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NLS ground states on graphs

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.


Introduction
In this paper we investigate the existence of a ground state for the NLS energy functional with the mass constraint where µ > 0 and p ∈ (2, 6) are given numbers and G is a 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 2, A metric graph G ( [6,12,16]) 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) ≤ E(u, G) and any ground state is in fact real valued, up to multiplication by a constant phase e iθ .
When G = R the minimization problem is well understood and the minimizers, called solitons, are known explicitly. The same is true when G = [0, +∞) is a half-line (the minimizer being "half a soliton" of mass 2µ) 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 2). The aim of this paper is that of studying existence and qualitative properties of solutions to (3), under quite general assumptions on G (for papers devoted to particular graphs, see [1,3,10]). Since when G is compact existence of minimizers for (3) is immediate, we focus on graphs where at least one edge is unbounded (a half-line), so that the embeddings H 1 (G) ֒→ L r (G) are not compact and existence for (3) is non-trivial. In fact, even though the infimum of E(u, G) is always trapped between two finite values (Theorem 2.2), 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 1.a) then (3) has no solution, while if G is a straight line with one pendant attached to it (Figure 1.b) then minimizers do exist.
Our results extend in two directions. First, we prove a nonexistence result (Theorems 2.3 and 2.5) for a broad family of non-compact graphs (see condition (H) in Section 2): roughly speaking, if no cut-edge of G segregates all the half-lines of G in the same connected component, then (3) has no solution, the only exceptions being certain graphs with a particular topology which we characterize completely (see Example 2.4). Moreover, for all these graphs the unattained infimum of E(u, G) coincides with the NLS energy of a soliton on the real line, for the same values of µ and p.
The mentioned condition on G, that prevents existence of minimizers in (3), 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 1.b: we prove that for this particular G problem (3) does have a solution (Theorem 2.6) and we establish some qualitative properties of the minimizers (Theorem 2.7). In fact, in this case the energy level of the minimizers is 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 6.1. The interesting feature is that a hybrid rearrangement of a function u ∈ H 1 (G) is needed, adapted to the topology of G: high values of u are rearranged increasingly on the pendant, while small values of u are rearranged 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 (3). The problem of characterizing all non-compact G such that (3) 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 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.
Among the physical motivations for this problem (see [17] and references therein), nowadays the most topical is probably given by the Bose-Einstein condensation (see [11]). 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 wave function that minimizes the Gross-Pitaevskii energy functional under the the normalization condition ϕ 2 L 2 (Ω) = N. The real number α is the scattering length associated with the two-body interaction between the particles in the gas. The functional in (1) corresponds to the case of a negative scattering length, that is realized, for instance, by an attractive two-body interaction. Besides, in (1) we consider a general subcritical nonlinearity power.
In (4), the domain Ω 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 ( [15,19]) has been envisaged theoretically, even though, at least to our knowledge, they have not been experimentally realized so far. To give a mathematical description of such an experimental setting, one should choose a spatial domain Ω that reproduces the shape of the trap and then minimize the energy in (4). One would expect that, for branched traps, the domain Ω may be replaced by a suitable graph G. The possible ground state then provides the state of the condensate in the trap Ω, 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 2.2, 2.3, 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. [4,8,13,18].
The paper is organized as follows: Section 2 contains a precise setting of the problem and the statements of the main results. In Sections 3 and 4 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 5 and 6 contain the proofs of the results stated in Section 2.
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 ∞, while ordinary vertices (i.e. junctions of two or more edges) are denoted by a bullet.

Setting, notation and main results
Although we shall not need deep results from graph theory, the notion of graph is central to this paper: we refer the reader to [5,7] for a modern account on the subject.
