L 2 -critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features
LL -CRITICAL NLS ON NONCOMPACT METRIC GRAPHS WITH LOCALIZEDNONLINEARITY: TOPOLOGICAL AND METRIC FEATURES SIMONE DOVETTA AND LORENZO TENTARELLI
Abstract.
Carrying on the discussion initiated in [12], we investigate the existence of ground states ofprescribed mass for the L -critical NonLinear Schrödinger Equation (NLSE) on noncompact metric graphswith localized nonlinearity. Precisely, we show that the existence (or nonexistence) of ground states mainlydepends on a parameter called reduced critical mass, and then we discuss how the topological and metricfeatures of the graphs affect such a parameter, establishing some relevant differences with respect to thecase of the extended nonlinearity studied by [4]. Our results rely on a thorough analysis of the optimalconstant of a suitable variant of the L -critical Gagliardo-Nirenberg inequality.AMS Subject Classification: 35R02, 35Q55, 81Q35, 35Q40, 49J40.Keywords: metric graphs, NLS, ground states, localized nonlinearity, L -critical case. Introduction
In this paper, we aim at discussing the existence of ground states for the NonLinear Schrödinger Equation(NLSE) on metric graphs with localized nonlinearities.We briefly recall that a metric graph is the locally compact metric space which arises as one endows a multigraph G = (V , E) with a parametrization that associates each bounded edge e ∈ E with a closed andbounded interval I e = [0 , (cid:96) e ] of the real line, and each unbounded edge e ∈ E with a half-line I e = R + (fordetails see [2, 5] and references therein). Consistently, functions on metric graphs u = ( u e ) e ∈ E : G → R arefamilies of functions defined on each edge u e : I e → R in such a way that u | e = u e . Lebesgue spaces are,then, given by L p ( G ) := (cid:77) e ∈ E L p ( I e ) , p ∈ [1 , ∞ ] , while H ( G ) := (cid:26) u ∈ (cid:77) e ∈ E H ( I e ) : u is continuous on G (cid:27) , both equipped with the natural norms, denoted by (cid:107) u (cid:107) p, G and (cid:107) u (cid:107) respectively. In addition, in the followingwe limit ourselves to the study of those graphs G such that (A) G is connected , noncompact , with a finite number of edges and with non-empty compact core K (which is the subgraph of the bounded edges of G ).In view of this, the central issue discussed in the present paper is that of the existence of minimizers forthe L - critical NLS energy functional E ( u, K ) := 12 (cid:90) G | u (cid:48) | dx − (cid:90) K | u | dx (1)under the mass constraint (cid:107) u (cid:107) , G = µ > , that is, the so-called ground states . In other words, we discuss the existence of functions u ∈ H µ ( G ) suchthat E ( u, K ) = E G ( µ, K ) := inf v ∈ H µ ( G ) E ( v, K ) , (2) Date : May 9, 2019. a r X i v : . [ m a t h . A P ] M a y S. DOVETTA AND L. TENTARELLI
Figure 1. tadpole graph.with H µ ( G ) = (cid:8) u ∈ H ( G ) : (cid:107) u (cid:107) , G = µ (cid:9) . Such functions are known (the proof is completely analogous to that of [2, Proposition 3.3]) to be L -solutionsof − ∆ G u + λu = χ K | u | u, λ ∈ R + , where χ K is the characteristic function of K and − ∆ G is the so-called Kirchhoff
Laplacian, namely theoperator defined by − ∆ G u | ˚ I e := − u (cid:48)(cid:48) e dom( − ∆ G ) := (cid:26) u ∈ H ( G ) : u e ∈ H ( I e ) , ∀ e ∈ E , and (cid:88) e (cid:31) v du e dx e (v) = 0 , ∀ v ∈ K (cid:27) (3)with e (cid:31) v meaning that the edge e is incident at the vertex v and du e dx e (v) standing for u (cid:48) e (0) or − u (cid:48) e ( (cid:96) e ) according to whether x e is equal to or (cid:96) e at v . It is, also, straightforward that ψ ( t, x ) := e ıλt u ( x ) , with u ground state, is a stationary solution of the time-dependent nonlinear Schrödinger equation ı ∂ψ∂t = − ∆ G ψ − χ K | ψ | ψ. (4)Problem (2) was first proposed in [12], in the “toy” model of the tadpole graph (see Figure 1). Here weaim at extending that seminal result in order to present an (almost) complete classification of the wholephenomenology.The study of the NLSE on graphs has recently become a quite popular research topic and then theliterature has hugely increased. We mention, e.g., [1, 2, 3, 8, 14, 15] (and the references therein) for problemsinvolving the nonlinearity extended to the whole graph in the L -subcritical case, and [7, 11, 13, 16] forthe discussion of the Schrödinger equation on compact graphs. We also mention two recent works on someother dispersive equations: [17] for the KdV equation and [6] for the
Nonlinear Dirac equation .Furthermore, with respect to the present paper, it is worth recalling [4], which deals with the existenceof L -critical ground states in the extended case, where (1) is replaced by E ( u ) := 12 (cid:90) G | u (cid:48) | dx − (cid:90) G | u | dx, (5)and [18, 19, 20], which deals with the localized L - subcritical problem, where (1) is replaced by E ( u, K , p ) := 12 (cid:90) G | u (cid:48) | dx − p (cid:90) K | u | p dx, p ∈ (2 , . Before stating our main theorems, let us mention some classical results on the L -critical issue withextended nonlinearity on the real line and the half-line (see [9]). Precisely, it is well known that, letting E G ( µ ) := inf u ∈ H µ ( G ) E ( u ) , with E ( · ) defined by (5) and G = R , then a threshold phenomenon shows up, that is E R ( µ ) = (cid:40) if µ ≤ µ R −∞ if µ > µ R , (6) -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 3 where µ R = π √ / and is usually called critical mass of the real line. Moreover, E R ( µ ) is attained if andonly if µ = µ R , and a whole family of ground states exists, the solitons { φ λ } λ> , given by φ λ ( x ) := √ λφ ( λx ) , φ ( x ) := sech / (2 x/ √ , (7)and satisfying E ( φ λ , R ) = 0 , for every λ > .Analogously, when G = R + , the portrait is the same with µ R replaced by the critical mass of the half-line µ R + = µ R / √ π/ . Furthermore, ground states exist (again) only at the critical mass and they are theso-called half-solitons , i.e. the restrictions of φ λ to R + .We can now state the first result of the paper, which provides a topological classification of metric graphs.In the following, recall that a metric graph G is said to admit a cycle covering if and only if each edge of G belongs to a cycle, where a cycle can be either a loop, that is a closed path of consecutive bounded edges,or an unbounded path joining the endpoints of two distinct half-lines, which are then identified as a singlevertex at infinity (see [3, 4] for further details). Finally, by a terminal edge we mean any edge ending witha vertex of degree 1. Theorem 1.1.
