains at least one half-line,
then clearly $\omega=+\infty$ in \eqref{defomega}, so that
$I^*=\R^+$ and $\widehat{I}=\R$. Thus, in particular, $u^*\in H^1(\R^+)$
while $\widehat{u}\in H^1(\R)$.
\end{remark}
The following standard result deals with the optimality conditions satisfied
by
any solution to \eqref{minE2}.
\begin{proposition}
\label{necess} Let $\G$ be a metric graph, and $u \in H_\mu^1(\G)$
a solution to \eqref{minE2}.
Then
\begin{itemize}
\item[(i)] there exists $\lambda \in\R$ such that
\begin{equation}
\label{eulero}
u_e'' +u_e |u_e|^{p-2} = \lambda u_e\quad\text{for every edge $e$;}
\end{equation}
\item[(ii)] for every vertex $\vv$ (that is not a vertex at infinity)
\begin{equation}
\label{kir}
\sum_{e\succ \vv}
\frac{d u_e}{d x_e}(\vv)=0
\qquad\text{ (Kirchhoff conditions),}
\end{equation}
where the condition $e\succ \vv$ means that edge $e$ is incident at $\vv$;
\item[(iii)] up to replacing $u$ with $-u$, one has that $u>0$ on $\G$.
\end{itemize}
\end{proposition}
The Kirchhoff condition \eqref{kir} is well known (see \cite{Friedlander,Kuchment}) and
is a natural form of continuity of $u'$ at the vertices of $\G$.
Observe that, by \eqref{eulero}, $u_e\in H^2(I_e)$ for every edge $e$, so that $u'_e$ is well defined
at both endpoints of $I_e$: in \eqref{kir}, of course, the symbol
$d u_e/d x_e(\vv)$ is a shorthand notation for $u_e'(0)$ or $-u_e'(\ell_e)$, according to
whether the coordinate $x_e$ is equal to $0$ or $\ell_e$ at $\vv$.
\begin{proof} Since both the energy $E(u,\G)$ and the $L^2$ constraint in
\eqref{h1m} are differentiable in $H^1(\G)$
and $u$ is a constrained critical point, computing G\^{a}teaux derivatives
one has
\begin{equation}
\label{weak}
\int_\G \left(u'\eta' - u |u|^{p-2} \eta\right)\dx
+ \lambda \int_\G u \eta\dx =0\qquad \forall \eta \in H^1(\G)
\end{equation}
where $\lambda$ is a Lagrange multiplier.
Fixing an edge $e$, choosing $\eta\in C^\infty_0(I_e)$ and integrating by parts,
one obtains \eqref{eulero}.
Now fix a vertex $\vv$ (not at infinity) and choose $\eta\in H^1(\G)$,
null at every vertex of $\G$ except at $\vv$: integrating by parts in \eqref{weak}
and using (i), only the boundary terms at $\vv$ are left, and one finds
\[
-\sum_{e\succ \vv}
\frac{d u_e}{d x_e}(\vv)\eta(\vv)=0,
\]
and \eqref{kir} follows since $\eta(\vv)$ is arbitrary.
To prove (iii), observe that if $u$ is a minimizer so is $|u|$, hence we may assume
that $u\geq 0$.
First assume that $u$ vanishes at a vertex
$\vv$.
Since $u\ge 0$ on $\G$,
no term
involved in \eqref{kir}
can be negative: since their sum is zero,
every derivative in \eqref{kir} is in fact zero.
Then, by uniqueness for the ODE \eqref{eulero}, we see that
$u_e\equiv 0$ along every edge $e$ such that $e\succ \vv$: since
$\G$ is connected, this argument can be iterated through neighboring vertices
and one obtains that $u\equiv 0$ on $\G$, a contradiction since $u\in H^1_\mu(\G)$.
If, on the other hand, $u_e(x)=0$ at some point $x$ interior to some edge $e$, from $u\geq 0$
we see that also $u'_e(x)=0$ and, as before, from \eqref{eulero} we deduce that
$u_e\equiv 0$ along $e$. Thus, in particular, $u(\vv)=0$ at a vertex $\vv \prec e$,
and one can argue as above.
\end{proof}
Another useful result, valid for any metric graph $\G$, is the
following Gagliardo-Nirenberg inequality:
\begin{equation*}
\|u\|_{L^p(\G)}^p \le C\|u\|_{L^2(\G)}^{\frac p 2 +1}
\|u\|_{H^1(\G)}^{\frac p 2-1}\qquad \forall u \in H^1(\G),
\end{equation*}
where $C=C(\G,p)$. This is well known when $\G$ is an
interval (bounded or not, see \cite{brezis}): for the general case,
it suffices to write the inequality for each edge of $\G$, and take the sum.