Throughout the paper a graph is always meant as a (connected) 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 metric graphs (see [16,12]), i.e. (connected) graphs G = (V, E) where each edge e ∈ E is associated with either a closed bounded interval I e = [0, ℓ e ] of length ℓ e > 0, or a closed half-line I e = [0, +∞), letting ℓ e = +∞ in this case. Two edges e, f ∈ E joining the same pair of vertices, if present, are 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 distinct even in the case where ℓ e = ℓ f . For every e ∈ E joining two vertices v 1 , v 2 ∈ V , a coordinate ∞ ∞ ∞ ∞ ∞ Figure 2: A metric graph with 5 half-lines and 13 bounded edges, one of which forms a self-loop.
x e is chosen along I e , in such a way that v 1 corresponds to x e = 0 and v 2 to x e = ℓ e , or viceversa: if ℓ e = +∞, 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 = +∞ is called a vertex at infinity. The subset of V consisting of all vertices at infinity will be denoted by V ∞ . With this respect, we shall always assume that (5) all vertices at infinity of G (if any) have degree one where, as usual, the degree of a vertex v ∈ V is the number of edges incident at v (counting twice any self-loop at v, of course). Finally, the cardinalities of E and V are assumed to be finite. An example of a typical metric graph G is given in Figure 2.
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 With some abuse of notation, we often identify an edge e with the corresponding interval I e : thus, topologically, the metric space G is the disjoint union I e of its edges, with some of their endpoints glued together into a single point (corresponding to a vertex v ∈ V \ V ∞ ), according to the topology of the 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 v ∈ V ∞ is of course a vertex of the graph G, but is not a point of the metric space G (this is consistent with (5)). With G as above, a function u : G → R can be regarded as a bunch of functions (u e ) e∈E , where u e : I e → 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 Similarly, the Sobolev space H 1 (G) is defined as the set of those functions u : G → R such that (7) u = (u e ) is continuous on G, and u e ∈ H 1 (I e ) for every edge e ∈ E, with the natural norm 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 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 v of G, the corresponding branches of u satisfy a no-jump condition of the kind u e (0) = u f (0) (or u e (ℓ e ) = u f (ℓ f ), or u e (ℓ 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 (5), vertices at infinity are never involved in these continuity conditions: on the other hand, if u ∈ H 1 (G), then automatically Within this framework, we are now in a position to state our main results. Fix G as above, and numbers µ, p satisfying (9) µ > 0 and 2 < p < 6.
For u ∈ H 1 (G), the NLS energy in (1) is finite and takes the concrete form If we let the minimization problem (3) takes the compact form Remark 2.1. The classical instance of (11) where G is the real line R falls within our framework as a particular case, when G is made up of two unbounded edges (half-lines), joined at their initial point (Figure 3.a). In this case, the solutions to (11) are called solitons, and are known to be unique up to translations and a change of sign. In particular, there is a unique minimizer which is a positive and even function: we shall denote this function by φ µ (the dependence on p being understood), and of course (12) E(φ µ , R) = min It is well known that solitons obey the scaling rule where α, β > 0 by (9), and φ 1 ( The following is a general result for non-compact graphs.
Theorem 2.2. If G contains at least one half-line, then and, in the other direction, In order to investigate whether the infimum in (14) is attained or not, the following structure assumption on the graph G will play a crucial role: (H) After removal of any edge e ∈ E, every connected component of the Some remarks are in order. Firstly, (H) entails that G has at least one vertex at infinity v 1 ∈ V ∞ , whence G is not compact by (6). Secondly, the condition on e is relevant only when e is a cut-edge for G (i.e. when the removal of e disconnects G), because when (V, E \ {e}) is connected the presence of v 1 ∈ V ∞ makes the condition trivial. On the other hand, the edge e (half-line) that has v 1 as vertex at infinity is necessarily a cut-edge, since by (5) its removal leaves vertex v 1 isolated in the graph (V, E \ {e}): therefore, the other connected component necessarily contains a vertex at infinity v 2 = v 1 . Hence we see that, in particular, (16) (H) ⇒ G has at least two vertices at infinity.
Roughly speaking, assumption (H) says that there is always a vertex at infinity on both sides of any cut-edge. Any cut-edge e that violates (H), would therefore leave all the vertices at infinity on the same connected component thus forming a sort of "bottleneck", as regards the location of V ∞ 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 2 satisfies (H).
Under assumption (H), the inequality in (14) is in fact an equality:

Remark. Hypothesis (H) and hence Theorem 2.3 apply to examples of graphs
previously treated in the literature: star-graphs with unbounded edges ( [1]) and general multiple bridges ( [3]). Furthermore, they apply to any semieulerian graph with two vertices at infinity, as well as to more complicated networks like the one represented in Figure 2.
It is easy to construct examples of graphs G satisfying (H), for which the infimum in (17) is achieved.  (17) is achieved.