Let G satisfy (A) . Then, there exists µ K ∈ [ µ R + , µ R ] such that E G ( µ, K ) = 0 if µ ≤ µ K < if µ ∈ ( µ K , µ R ]= −∞ if µ > µ R . (8) Moreover, (i) if G has at least one terminal edge (see, e.g., Figure 2(a)), then µ K = µ R + , E G ( µ, K ) = −∞ for all µ > µ K , and ground states never exist; (ii) if G admits a cycle-covering (see, e.g., Figure 2(b)), then µ K = µ R and ground states never exist; (iii) if G has exactly one half-line and no terminal edges (see, e.g., Figure 2(c)), then µ R + < µ K < √ (9) and ground states of mass µ exist if and only if µ ∈ [ µ K , µ R ] . (iv) if G has neither a terminal edge, nor a cycle-covering, and at least two half-lines (see, e.g., Figure2(d)), then µ R + < µ K ≤ µ R (10) and ground states of mass µ exist if and only if µ ∈ [ µ K , µ R ] , provided that µ K (cid:54) = µ R . Preliminarily, let us point out that assumption µ K (cid:54) = µ R in case (iv) is consistent, in the sense that onecan easily exhibit examples of graphs fulfilling it (see, for instance, the signpost graph of Figure 6, whenthe vertical edge is large enough, as in Proposition 4.3).It turns out that the actual value of µ K , that we refer to as the reduced critical mass of G in the following,is strictly related to a Gagliardo-Nirenberg-type inequality, i.e. µ K := (cid:114) C K , (11)where C K denotes the sharpest constant of (cid:107) u (cid:107) , K ≤ C K (cid:107) u (cid:107) , G (cid:107) u (cid:48) (cid:107) , G , ∀ u ∈ H ( G ) , (12)namely C K := sup u ∈ H ( G ) Q ( u ) , where Q ( u ) := (cid:107) u (cid:107) , K (cid:107) u (cid:107) , G (cid:107) u (cid:48) (cid:107) , G . (13) S. DOVETTA AND L. TENTARELLI (a) a graph with a terminal edge (b) a graph with a cycle-covering (c) a graph with one half-line and noterminal edge (d) a graph with no terminal edge,nor cycle-covering and at least twohalf-lines
Figure 2. examples of cases (i)-(iv) in Theorem 1.1.The dependence of the existence of ground states on a critical mass is a common feature with the issueof the extended nonlinearity discussed in [4]. However, some major differences appear. First, in [4] µ K isreplaced by µ G := (cid:115) C G ,C G being the optimal constant of the standard Gagliardo-Nirenberg inequality (cid:107) u (cid:107) , G ≤ C G (cid:107) u (cid:107) , G (cid:107) u (cid:48) (cid:107) , G , ∀ u ∈ H ( G ) . (14)Clearly, C K ≤ C G , so that, by definition, µ G ≤ µ K . Furthermore, even though [4] shares the same classification of the problem according to (i)–(iv), a ratherdifferent phenomenology shows up in the localized setting. • Cases (i) and (ii) are almost identical for the localized and the extended issues if one replaces µ K with µ G , up to the fact that in [4] there exist graphs supporting ground states at µ = µ G . Here, onthe contrary, the localization of the nonlinearity prevents the existence of ground states. • In case (iii) the difference with respect to the extended nonlinearity is more remarkable. Indeed,in [4], independently of further properties of the graphs, the critical mass satisfies µ G = µ R + andthe interval of masses for which one has existence of ground states is ( µ R + , µ R ] . On the contrary,in Theorem 1.1 the reduced critical mass µ K is strictly greater than the critical mass of the half-line and hence the existence interval is smaller. Nevertheless, it is provided to be nonempty, since µ K < √ < µ R . In addition, it is worth mentioning that the fact that µ K > µ R + allows us to treatalso the endpoint case µ = µ K , which is instead open for the extended problem. • Case (iv) is analogous to the extended case, again with µ K in place of µ G . In particular, in boththe situations one has to prescribe the (reduced) critical mass to be different from µ R in order to -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 5 guarantee the existence of ground states. However, one can see that if µ K (cid:54) = µ R , then µ G < µ K .Indeed, assume by contradiction µ G = µ K , so that E G ( µ K , K ) = E G ( µ G ) = 0 . Item (iv) of Theorem1.1 entails that there exists a minimizer u ∈ H µ G ( G ) , and thus E ( u, K ) = E ( u ) = 0 , whence (cid:107) u (cid:107) , K = (cid:107) u (cid:107) , G . Hence, u is a ground state of the extended problem, as well, and it is supportedon the sole compact core K . Since this is impossible due to standard regularity properties of globalminimizers (see [2]), there results µ G < µ K . Consequently, as for case (iii), the existence interval isstrictly smaller for the problem with a concentrated nonlinearity.It is also worth highlighting that, while the fact that the topology of the graph plays a crucial role isa common feature in the context of extended nonlinearities, it is new in the localized setting. Indeed, inthe subcritical case the problem of the ground states with localized nonlinearity is affected only by metricproperties of the graph (see, e,g, [20, Theorems 3.3 and 3.4, and Remark 3.8]).Nevertheless, it turns out that the metric features preserve their importance even in the critical problem,at least in cases (iii) and (iv), as shown by the following Theorem 1.2.
Estimates (9) are sharp in general; i.e., for every ε > there exist two non-compact metricgraphs G ε , G ε (with compact cores K ε , K ε , respectively), with exactly one half-line and no terminal edges,such that µ K ε ≤ µ R + + ε and µ K ε ≥ √ − ε . Theorem 1.3.