In particular, when $\|u\|_{L^2(\G)}^2=\mu$ is fixed, we obtain
\begin{equation*}
%\label{GN2}
\|u\|_{L^p(\G)}^p \le C+
C\|u'\|_{L^2(\G)}^{\frac p 2-1}\qquad \forall u \in H^1_\mu(\G),
\end{equation*}
where now $C=C(\G,p,\mu)$. Since our $p$ satisfies \eqref{mup}, this shows
that the negative term in \eqref{NLSe} grows \emph{sublinearly}, at infinity,
with respect to the positive one. As a consequence, Young's inequality gives
$\|u'\|_{L^2(\G)}^2\leq C+CE(u,\G)$ when $u \in H^1_\mu(\G)$, and hence also
\begin{equation}
\label{coerc}
\|u\|_{H^1(\G)}^2 \leq C+CE(u,\G)
\qquad \forall u \in H^1_\mu(\G),\quad
C=C(\G,p,\mu).
\end{equation}
\section{Some auxiliary results}\label{sec:aux}
In this section we discuss two auxiliary
\emph{double-constrained} problems,
on $\R^+$ and on $\R$ respectively, that
will be useful in Section~\ref{sec:baffo}
and may be of some interest in themselves.
We begin with the double-constrained problem on the half-line
\begin{equation}
\label{minhalf}
\min E(\phi,\R^+),\quad
\phi \in H^1(\R^+),
\quad
\int_0^\infty |\phi|^2\,dx=\frac m 2,\quad
\phi(0)=a
\end{equation}
for fixed $m,a>0$. This corresponds to \eqref{minE} when $\G=\R^+$ and $\mu=m/2$,
with the
\emph{additional} Dirichlet condition
\begin{equation*}
%\label{bicond}
\phi(0) = a.
\end{equation*}
\begin{theorem}
\label{unic}
For every $a,m>0$ there exist unique
$M>0$ and $y\in\R$ such that the soliton $\phi_M$ satisfies the two conditions
\begin{equation}
\label{cond}
\phi_{M}(y) =a\qquad\text{and}\qquad \int_0^{+\infty} \phi_{M}(y+x)^2\dx = \frac m 2.
\end{equation}
Moreover, the function $x\mapsto\phi_M(y+x)$ is the unique solution to \eqref{minhalf}.
Finally, there holds
\begin{equation}
\label{ypos}
a>\phi_m(0)\iff y>0.
\end{equation}
\end{theorem}
\begin{proof}
Recalling the scaling rule of solitons
\eqref{scalingrule},
let $z=\mu^\beta y$ (to be determined).
Then, changing variable $x=M^{-\beta} t$ in the integral, the conditions in \eqref{cond} become
\begin{equation}
\label{condbis}
M^\alpha \phi_1(z)=a,\quad\text{and}\quad
M \int_0^{+\infty} \phi_1(z+t)^2\dt = \frac m 2.
\end{equation}
Using the first condition, we can eliminate $M$ from the second and obtain
\begin{equation}
\label{tosolve}
\phi_1(z)^{-\frac{1}\alpha} \int_0^{+\infty} \phi_1(z+t)^2\dt = \frac{m a^{-\frac{1}\alpha}}{2}.
\end{equation}
Denoting by $g(z)$ the function on the left-hand side, we claim that
\begin{equation}
\label{limiti}
\lim_{z\to -\infty}g(z) = +\infty \qquad\hbox{and}\qquad \lim_{z\to +\infty}g(z) =0.
\end{equation}
The first limit is clear as
$\phi_1(z)\to 0$ while the integral tends to $\| \phi_1\|_{L^2(\R)}^2=1$.
For the second, since $\phi_1$ is decreasing on $\R^+$ one can estimate
\[
\phi_1(z+t)^2\leq
\phi_1(z)^{\frac 1 \alpha}
\phi_1(z+t)^{2-\frac 1 \alpha}\quad\forall z,t\geq 0
\]
in the integral and observe that $2-1/\alpha>0$ by \eqref{mup}.
Moreover, $g$ is strictly decreasing: this is clear when $z<0$,
since in this case $g(z)$ is the product of two strictly decreasing functions.
When $z>0$,
differentiation yields
\begin{align*}
g'(z) &= 2\int_0^{+\infty}
\frac{\phi_1(z+t)^2}{\phi_1(z)^{\frac 1\alpha}}
\left[\frac{\phi_1'(z+t)}{\phi_1(z+t)}-
\frac 1{2\alpha}\frac{\phi_1'(z)}{\phi_1(z)}\right]\dt\\
&<
2\int_0^{+\infty}
\frac{\phi_1(z+t)^2}{\phi_1(z)^{\frac 1\alpha}}
\left[\frac{\phi_1'(z+t)}{\phi_1(z+t)}-
\frac{\phi_1'(z)}{\phi_1(z)}\right]\dt
<0,
\end{align*}
having used $1/2\alpha<1$ and $\phi_1'(z)<0$ in the first inequality, and
the log--concavity of $\phi_1$ (see Remark~\ref{remclassic})
in the second. This and \eqref{limiti} show that,
given $a,m>0$, there exists a unique $z\in \R$ (hence a unique $y$)
satisfying
\eqref{tosolve}, while $M$ is uniquely determined by the first
condition in \eqref{condbis}.