(b) The symmetry of the soliton φ µ ∈ H 1 (R) can be exploited to construct other examples. Given a 1 > 0, let G be the quotient space R/{±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 3.b, the length of the loop being 2a 1 . Since φ µ (a 1 ) = φ µ (−a 1 ), φ µ can be seen as an element of H 1 µ (G), letting x = 0 correspond to the north pole of the loop in Figure 3.b. As before, by (12) the infimum in (17) is achieved.
(c) More generally, for n ≥ 2 fix a n > . . . > a 1 > 0, and let G be obtained from R by gluing together each pair of points ±a 1 , . . . , ±a n , the corresponding new points being denoted {x j }. As a metric graph, G is as in Figure 3.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 ≤ j ≤ n). Since φ µ (a i ) = φ µ (−a i ), reasoning as in (b) we see that the infimum in (17) is attained.
In fact, the graphs of the previous example are the only ones for which the infimum is attained. Theorem 2.5. If G satisfies (H) then (17) holds true, but the infimum is never achieved unless G is isometric to one of the graphs discussed in Example 2.4.
Thus, with the only exception of the graphs of Example 2.4, 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 1.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.
Theorem 2.6. Let G consist of two half-lines and one bounded edge (of arbitrary length ℓ > 0) joined at their initial point. Then and the infimum is achieved.
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.
Theorem 2.7. Let G be as in Theorem 2.6, and let u ∈ H 1 µ (G) be any minimizer that achieves the infimum in (18). Then, up to replacing u with −u, we have u > 0 and (i) u is strictly monotone along the pendant, with a maximum at the tip.
(ii) If u 1 , u 2 denote the restrictions of u to the two half-lines, with coordinates x ≥ 0 starting both ways at the triple junction, then for suitable y > 0 and µ * > µ that depend on the mass µ and the length of the pendant ℓ. In particular, the restriction of u to the straight line is symmetric and radially decreasing, with a corner point at the origin.
(iii) For fixed µ, the infimum in (18) is a strictly decreasing function of ℓ.
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 portion of the soliton φ µ * .

Some preliminary results
The decreasing rearrangement u * of a function u ∈ H 1 (G), where G is a metric graph, was first used in [12] where it is proved that, as in the classical case where G is an interval (see [14]), this kind of rearrangement does not increase the Dirichlet integral (see also [2]). Besides the increasing rearrangement u * , we shall also need the symmetric rearrangement u, whose basic properties we now recall.
Given u ∈ H 1 (G), assume for simplicity that and, as in [12], let ρ(t) denote the distribution function of u: where the u e 's are the branches of u as in (7). Set where ω ∈ [0, ∞] is the total length of G. As usual, one can define the following rearrangements of u: (i) the decreasing rearrangement u * : I * → R as the function (ii) the symmetric decreasing rearrangement u : I → R as the function Since u, u * and u are equimeasurable, one has (22) As in the classical case where G is an interval (see [14]), when G is a connected metric graph it turns out (see [12]) that u * ∈ H 1 (I * ) and u ∈ H 1 ( 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 ( [12]), this is not always true for u, a sufficient condition being that the number of preimages is at least two (see Remark 2.7 in [14]). More precisely, we have Proposition 3.1. Let G be a connected metric graph, and let u ∈ H 1 (G) satisfy (19). Then with strict inequality unless N (t) = 1 for a.e. t ∈ (m, M ). Finally, where equality implies that N (t) = 2 for a.e. t ∈ (m, M ).
The part concerning u * can be found in [12], while the corresponding statements for u can be proved in exactly the same way.
Remark 3.2. If G is non-compact, i.e. if G contains at least one half-line, then clearly ω = +∞ in (20), so that I * = R + and I = R. Thus, in particular, The following standard result deals with the optimality conditions satisfied by any solution to (11). (ii) for every vertex v (that is not a vertex at infinity) where the condition e ≻ v means that edge e is incident at v; (iii) up to replacing u with −u, one has that u > 0 on G.
The Kirchhoff condition (26) is well known (see [12,16]) and is a natural form of continuity of u ′ at the vertices of G. Observe that, by (25), u e ∈ H 2 (I e ) for every edge e, so that u ′ e is well defined at both endpoints of I e : in (26), of course, the symbol du e /dx e (v) is a shorthand notation for u ′ e (0) or −u ′ e (ℓ e ), according to whether the coordinate x e is equal to 0 or ℓ e at v.