Estimates (10) are sharp in general; i.e., for every ε > there exist two non-compact metricgraphs G ε , G ε (with compact cores K ε , K ε , respectively), without terminal edges and cycle coverings and withat least two half-lines, such that µ K ε ≤ µ R + + ε and µ K ε ≥ µ R − ε . Some comments are in order. First, we underline that Theorem 1.2 shows a completely new phenomenonwith respect to the extended case. Indeed, in [4] for the topologies that fall into case (iii) the actual valueof the critical mass µ G is completely insensitive to metric properties of G , as µ G = µ R + , whereas this is notthe case when a localization of the nonlinearity occurs.Furthermore, we can actually characterize the structure of graphs approaching either µ R + or √ inTheorem 1.2. The former case is realized when in the compact core there is a sufficiently long cut edge , thatis a single edge connecting two disjoint subgraph (Proposition 4.1). On the contrary, the latter situationoccurs when the compact core is extremely intricate, in the sense that it has a very large total length withrespect to its diameter (Proposition 4.2).It is also worth mentioning the role of the constant √ , which is new with respect to the extended case.For graphs with one half-line and no terminal edges, indeed, it is a kind of critical mass obtained by (11)if one replaces C K with the maximum of Q restricted to functions which are constant on the compact coreand exponentially fast decaying on the half-line (see Proposition 3.4). Albeit not being optimizers of theproblem, these functions seem to play a significant role when the compact core thickens, as non-constantsfunction would necessarily originate a large kinetic energy (cid:107) u (cid:48) (cid:107) , G .On the other hand, Theorem 1.3 shows a similar asymptotic behaviour induced by the metric. Since thiskind of graphs have at least one cut edge in their compact core, by extending or shrinking such an edge, onecan recover the two limiting optimal constants (see Propositions 4.3-4.7). In this case it is an open problemwhether an analogous phenomenon could occur for an extended nonlinearity.In conclusion, we recall that, as in the extended case, the existence of negative energy ground states,namely negative energy stationary solutions of (4), is something in sharp contrast with the usual behaviorof the time-dependent NLSE on standard domains, where solutions with negative energy blow up in a finitetime (see [9]).The paper is organized as follows. In Section 2 we recall some known facts on Gagliardo-Nirenberginequalities and show some useful results on boundedness from below of E ( · , K ) and pre-compactness ofthe minimizing sequences. In Section 3 we prove Theorem 1.1, stressing the connection between existenceresults, the value of C K and the topology of the graph. Finally, Section 4 provides the proof of Theorem S. DOVETTA AND L. TENTARELLI C K andthe metric structure of the graph.2. Preliminary results: Gagliardo-Nirenberg inequalities and ground states
In this section we analyse the connection between the Gagliardo-Nirenberg inequalities and the existenceof ground states of (2).First, in addition to (12) and (14), we recall another well known Gagliardo-Nirenberg inequality ongraphs: (cid:107) u (cid:107) ∞ , G ≤ C ∞ (cid:107) u (cid:107) / , G (cid:107) u (cid:48) (cid:107) / , G , ∀ u ∈ H ( G ) (15)(see [3] for a proof), where C ∞ denotes the smallest constant for which the inequality is satisfied. It is wellknown (see e.g. [20]) that, if G = R , then C ∞ = 1 , while if G = R + , then C ∞ = √ , and that in both casesit is attained by u ( x ) = e −| x | . On the other hand, for general non-compact graphs, although it is alwaystrue that C ∞ ≤ √ , the actual value of this constant depends both on topological and on metric propertiesof G . However, if G is a graph with exactly one half-line, we recover C ∞ = √ . Indeed, denoting by H theunique half-line of G and setting, for every ε > , u ε ( x ) := (cid:40) √ εu ( εx ) if x ∈ H√ ε if x ∈ K with u ( x ) = e − x , we get that u ε ∈ H ( G ) and that √ ≥ C ∞ ≥ (cid:107) u ε (cid:107) ∞ , G (cid:107) u ε (cid:107) / , G (cid:107) u (cid:48) ε (cid:107) / , G = √ ε (cid:16) (cid:107) u ε (cid:107) , H + ε |K| (cid:17) / (cid:107) u (cid:48) ε (cid:107) / , H = √ ε (cid:16) + ε |K| (cid:17) / (cid:16) ε (cid:17) / −→ √ , as ε → . Another important preliminary remark concerns a modified version of the Gagliardo-Nirenberg inequality(14), which has been first established in [4, Lemma 4.4]. Let G satisfy (A) and assume also that it does notcontain any terminal edge. Then, if µ ∈ (0 , µ R ] , for every u ∈ H µ ( G ) there exists θ u ∈ [0 , µ ] (depending on u ) such that (cid:107) u (cid:107) , G ≤ (cid:16) µ − θ u µ R (cid:17) (cid:107) u (cid:48) (cid:107) , G + C (cid:112) θ u , (16)with C > depending only on G . In addition, note that if { u n } ⊂ H µ ( G ) is a sequence of functions suchthat E K ( u n , G ) ≤ − α < , for some α > , then inf n θ u n > . Indeed, by (16) (cid:107) u (cid:48) n (cid:107) , G (cid:16) − ( µ − θ u n ) µ R (cid:17) − C (cid:112) θ u n ≤ E ( u n , K ) ≤ − α < , (17)so that θ u n > uniformly on n .As mentioned in the Introduction, the key Gagliardo-Nirenberg inequality of the problem with localizednonlinearity is not the standard one given by (14), but instead (12) where the L term affects only thecompact core of the graph. In particular, it is evident by (11) that the value of the best constant C K is thecrucial parameter in order to determine whether solutions of (2) exist or not. Indeed, plugging (12) into(1), E ( u, K ) ≥ (cid:107) u (cid:48) (cid:107) , G (cid:18) − C K µ (cid:19) ∀ u ∈ H µ ( G ) (18)showing that C K plays a role in establishing the lower boundedness of the energy. More precisely, we canstate the following Lemma 2.1.
Let G satisfy (A) and µ K be the reduced critical mass defined by (11) - (13) . The followingclassification holds: -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 7 (i) if µ ≤ µ K , then E G ( µ, K ) = 0 ; (ii) if µ ∈ ( µ K , µ R ] , then E G ( µ, K ) < ; (iii) if µ > µ R , then E G ( µ, K ) = −∞ .Proof. First, note that, for every µ > , if { u n } ⊂ H µ ( G ) is a sequence such that (cid:107) u (cid:48) n (cid:107) , G → , then E G ( µ, K ) ≤ E ( u n , K ) ≤ (cid:107) u (cid:48) n (cid:107) , G −→ as n → ∞ so that E G ( µ, K ) ≤ . Furthermore, if µ ≤ µ K , then (18) entails that E ( u, K ) ≥ for all u ∈ H µ ( G ) , thusproving (i).On the other hand, assume µ > µ K ; for instance, µ = (1 + δ ) µ K , for some δ > . Now, by (13), thereexists u ∈ H µ ( G ) such that Q ( u ) > C K (1 + δ ) , whence E ( u, K ) < (cid:107) u (cid:48) (cid:107) , G (cid:16) − C K µ δ ) (cid:17) = 0 by (11) and the assumption on µ , which yields (ii).Finally, let µ > µ R and v ∈ H µ ( R ) such that supp { v } = [0 , and E ( v, R ) < (the existence of v beingguaranteed by (6)). For every λ > define then v λ ( x ) := √ λv ( λx ) , so that, v λ ∈ H µ ( R ) and supp { v λ } = [0 , /λ ] . Clearly, when λ is large enough, v λ can be regarded as afunction on G supported on a single edge of the compact core K . As a consequence, E G ( µ, K ) ≤ E ( v λ , K ) = λ E ( v, R ) −→ −∞ as λ → ∞ , which concludes the proof. (cid:3) As boundedness from below is not enough in order to prove ground states to exist, the following lemmaprovides some sufficient conditions for existence/nonexistence.