To prove \eqref{ypos} observe that, by \eqref{scalingrule}, $a>\phi_m(0)$
is equivalent to $a>m^\alpha\phi_1(0)$, which in turn is equivalent to
$g(0)>ma^{-1/\alpha}/2$ since
\[
g(0)=
\phi_1(0)^{-\frac{1}\alpha} \int_0^{+\infty} \phi_1(t)^2\dt =
\frac{\phi_1(0)^{-\frac{1}\alpha}}2.
\]
But since $g(z)=ma^{-1/\alpha}/2$ by \eqref{tosolve} and $g$ is decreasing, the last inequality
is equivalent to $z>0$, which proves \eqref{ypos}.
For the last part of the claim,
by Remark~\ref{remclassic}
%recall that
%the soliton $\phi_M$ and its translates are the unique positive minimizers
%of $E(\phi,\R)$ with mass constraint $\Vert\phi\Vert_{L^2(\R)}^2=M$:
%in particular,
we see that the function
$x\mapsto \phi_M(x+y)$ minimizes $E(\phi,\R)$
under the \emph{two constraints} $\Vert\phi\Vert_{L^2(\R)}^2=M$ and
$\phi(0)=a$. If a competitor $\varphi(x)$
better than $x\mapsto \phi_M(x+y)$ could be found for \eqref{minhalf}, then
the function equal to $\phi_M(x+y)$ for $x<0$ and to $\varphi(x)$ for $x\geq 0$ would
violate the mentioned optimality of $\phi_M(x+y)$. Finally, uniqueness
follows from the uniqueness of $M$ and $y$ satisfying \eqref{cond}: indeed,
any other solution $\varphi(x)$ to \eqref{minhalf} not coinciding on $\R^+$ with
any soliton, arguing as before would give rise to a \emph{non-soliton} minimizer
of $E(\phi,\R)$ with mass constraint $\Vert\phi\Vert_{L^2(\R)}^2=M$.
\end{proof}
\noindent Now we consider the analogue problem on the \emph{whole} real line, namely
\begin{equation}
\label{pretta}
\min E(\phi,\R),\quad
\phi \in H^1(\R),
\quad
\int_{-\infty}^\infty |\phi|^2\,dx=m,\quad
\phi(0)=a
\end{equation}
for fixed $m,a>0$. Its solutions are characterized as follows, a
special role being played by the soliton $\phi_m$ of mass $m$.
\begin{theorem}
\label{thretta} Let $a,m > 0$ be given.
\begin{enumerate}
\item[(i)] If $a<\phi_m(0)$ then problem \eqref{pretta}
has exactly two solutions, given by $x\mapsto \phi_m(x\pm y)$ for a suitable $y>0$.
\item[(ii)] If $a=\phi_m(0)$, then problem
\eqref{pretta} has $\phi_m(x)$ as unique solution.
\item[(iii)] if $a>\phi_m(0)$, then
problem \eqref{pretta} has exactly one solution, namely
$x\mapsto \phi_M(|x|+y)$ for suitable $M, y>0$.
\end{enumerate}
\end{theorem}
\begin{proof} Cases (i) and (ii) are immediate since $a$ is in the range of $\phi_m$:
as $\phi_m$ and its translates are the only positive minimizers of $E(\cdot,\R)$ in $H^1_m(\R)$,
the second constraint in \eqref{pretta}
can be matched for free by
a translation, with $y\geq 0$ such that $\phi_m(\pm y)=a$.
Case (ii) is when $a=\max\phi_m$, and so $y=0$.
Now consider (iii), where the value $a$ is not in the range of $\phi_m$.
Let $M,y$ be the numbers provided by Theorem~\ref{unic}, and observe that
$y>0$ according to \eqref{ypos}. Since $\phi_M(y)=a$ by
\eqref{cond}, the soliton $\phi_M$ is clearly the
unique solution of the constrained problem
\begin{equation}
\label{aux1}
\min E(\phi,\R),\quad
\phi\in H^1_M(\R), \quad
\phi(-y)=\phi(y)=a.
\end{equation}
Moreover, as $y>0$, the second condition in \eqref{cond} implies that
\[
\int_{\R\setminus (-y,y)}|\phi_M(x)|^2\,dx
=
2\int_{y}^\infty |\phi_M(x)|^2\,dx=m.
\]
Therefore, the function $w(x)=\phi_M(|x|+y)$ is an admissible competitor
for \eqref{pretta}, and obviously
\[
E(w,\R)=E\bigl(\phi_M,(-\infty,-y)\bigr)
+E\bigl(\phi_M,(y,+\infty)\bigr).
\]
The existence of a $v(x)$ admissible for \eqref{pretta} and such that
$E(v,\R)