Proof. Since both the energy E(u, G) and the L 2 constraint in (10) are differentiable in H 1 (G) and u is a constrained critical point, computing Gâteaux derivatives one has where λ is a Lagrange multiplier. Fixing an edge e, choosing η ∈ C ∞ 0 (I e ) and integrating by parts, one obtains (25). Now fix a vertex v (not at infinity) and choose η ∈ H 1 (G), null at every vertex of G except at v: integrating by parts in (27)  To prove (iii), observe that if u is a minimizer so is |u|, hence we may assume that u ≥ 0. First assume that u vanishes at a vertex v. Since u ≥ 0 on G, no term involved in (26) can be negative: since their sum is zero, every derivative in (26) is in fact zero. Then, by uniqueness for the ODE (25), we see that u e ≡ 0 along every edge e such that e ≻ v: since G is connected, this argument can be iterated through neighboring vertices and one obtains that u ≡ 0 on G, a contradiction since u ∈ H 1 µ (G). If, on the other hand, u e (x) = 0 at some point x interior to some edge e, from u ≥ 0 we see that also u ′ e (x) = 0 and, as before, from (25) we deduce that u e ≡ 0 along e. Thus, in particular, u(v) = 0 at a vertex v ≺ e, and one can argue as above.
Another useful result, valid for any metric graph G, is the following Gagliardo-Nirenberg inequality: where C = C(G, p). This is well known when G is an interval (bounded or not, see [9]): for the general case, it suffices to write the inequality for each edge of G, and take the sum. In particular, when u 2 L 2 (G) = µ is fixed, we obtain where now C = C(G, p, µ). Since our p satisfies (9), this shows that the negative term in (1) grows sublinearly, at infinity, with respect to the positive one. As a consequence, Young's inequality gives u ′ 2 L 2 (G) ≤ C + CE(u, G) when u ∈ H 1 µ (G), and hence also

Some auxiliary results
In this section we discuss two auxiliary double-constrained problems, on R + and on R respectively, that will be useful in Section 6 and may be of some interest in themselves.
We begin with the double-constrained problem on the half-line for fixed m, a > 0. This corresponds to (3) when G = R + and µ = m/2, with the additional Dirichlet condition Using the first condition, we can eliminate M from the second and obtain Denoting by g(z) the function on the left-hand side, we claim that (34) lim z→−∞ g(z) = +∞ and lim z→+∞ g(z) = 0.
The first limit is clear as φ 1 (z) → 0 while the integral tends to φ 1 2 L 2 (R) = 1. For the second, since φ 1 is decreasing on R + one can estimate in the integral and observe that 2 − 1/α > 0 by (9).
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 having used 1/2α < 1 and φ ′ 1 (z) < 0 in the first inequality, and the logconcavity of φ 1 (see Remark 2.1) in the second. This and (34) show that, given a, m > 0, there exists a unique z ∈ R (hence a unique y) satisfying (33), while M is uniquely determined by the first condition in (32).
To prove (31) observe that, by (13), a > φ m (0) is equivalent to a > m α φ 1 (0), which in turn is equivalent to g(0) > ma −1/α /2 since But since g(z) = ma −1/α /2 by (33) and g is decreasing, the last inequality is equivalent to z > 0, which proves (31). For the last part of the claim, by Remark 2.1 we see that the function x → φ M (x + y) minimizes E(φ, R) under the two constraints φ 2 L 2 (R) = M and φ(0) = a. If a competitor ϕ(x) better than x → φ M (x + y) could be found for (29), then the function equal to φ M (x + y) for x < 0 and to ϕ(x) for x ≥ 0 would violate the mentioned optimality of φ M (x + y).
Moreover, as y > 0, the second condition in (30) implies that Therefore, the function w(x) = φ M (|x| + y) is an admissible competitor for (35), and obviously The existence of a v(x) admissible for (35) and such that E(v, R) < E(w, R), would allow the construction of a competitor better than φ M in (36), by redefining φ M (x) when |x| ≥ y, setting it equal to v(x − y) or v(x + y), according to whether x ≥ y or x ≤ −y. Thus w solves (35). Similarly, a competitor v ≡ w with E(v, R) = E(w, R) would violate the uniqueness of φ M as a solution of (36), and therefore w is the unique solution of (35).