Lemma 2.2.
Let G satisfy (A) and µ K be the reduced critical mass defined by (11) – (13) . Then: (i) whenever µ < µ K , E G ( µ, K ) is not attained; (ii) whenever µ ∈ ( µ K , µ R ] and E G ( µ, K ) > −∞ , E G ( µ, K ) is attained.Proof. If µ < µ K , then (11)–(18) imply that E ( u, K ) > for every u ∈ H µ ( G ) and, combining with Lemma2.1, (i) follows immediately.On the other hand, suppose µ ∈ ( µ K , µ R ] and E G ( µ, K ) > −∞ . Let, also, { u n } ⊂ H µ ( G ) be a minimizingsequence. By Lemma 2.1, there exists α > such that E ( u n , K ) ≤ − α as n is large enough, so that by (17),(16) holds with θ u n ≥ C > . Consequently, µ − θ u n < µ R , so that (16) entails that { u n } n ∈ N is boundedin H ( G ) and that u n (cid:42) u in H ( G ) and u n → u in L ( K ) , for some u ∈ H ( G ) . Moreover, by weak lowersemicontinuity E ( u, K ) ≤ lim inf n E ( u n , K ) = E G ( µ, K ) . (19)Therefore, we are left to prove that (cid:107) u (cid:107) , G =: m = µ . By weak lower semicontinuity again, m ≤ µ . Onthe other hand, it is immediate to see that m (cid:54) = 0 , since otherwise u ≡ and, by (19), E G ( µ, K ) ≥ , whichis a contradiction. Moreover, if m < µ , then there exists σ > such that (cid:107) σu (cid:107) , G = µ and E ( σu, K ) = σ (cid:107) u (cid:48) (cid:107) , G − σ (cid:107) u (cid:107) , K < σ E ( u, K ) < E ( u, K ) , which contradicts (19) and hence concludes the proof. (cid:3) Remark . Lemma 2.2 does not say anything about µ = µ K . Indeeds, this endpoint problem stronglydepends on the topology of the graph and must be addressed case by case. S. DOVETTA AND L. TENTARELLI Proof of Theorem 1.1: how the topology affects C K Throughout this section we provide the proof of Theorem 1.1. It is based on the study of the actual valueof C K in the four distinct topological classes (i)-(iv) defined by Theorem 1.1.As a first step, we derive upper and lower bounds for C K on a generic non-compact graph. In thefollowing, we denote by C R + and C R the best constant of (14) when G = R + and G = R , respectively. Proposition 3.1.
For every G satisfying (A) , the constant C K fulfills C R ≤ C K ≤ C R + . (20) Proof.
Let u ∈ H ( G ) and assume with no loss of generality u ≥ . Then, denote by u ∗ its decreasingrearrangement on R + , namely u ∗ ( x ) := inf { t ≥ ρ ( t ) ≤ x } x ∈ [0 , |G| ) , with ρ ( t ) := (cid:88) e ∈ E |{ x e ∈ I e : u e ( x e ) > t }| t ≥ . (21)By well known properties of rearrangements (see [2, Section 3] for details), (cid:107) ( u ∗ ) (cid:48) (cid:107) , R + ≤ (cid:107) u (cid:48) (cid:107) , G , (cid:107) u ∗ (cid:107) p, R + = (cid:107) u (cid:107) p, G ∀ p ≥ , (22)so that Q ( u ) ≤ (cid:107) u (cid:107) , G (cid:107) u (cid:107) , G (cid:107) u (cid:48) (cid:107) , G ≤ (cid:107) u ∗ (cid:107) , R + (cid:107) u ∗ (cid:107) , R + (cid:107) ( u ∗ ) (cid:48) (cid:107) , R + ≤ C R + , and recalling (13) C K ≤ C R + . On the other hand, there exists a sequence { (cid:101) v n } ⊂ H ( R ) , (cid:101) v n ≥ , such that (cid:107) (cid:101) v n (cid:107) , R (cid:107) (cid:101) v n (cid:107) , R (cid:107) (cid:101) v (cid:48) n (cid:107) , R −→ C R as n → ∞ (23)and such that (cid:107) (cid:101) v n (cid:107) , R (cid:107) (cid:101) v (cid:48) n (cid:107) , R ≥ C > (see, e.g. [10]). Now, simply truncating and lowering (cid:101) v n , onecan define another sequence v n ∈ H ( R ) , which is compactly supported, such that (cid:107) v n (cid:107) , R ≤ (cid:107) (cid:101) v n (cid:107) , R , (cid:107) v (cid:48) n (cid:107) , R ≤ (cid:107) (cid:101) v (cid:48) n (cid:107) , R and (cid:107) v n − (cid:101) v n (cid:107) → . As a consequence, (cid:107) (cid:101) v n (cid:107) , R (cid:107) (cid:101) v n (cid:107) , R (cid:107) (cid:101) v (cid:48) n (cid:107) , R ≤ (cid:107) v n (cid:107) , R (cid:107) v n (cid:107) , R (cid:107) v (cid:48) n (cid:107) , R + (cid:107) v n − (cid:101) v n (cid:107) , R (cid:107) (cid:101) v n (cid:107) , R (cid:107) (cid:101) v (cid:48) n (cid:107) , R ≤ C R + o (1) , and thus (23) still holds with (cid:101) v n replaced by v n . Then, for a fixed edge e in K , we can define u n ∈ H ( G ) as u n ( x ) := (cid:40) √ λ n v n ( λ n x ) if x ∈ I e elsewhere on G , where λ n > is chosen so that supp { u n } ⊂ I e , and since C K ≥ Q ( u n ) = (cid:107) v n (cid:107) , R (cid:107) v n (cid:107) , R (cid:107) v (cid:48) n (cid:107) , R , passing to the limit one obtains C K ≥ C R . (cid:3) In general, inequalities in (20) are not sharp. However, in order to prove this, it is necessary to prelimi-narily detect under which assumptions the optimal constant is attained.
Lemma 3.2.
Let G satisfy (A) and assume also that it does not contain any terminal edge. If, in addition, C K (cid:54) = C R , then there exists (cid:54)≡ u ∈ H ( G ) such that Q ( u ) = C K . -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 9 Proof.
Let { u n } ⊂ H ( G ) be a maximizing sequence for C K , so that there exists ε n ↓ such that (cid:107) u n (cid:107) , K = ( C K − ε n ) (cid:107) u n (cid:107) , G (cid:107) u (cid:48) n (cid:107) , G . Moreover, as the functional Q is homogeneous, we can set without loss of generality (cid:107) u n (cid:107) , G =: µ n = (cid:114) C K − ε n , so that (cid:107) u n (cid:107) , K (cid:107) u (cid:48) n (cid:107) , G = 3 . (24)Note also that, as ε n → and C K > C R by assumption, (for large n ) there results µ n < µ R + δ , for some fixed δ > . Therefore, combining (24) and (16), (cid:107) u (cid:48) n (cid:107) , G = 13 (cid:107) u n (cid:107) , K ≤ (cid:107) u n (cid:107) , G ≤ µ n µ R (cid:107) u (cid:48) n (cid:107) , G + C √ µ R and rearranging terms (cid:16) − µ n µ R (cid:17) (cid:107) u (cid:48) n (cid:107) , G ≤ C √ µ R . Hence, { u n } is bounded in H ( G ) and there exists u ∈ H ( G ) such that u n (cid:42) u in H ( G ) and (cid:107) u n (cid:107) p, K → (cid:107) u (cid:107) p, K , ∀ p ∈ [1 , ∞ ] . (25)In addition, by weak lower semicontinuity (cid:107) u (cid:107) , G ≤ lim inf n (cid:107) u n (cid:107) , G and (cid:107) u (cid:48) (cid:107) , G ≤ lim inf n (cid:107) u (cid:48) n (cid:107) , G . (26)On the other hand, by (15), (cid:107) u (cid:48) n (cid:107) , G = 13 (cid:107) u n (cid:107) , K ≤ (cid:107) u n (cid:107) ∞ , G |K| ≤ C ∞ (cid:107) u n (cid:107) , G (cid:107) u (cid:48) n (cid:107) , G |K| and thus (cid:107) u (cid:48) n (cid:107) , G ≥ |K| C ∞ µ n ≥ |K| C ∞ ( µ R ) / > . (27)Assume then u ≡ on G . As (cid:107) u n (cid:107) ∞ , K → , we have (cid:107) u (cid:48) n (cid:107) , G ≤ (cid:107) u n (cid:107) , K ≤ (cid:107) u n (cid:107) ∞ , K |K| −→ as n → ∞ , which contradicts (27). As a consequence, u (cid:54)≡ and, combining with (25) and (26) C K ≥ Q ( u ) ≥ lim sup n Q ( u n ) = C K , which concludes the proof. (cid:3) Now, we can improve the estimates in Proposition 3.1, owing to the topological features of the graphs.Precisely, the following proposition distinguishes the cases when the graph does possess a terminal edge, i.e.(i) of Theorem 1.1, and when it does not, i.e. (ii)-(iv) of Theorem 1.1. Furthermore, for cases (i) and (ii) itprovides the exact value of C K . Proposition 3.3.
For every G satisfying (A) : (i) if G has (at least) a terminal edge, then C K = C R + ; (ii) if G has no terminal edge, then C K < C R + ; if G admits also a cycle-covering, then C K = C R . Proof.
Let us split the proof in two parts.
Part (i): graphs with a terminal edge.
Arguing as in the proof of Proposition 3.1, one can see that thereexists { v n } ⊂ H ( R + ) , v n ( x ) ≡ for large x , such that (cid:107) v n (cid:107) , R + (cid:107) v n (cid:107) , R + (cid:107) v (cid:48) n (cid:107) , R + −→ C R + , as n → ∞ , and that from { v n } one can construct a sequence { u n } ⊂ H ( G ) , supported only on a terminal edge of G ,such that Q ( u n ) = (cid:107) v n (cid:107) , R + (cid:107) v n (cid:107) , R + (cid:107) v (cid:48) n (cid:107) , R + . Hence, as C K ≥ Q ( u n ) , passing to the limit in n yields C K ≥ C R + , which in view of (20) completes theproof. Part (ii): graphs without terminal edges.
Owing to Proposition 3.1, it is sufficient to show that C K (cid:54) = C R + .Suppose by contradiction that C K = C R + . By Lemma 3.2 C K is attained, i.e. there exists u ∈ H ( G ) suchthat Q ( u ) = C R + . It follows by (22) that C R + = Q ( u ) ≤ (cid:107) u (cid:107) , G (cid:107) u (cid:107) , G (cid:107) u (cid:48) (cid:107) , G ≤ (cid:107) u ∗ (cid:107) , R + (cid:107) u ∗ (cid:107) , R + (cid:107) ( u ∗ ) (cid:48) (cid:107) , R + ≤ C R + , (28)with u ∗ being the decreasing rearrangement of u (which is again not restrictive to assume nonnegative).Thus (cid:107) u (cid:107) , K = (cid:107) u (cid:107) , G , so that u ≡ outside of K . As a consequence, ( u | K ) ∗ ∈ H ( R + ) is null for large x and at the same timeattains C R + , which is impossible. Then C K (cid:54) = C R + .Suppose, now, that G admits a cycle-covering. Given u ∈ H ( G ) , u ≥ , denote by (cid:98) u ∈ H ( R ) itssymmetric rearrangement, i.e. (cid:98) u ( x ) := inf { t ≥ ρ ( t ) ≤ | x |} x ∈ ( −|G| / , |G| / , with ρ defined by (21). The presence of a cycle-covering entails that u attains at least twice almost everyvalue in its image, and thus, from [2, Section 3], (cid:107) ( (cid:98) u ) (cid:48) (cid:107) , R ≤ (cid:107) u (cid:48) (cid:107) , G , (cid:107) (cid:98) u (cid:107) p, R = (cid:107) u (cid:107) p, G ∀ p ≥ . Hence Q ( u ) ≤ (cid:107) (cid:98) u (cid:107) , R (cid:107) (cid:98) u (cid:107) , R (cid:107) ( (cid:98) u ) (cid:48) (cid:107) , R ≤ C R , and, passing to the supremum on H ( G ) , in view of (20) we have C K = C R . (cid:3) The next proposition pushes forward the analysis of C K in the case of graphs with exactly one half-lineand no terminal edge. Proposition 3.4.