Proof of the nonexistence results
Proof of Theorem 2.2. Consider u ε ∈ H 1 (R) with compact support, such that u ε 2 L 2 = µ and u ε → φ µ strongly in H 1 (R) as ε → 0. Since u ε → φ µ also in L p (R), we see that Now, by a translation, we may assume that u ε is supported in [0, +∞): identifying this interval with one of the half-lines of G, we may consider u ε as a function in H 1 µ (G), by extending it to zero on any other edge of G. Then we have from the previous equation and (14) follows from (12). The inequality in (15) is immediate from (22) and (23) (see Remark 3.2), by rearranging an arbitrary u ∈ H 1 µ (G). Finally, the equality in (15) is well known.
To prove of Theorems 2.3 and 2.5 we need to investigate how assumption (H), which is purely graph-theoretical, reflects on G as a metric-space.
Proof. We recall that a trail in a graph (see [7]) is a finite sequence of consecutive edges, in which no edge is repeated (note that a vertex, instead, may be repeated, if the trail contains cycles or self-loops). Let T be the class of all those trails T in the graph G, such that the initial edge and the final edge of T are half-lines (observe that, due to (5), no edge of T other than the initial and the final edge can be a half-line: thus, in a sense, every T ∈ T provides an immersion of R in G -not an embedding, however, as T may have cycles). Recalling (16), since G is connected we see that T = ∅. Moreover, any point x 0 ∈ G that is covered by at least one trail T ∈ T , satisfies condition (H ′ ). Indeed, to construct γ 1 and γ 2 , it suffices to start at x 0 and move along T both ways, according to arclength: the two obtained curves can then be made injective, by removing any useless loop that each of them, separately, may form. Moreover, since no edge is repeated in T , γ 1 and γ 2 may intersect only at finitely many vertices of G. Since the initial and the final edges of T are half-lines, each γ i (t) is eventually trapped in a half-line and thus each γ i has infinite length, which makes it possible to parameterize γ i (t) injectively by arclength with t ∈ [0, +∞). Therefore, it suffices to prove that the trails in T cover G. Let E 0 ⊆ E denote the set of those edges that do not belong to any trail T ∈ T . Assuming E 0 = ∅, since G is connected there must be some edge e ∈ E 0 with one vertex w on some trail T ∈ T (w ∈ V \ V ∞ , by (5)). If e were a cut-edge for G, then by (H) there would be a vertex v 1 ∈ V ∞ in the connected component of (V, E \ {e}) disjoint from T : then it would be possible to go from v 1 to w along a trail that crosses e, and then proceed over a portion of T up to another vertex v 2 ∈ V ∞ , thus constructing a path T ′ ∈ T that contains e, which is impossible since e ∈ E 0 . Now, as e is not a cut-edge, it necessarily belongs to a cycle C (made up of bounded edges only, due to (5)): then, by inserting C in the middle of T at vertex w, we can construct a trail T ′ ∈ T that contains e. Since this is a contradiction, we see that E 0 = ∅ and therefore T covers G.
Proof of Theorem 2.3. We first prove that for every u ∈ H 1 µ (G), where u is the symmetric rearrangement of u as defined in (21). As E(u, G) = E(|u|, G), we may assume that u ≥ 0. In fact we have that M := max G u > 0 by (2), and that m := inf G u = 0 by (8), as G contains at least two half-lines due to (16). From Remark 3.2 u ∈ H 1 (R) and, in fact, u ∈ H 1 µ (R) by (22) with r = 2, which proves the second inequality in (37).
The first inequality, due to (22) written with r = p, is in fact equivalent to the integral inequality in (24), which follows as soon as we show that This, in turn, follows from (H ′ ) of Lemma 5.1. If γ 1 , γ 2 are as in (H ′ ), relative to a point x 0 ∈ G where u(x 0 ) = M , we may define the continuous function Clearly v(0) = u(x 0 ) = M . Moreover, as each γ i (z) parameterizes a halfline of G for z large enough, from (8) we have that v(z) → 0 as |z| → ∞, hence v has at least two distinct preimages (in R) for every value t ∈ (0, M ). But as the images of γ 1 , γ 2 are disjoint except for finitely many points of G, (38) is established and (37) follows. Now (37), combined with (14), proves Theorem 2.3.
Proof of Theorem 2.5. Carrying on with the previous proof, assume that some u ∈ H 1 µ (G) achieves the infimum in (17) (i.e. in (37)). Then, both inequalities in (37) are equalitites. From them (combining the former with Proposition 3.1) we infer that Indeed, in the proof of (38), we proved a slightly stronger statement, namely that #(u −1 (t) ∩ Γ) ≥ 2 for a.e. t (while N (t) counts preimages in the whole G). Then, from (i), it follows that those values t ∈ (0, M ) attained by u on G \ Γ (if any) form a set of measure zero. Since u is continuous on G and G \ Γ is open (Γ is, in fact, the trail mentioned in assumption (H)), this implies that u is constant on any edge e of G not belonging to Γ (as Γ is a trail, we may regard it is as a subgraph of G). But u and u are equimeasurable and, by (ii), every level set of u has measure zero, hence the same is true for u: therefore, for any edge e of G not in Γ, the only possibility is that u ≡ 0 on e (for the moment, this cannot be excluded: as meas({ u > 0}) = +∞, the quantity meas({u = 0}) cannot be computed by complementation). However, since u ≡ 0 outside Γ, considering Γ as a subgraph, the restriction u |Γ satisfies Since the γ i : [0, +∞) → Γ are injective and parameterized by arclength, it follows from (39) and the previous relations that Then, by (ii), v is necessarily the translate of a soliton of mass µ but, since v(0) = M = u(0), v is centered at the origin and so v = u = φ µ . In particular, v > 0 whence also u |Γ > 0: then, by continuity, no edge of G where u ≡ 0 can be attached to Γ, hence (iii) is proved since G is connected.
which shows that each γ i is a simple curve in G. Recall that, by (H ′ ), the images of the γ i 's intersect at at their starting point x 0 and, possibly, also at finitely many other points x 1 , . . . , x n of G. If n = 0 (i.e. if x 0 is the only intersection), then clearly Γ is isometric to the real line R, and by (iii) we see that G is necessarily the graph of Example 2.4 (a). If n = 1, then Γ is a straight line with two points glued together, hence G is necessarily the graph of Example 2.4 (b).
Finally, if n > 1, since v = ϕ µ is an even and strictly radially decreasing function, we have Since the γ i 's are parameterized by arclength, this shows that for every j ∈ {1, . . . , n} the distance a j of x j from x 0 , measured along γ 1 , is the same as the distance measured along γ 2 . And this forces G to be isometric to the graph in Example 2.4 (c).

Proof of the existence results
Throughout this section, G is the graph described in Theorem 2.6 (see Figure 4). In dealing with (11), however, it is convenient to let u = (φ, ψ), and identify each u ∈ H 1 (G) with a pair of functions φ, ψ, with φ ∈ H 1 (R), ψ ∈ H 1 (I) with I = [0, ℓ], satisfying the continuity condition φ(0) = ψ(0): here, of course, the interval I = (0, ℓ) represents the pendant of G, while R represents the union of its two half-lines. Then (11)  As mentioned in the Introduction, the key to the proofs of Theorems 2.6 and 2.7 is the following lemma, that shows how the value of E(u, G) can be reduced, for any u ∈ H 1 µ (G), by a proper hybrid rearrangement operation, tailored to the topology of G. (i) φ : R → R is even and radially decreasing; G) and, if equality occurs, then letting u = (φ, ψ) we necessarily have that ψ is increasing on [0, ℓ] and min ψ = max φ.