For every G satisfying (A) with exactly one half-line and no terminal edge C K > . Proof.
We divide the proof in two steps.
Step(i): C K ≥ . Denote by H the half-line of G and L the measure of K . For every c, α > , we definethe function u c,α ∈ H ( G ) such that u c,α ( x ) := (cid:40) c if x ∈ K ce − αx if x ∈ H , (29)(see for instance Figure 3). -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 11 ce − αx c Figure 3. example of the function u c,α as in the proof of Proposition 3.4.Direct computations yield (cid:107) u c,α (cid:107) , G = (cid:90) K c dx + (cid:90) H c e − αx dx = c L + c α = c (cid:16) L + 12 α (cid:17) (cid:107) u (cid:48) c,α (cid:107) , G = (cid:90) H c α e − αx dx = c α (cid:107) u c,α (cid:107) , K = (cid:90) K c dx = c L , so that Q ( u c,α ) = c Lc (cid:16) L + α (cid:17) c α = 8 αL (2 αL + 1) =: F ( α ) . Now, F (cid:48) ( α ) = 8 L (1 − Lα )(2 αL + 1) and then F has a maximum point at α = α := L . Hence, for every c, α > Q ( u c,α ) ≤ F ( α ) = 1 , (30)and thus, by definition, C K ≥ . Step(ii): C K (cid:54) = 1 . Assume, by contradiction, that C K = 1 . Since F ( α ) = Q ( u c,α ) , u c,α as in (29) with α = α := L is an optimizer of the functional Q on H ( G ) \{ } . As a consequence, for every ϕ ∈ H ( G ) , ddε Q ( u c,α + εϕ ) (cid:12)(cid:12) ε =0 = 0 , namely, after standard computations A ( u c,α ) (cid:90) G u (cid:48) c,α ϕ (cid:48) dx + B ( u c,α ) (cid:90) G u c,α ϕ dx = C ( u c,α ) (cid:90) K | u c,α | u c,α ϕ dx, with A ( u c,α ) := 2 (cid:107) u c,α (cid:107) , K (cid:0) (cid:107) u c,α (cid:107) , G (cid:107) u (cid:48) c,α (cid:107) , G (cid:1) , B ( u c,α ) := 4 (cid:107) u c,α (cid:107) , K (cid:107) u c,α (cid:107) , G (cid:107) u (cid:48) c,α (cid:107) , G C ( u c,α ) := 6 (cid:107) u c,α (cid:107) ,G (cid:107) u (cid:48) c,α (cid:107) , G . Now, arguing as in the proof of [2, Proposition 3.3], we get that − A ( u c,α ) ( u c,α ) (cid:48)(cid:48) e + B ( u c,α ) ( u c,α ) e = χ K C ( u c,α ) | ( u c,α ) e | ( u c,α ) e , ∀ e ∈ E , (31) and that u c,α ∈ dom( − ∆ G ) . However, this is impossible, since the Kirchhoff conditions (cid:88) e (cid:31) v d ( u c,α ) e dx e (v) = 0 , ∀ v ∈ K , are not fulfilled by u c,α at the vertex joining the compact core to the half-line. As a consequence, u c,α cannot be an optimizer of Q , so that C K (cid:54) = 1 . (cid:3) Remark . Note that, whenever G is a graph with exactly one half-line and no terminal edges, Q ( u ) ≤ for every u ∈ H ( G ) which is constant on the compact core, independently of its specific form on the half-line H . Indeed, letting c := u | K and m := (cid:107) u (cid:107) , H , Proposition 4.3 in [20] ensures that inf { (cid:107) v (cid:48) (cid:107) , H : v ∈ H ( H ) , v (0) = c and (cid:107) v (cid:107) , H = m } is attained by the exponential function ϕ ( x ) = ce − c xm . Thus, Q ( u ) = c |K| ( c |K| + m ) (cid:107) u (cid:48) (cid:107) , H ≤ c |K| ( c |K| + m ) (cid:107) ϕ (cid:48) (cid:107) , H ≤ by the proof of the previous proposition.We are now ready to prove Theorem 1.1. Proof of Theorem 1.1.
The first part of Theorem 1.1, i.e. the validity of (8), is proved by Lemma 2.1setting µ K as in (11)-(13). On the other hand, the values and the estimates on µ K in cases (i)-(iv) are astraightforward application of the results of Propositions 3.1, 3.3 and 3.4.It is then left to discuss the sole existence of the ground states. Since for µ > µ R nonexistence is immediateby (8) and for µ < µ K it is straightforward by Lemma 2.2, we only consider masses µ K ≤ µ ≤ µ R . Let ussplit the proof according to the four classes. Case (i): graphs with at least a terminal edge.
Recall that in this case µ K = µ R + . Assume first that µ > µ R + . It is well known that there exists v ∈ H ( R + ) , with v ( x ) ≡ for large x , such that E ( v, R + ) < .Then, arguing as in the proof of Lemma 2.1, one can construct a sequence { u n } ⊂ H µ ( G ) supported on aterminal edge of G for which E ( u n , K ) → −∞ , as n → ∞ . Hence, E G ( µ, K ) = −∞ and no ground state mayexist.Now, assume µ = µ R + . Suppose also, by contradiction, that there exists a ground state u ∈ H µ R + ( G ) .Since, by Lemma 2.1, E ( u, K ) = E G ( µ R + , K ) = 0 , we get (cid:107) u (cid:107) , K (cid:107) u (cid:48) (cid:107) , G = 3 and, dividing both terms by µ R + , Q ( u ) = 3 µ R + = C R + . Hence, since C K = C R + by Proposition 3.3, u is an optimizer for C K too. However, arguing as in (28)and the subsequent paragraph, this entails that the decreasing rearrangement on R + of u | K is a compactly-supported function attaining C R + , which is well known to be impossible. Hence, ground states do not existalso when µ = µ R + . Case (ii): graphs admitting a cycle-covering.
Recall that in this case µ K = µ R , which is then the uniquevalue of the mass to discuss. This case can be dealt with repeating the previous argument with µ R in placeof µ R + , C R in place of C + R and the symmetric rearrangement (cid:98) u in place of the decreasing rearrangement u ∗ . Case (iii): graphs with exactly one half-line.
Recall that in this case µ K ∈ ( µ R + , √ . By (16) we get E ( u, K ) ≥ (cid:107) u (cid:48) (cid:107) , G (cid:16) − µ µ R (cid:17) − C √ µ , and, since √ < µ R , E G ( µ, K ) > −∞ . Then, if µ > µ K , by Lemma 2.2 ground states exist. -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 13 B (cid:96) H Γ Γ Figure 4.
The graph G (cid:96) of Proposition 4.1On the other hand, assume µ = µ K and let u ∈ H µ K ( G ) be an optimizer of C K provided by Lemma 3.2(note that here C K > > C R ). Then (cid:107) u (cid:107) , K (cid:107) u (cid:48) (cid:107) , G = C K µ K = 3 , so that, recalling (8), E ( u, K ) = 12 (cid:107) u (cid:48) (cid:107) , G (cid:16) − (cid:107) u (cid:107) , K (cid:107) u (cid:48) (cid:107) , G (cid:17) = 0 = E G ( µ K , K ) , which entails that u is a ground state. Case (iv): graphs without terminal edges and cycle-coverings, and with at least two half-lines.
Recall thatin this case µ K ∈ ( µ R + , µ R ] . Here, if one assumes µ K (cid:54) = µ R (that is, C K (cid:54) = C R ), then it is possible to recoverthe argument in the proof of case (iii). (cid:3) Proofs of Theorems 1.2-1.3: how the metric affects C K In this section, we present the proofs of Theorems 1.2 (Section 4.1) and 1.3 (Section 4.2). Precisely, weshow that in cases (iii) and (iv) of Theorem 1.1, the estimates on the reduced critical mass µ K ((9) and(10), respectively) cannot be improved in general. In particular, we explain how upper/lower bounds givenby (9) and (10) can be asymptotically obtained suitably modifying the metric of the compact core of thegraph.As in the previous section, the bulk of the analysis focuses on C K . In addition, we will tacitly assume,in the following, that any graph satisfies (A) .4.1. Graphs with exactly one half-line and no terminal edges.
We first focus on graphs with exactlyone half-line and no terminal edges, for which we already proved that µ R + < µ K < √ , i.e. < C K < C R + ,and that ground states do exist if and only if µ ∈ [ µ K , µ R ] .The first preliminary result concerns the existence of a sequence of graphs whose optimal constantsconverge to C R + . Proposition 4.1.
For every (cid:96) > , let G (cid:96) be a graph consisting of one half-line and a compact core K (cid:96) givenby a cut edge B (cid:96) , of length (cid:96) , joining two disjoint subgraphs Γ , Γ , such that Γ ∩ H = ∅ (see, e.g., Figure4). Let also C K (cid:96) be the optimal constant (13) when G = G (cid:96) . Then, C K (cid:96) −→ C R + , as (cid:96) → ∞ . (32) Proof.
First, identify, for the sake of simplicity, B (cid:96) with the interval [0 , (cid:96) ] and H with the interval [ (cid:96), ∞ ) .With a little abuse of notation, let also { φ λ } be the sequence of the half-solitons on R + , i.e. the restrictionsto R + of the functions given by (7).Then, set λ := (cid:96) − / , for every (cid:96) > , and define the sequence { u λ } ⊂ H ( G (cid:96) ) such that u λ ( x ) := φ λ ( x ) if x ∈ B (cid:96) ∪ H φ λ (0) if x ∈ Γ φ λ ( (cid:96) ) if x ∈ Γ . v v v H K n Figure 5. example of a graph G n as in Proposition 4.2. Here, for every n , K n consists of n + 1 edges, all of length smaller than a fixed constant, joining the vertices v , v , plustwo additional edges linking v and v to H .Recalling that φ (0) = 1 , we have (cid:107) u λ (cid:107) , G (cid:96) = (cid:107) φ (cid:107) , R + + (cid:96) − / (cid:0) | Γ | + φ ( (cid:96) / ) | Γ | (cid:1) (cid:107) u (cid:48) λ (cid:107) , G (cid:96) = (cid:96) − (cid:107) φ (cid:48) (cid:107) , R + (cid:107) u λ (cid:107) , K (cid:96) = (cid:96) − (cid:107) φ (cid:107) L (0 ,(cid:96) / ) + (cid:96) − / (cid:0) | Γ | + φ ( (cid:96) / ) | Γ | (cid:1) , so that, as (cid:96) → ∞ , Q ( u λ ) = (cid:107) φ (cid:107) L (0 ,(cid:96) / ) + (cid:96) − / (cid:0) | Γ | + φ ( (cid:96) / ) | Γ | (cid:1)(cid:16) (cid:107) φ (cid:107) , R + + (cid:96) − / | Γ | + (cid:96) − / φ ( (cid:96) / ) | Γ | (cid:17) (cid:107) φ (cid:48) (cid:107) , R + −→ (cid:107) φ (cid:107) , R + (cid:107) φ (cid:107) , R + (cid:107) φ (cid:48) (cid:107) , R + = C R + since φ is an optimizer of C R + . Hence, since Q ( u λ ) ≤ C K (cid:96) < C R + for every (cid:96) > , (32) follows. (cid:3) Now, we can focus on the existence of a sequence of graphs whose optimal constants converge, on thecontrary, to .Recalling the proof of Proposition 3.4, note that 1 is the value of C K if one restrict the maximization in(13) to functions that are constant on the compact core. As already pointed out in the previous section,such functions cannot be actual optimizers, but nevertheless the following proposition shows that when thecompact core becomes too intricate, i.e. it has a large total length but a small diameter, then the optimizerscannot exhibit a significantly different behaviour. Proposition 4.2.
For every n ∈ N , let G n be the graph given by a half-line H and a compact core K n attached to the origin of H at some point v (see, e.g, Figure 5). Let also C K n be the optimal constant (13) when G = G n . If diam ( K n ) ≤ C uniformly in n , for some C > , and |K n | → + ∞ as n → ∞ , then C K n −→ , as n → ∞ . (33) Proof.
We divide the proof in two steps.