Proof. By the assumptions on u, it is easy to see that there exists τ > 0 such that The idea of the proof is to rearrange the portion of u above τ increasingly on I, and the portion below τ symmetrically on R. More precisely, let and observe that f, g ∈ H 1 (G). Then one can define f and g * , the symmetric and the decreasing rearrangements of f and g, respectively, in such a way that f ∈ H 1 (R) and g * ∈ H 1 (R + ) by Remark 3.2. Note that: we have ψ(0) = τ + g * (ℓ) = τ and φ(0) = f (ℓ/2) = τ , and the first condition in (40) is satisfied. Moreover, as φ ∈ H 1 (R) and ψ ∈ H 1 (I), letting u = ( φ, ψ) we have that u ∈ H 1 (G). In fact, since by construction for t > 0 u and u are equimeasurable and therefore also u ∈ H 1 µ (G) (i.e., (40) is fully satisfied). For the same reason, Now we claim that The equality is immediate from the definition of φ(x), since f ≡ τ on [−ℓ/2, ℓ/2], while the inequality follows from (24), as soon as we prove that Indeed, by (41), the closed set {u ≥ τ } cannot be strictly contained inside the pendant edge of G, since the latter has measure ℓ: therefore, we have that u(x) ≥ τ for at least one point x 0 in the union of the two half-lines of G (possibly, only at the triple junction). But this makes (44) obvious, even for every t ∈ (0, τ ), since u → 0 as one approaches the two points at infinity of G, both ways from x 0 . Similarly, since g * is supported in [0, ℓ] and ψ ′ (x) = (g * ) ′ (ℓ − x), where now (23) is used. Taking the sum with (43), we find which combined with (42) shows that E( u, G) ≤ E(u, G). Finally, if equality holds, then it must hold in (45) as well: from the discussion after (23), then, we see that the function (u − τ ) + must have exactly one preimage x t ∈ G, for almost every value t > 0 that it achieves on G. Now split u = (φ, ψ), with φ ∈ H 1 (R) and ψ ∈ H 1 (I): if M = max φ > τ then, as φ(x) → 0 when |x| → ∞, for every t ∈ (τ, M ) we would have #φ −1 (t) ≥ 2, hence (u − τ ) + would achieve every value t ∈ (τ, M ) at least twice on G, which would be a contradiction. We deduce that φ ≤ τ , hence (41) implies that the set {u > τ } is contained in the pendant edge of G: indeed, since the pendant itself has measure ℓ, we obtain that ψ ≥ τ on I, so that ψ ≥ φ and (u − τ ) + is supported on the pendant edge of G. Since (u − τ ) + has one preimage for almost every value, we deduce that ψ is monotone, and this completes the proof.
To prove that the infimum in (18) is attained, let u n = (φ n , ψ n ) be a minimizing sequence, i.e.
As E(u n , G) = E(|u n |, G), we may assume that u n ≥ 0 and even more, by a density argument using e.g. piecewise linear functions, that each u n satisfies the assumptions of Lemma 6.1. Thus, we may assume that φ n , ψ n satisfy properties as in (i) and (ii) therein, so that in particular (47) φ n is even and radially decreasing.
Since I = [0, ℓ] is bounded, ψ n → ψ strongly in L p (I), so that because the positive term in (1) is lower semicontinuous. Concerning φ n , clearly φ n → φ uniformly on compact sets: however, combining (47) with the fact that φ(x) → 0 when |x| → ∞, one can easily check that φ n → φ in L ∞ (R). Since {φ n } is bounded in L 2 (R) and p > 2, by interpolation we infer that also φ n → φ strongly in L p (R), so that E(φ, R) ≤ lim inf n E(φ n , R).
Proof of Theorem 2.7, (iii). Given any two different lengths ℓ > ℓ ′ > 0 for the pendant, let G, G ′ denote the corresponding graphs, and let u ∈ H 1 µ (G ′ ), u > 0, be a minimizer on G ′ : proceeding exactly as in the proof of Lemma 6.1, we shall construct a function u ∈ H 1 µ (G) such that E(u, G ′ ) > E( u, G). Since u has qualitative properties as in (i) and (ii) of Theorem 2.7, we can certainly find τ = τ (ℓ) > 0 satisfying (41) (now u ∈ H 1 µ (G ′ ) and ℓ is unrelated to G ′ , but this is irrelevant to the computations following (41)). Then we proceed verbatim with the construction of u ∈ H 1 µ (G) satisfying (42) and (45), with the proviso that G be replaced with G ′ wherever u is involved (the same applies to (44) and (43), hereafter). Since the structure of u is as in (i)-(ii) of Theorem 2.7, and the pendant of G ′ has length ℓ ′ , from ℓ > ℓ ′ and (41) we see that f = u∧τ has exactly two preimages for every value t ∈ (0, τ ) so that also (44) and hence (43) are satisfied, and the same is true for g = (u − τ ) + for all those values t > 0 small enough, so that the inequality in (45) is now strict. Therefore, we have E(u, G ′ ) > E( u, G) hence the infimum in (18), as a function of ℓ, is strictly decreasing.