Step (i): behavior of the optimizers of C K n . By Proposition 3.4, C K n > > C R , and hence, for every n ∈ N , Lemma 3.2 entails that there exists u n ∈ H ( G n ) , u (cid:54) = 0 , such that Q ( u n ) = C K n . Moreover, bystandard regularity theory for (31) (see e.g. [2, Proposition 3.3]) we can assume u n > and, from thehomogeneity of Q , (cid:107) u n (cid:107) , G = µ K n so that (cid:107) u n (cid:107) , K n (cid:107) u (cid:48) n (cid:107) , G n = 3 , ∀ n ∈ N . (34)On the other hand, note that, (cid:107) u n (cid:107) ∞ , G n = (cid:107) u n (cid:107) ∞ , K n . Indeed, if this is not the case, simply setting v n ( x ) :=min { u n ( x ) , (cid:107) u n (cid:107) ∞ , K n } for every x ∈ G n , one can see that v n ∈ H ( G n ) and that Q ( u n ) < Q ( v n ) , which isa contradiction. As a consequence, denoting by x n ∈ K n a point such that u n ( x n ) = (cid:107) u n (cid:107) ∞ , G n =: M n , andby γ n ⊂ K n the smallest path from x n to vertex v , one can easily check that K n /γ n is connected, regardlessof the location of x n . -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 15 Now, consider the restriction of u n to γ n ∪ H , i.e. u n | γn ∪H , and let η n : [0 , + ∞ ) → R be its decreasingrearrangement on R + , so that η n (0) = M n , (cid:90) + ∞ | ( η n ) (cid:48) | dx ≤ (cid:90) γ n ∪H | u (cid:48) n | dx and (cid:90) + ∞ | η n | p dx = (cid:90) γ n ∪H | u n | p dx ∀ p ≥ . On the other hand, since K n /γ n is connected, the image of u n |K n/γn is connected in turn. Setting (cid:96) n := |K n /γ n | , we can define, therefore, the decreasing rearrangement of u n |K n/γn on the interval [0 , (cid:96) n ] , i.e. η n : [0 , (cid:96) n ] → R , which satisfies η n (0) = M n , (cid:90) (cid:96) n | ( η n ) (cid:48) | dx ≤ (cid:90) K n /γ n | u (cid:48) n | dx and (cid:90) (cid:96) n | η n | p dx = (cid:90) K n /γ n | u n | p dx ∀ p ≥ . Then, define the function η n : ( −∞ , (cid:96) n ] → R such that η n ( x ) := η n ( − x ) if x < η n ( x ) if x ∈ [0 , (cid:96) n ] . Exploiting the properties of η n and η n , one can easily see that η n ∈ H ( −∞ , (cid:96) n ) , (cid:107) η (cid:48) n (cid:107) L ( −∞ ,(cid:96) n ) ≤ (cid:107) u (cid:48) n (cid:107) , G n and (cid:107) η n (cid:107) L p ( −∞ ,(cid:96) n ) = (cid:107) u n (cid:107) p, G n for every p ≥ .As a further step, arguing as in [4, Proof of Lemma 4.4, Step 2–Step 3], one can construct a sequence offunctions v n ∈ H ( R ) such that, for some θ n := θ n ( u n ) ∈ (0 , µ K n ) (a) v n (0) = η n (0) = M n ;(b) (cid:82) R | v n | dx = (cid:82) (cid:96) n −∞ | η n | dx − θ n = µ K n − θ n ;(c) (cid:82) R | v (cid:48) n | dx ≤ (cid:82) (cid:96) n −∞ | η (cid:48) n | dx + C(cid:96) n θ / n ≤ (cid:107) u (cid:48) n (cid:107) , G n + C(cid:96) n θ / n ;(d) (cid:82) R | v n | dx ≥ (cid:82) (cid:96) n −∞ | η n | dx − C(cid:96) n θ n = (cid:107) u n (cid:107) , G n − C(cid:96) n θ n ;where C > does not depend on n .Hence, combining (a)-(d) with (14) for G = R , (cid:107) u n (cid:107) , G n − C(cid:96) n θ n ≤ (cid:107) v n (cid:107) , R ≤ µ K n − θ n ) µ R (cid:107) v (cid:48) n (cid:107) , R ≤ µ K n − θ n ) µ R (cid:16) (cid:107) u (cid:48) n (cid:107) , G n + C(cid:96) n θ / n (cid:17) and thus, rearranging terms and recalling that θ n < µ K n < √ < µ R (possibly redefining C ), (cid:107) u n (cid:107) , G n ≤ µ K n µ R (cid:107) u (cid:48) n (cid:107) , G n + C(cid:96) n . (35)Moreover, plugging (34) into (35), we get (cid:107) u (cid:48) n (cid:107) , G n = 13 (cid:107) u n (cid:107) , K n ≤ (cid:107) u n (cid:107) , G n ≤ µ K n µ R (cid:107) u (cid:48) n (cid:107) , G n + C(cid:96) n , that is (cid:16) − µ K n µ R (cid:17) (cid:107) u (cid:48) n (cid:107) , G n ≤ C(cid:96) n . (36)Since both |K n | → + ∞ as n → ∞ and | γ n | ≤ diam ( K n ) ≤ C for every n , then (cid:96) n ∼ |K n | provided n is largeenough, so that (36) implies (cid:107) u (cid:48) n (cid:107) , G n |K n | ≤ C (37)uniformly in n . Step (ii): proof of (33) . Let m n := min x ∈K n u n ( x ) and y n ∈ K n be such that u n ( y n ) = m n . Furthermore,let z n ∈ H be the closest point (of the half-line) to the compact core such that u n ( z n ) = m n (possibly z n = v if the minimum is attained at the vertex joining H and K n ).We then consider the functions defined as (cid:101) u n ( x ) := (cid:40) m n if x ∈ K n u n ( x + z n ) if x ∈ H . Clearly, (cid:101) u n ∈ H ( G n ) and, by construction, (cid:107) (cid:101) u (cid:107) , G n ≤ (cid:107) u n (cid:107) , G n , (cid:107) (cid:101) u (cid:48) n (cid:107) , G n ≤ (cid:107) u (cid:48) n (cid:107) , G n , so that C K n ≤ (cid:107) u n (cid:107) ∞ , G n |K n |(cid:107) u n (cid:107) , G n (cid:107) u n (cid:107) , G n = (cid:107) u n (cid:107) ∞ , G n m n (cid:107) (cid:101) u n (cid:107) , K n (cid:107) (cid:101) u n (cid:107) , G n (cid:107) (cid:101) u (cid:48) n (cid:107) , G n ≤ (cid:107) u n (cid:107) ∞ , G n m n , (38)the last inequality being motivated by Remark 3.1.As C K n > , it is then left to estimate (cid:107) u n (cid:107) ∞ , G n /m n . First, recall that (cid:107) u n (cid:107) ∞ , G n = (cid:107) u n (cid:107) ∞ , K n = u n ( x n ) ,with x n ∈ K n defined in Step (i), whereas m n = u n ( y n ) . Let Γ n ⊂ K n be the smallest path from y n to x n .Then we have m n (cid:107) u n (cid:107) ∞ , G n = 1 − u n (¯ x n ) − u n (¯ y n ) (cid:107) u n (cid:107) ∞ , G n = 1 − (cid:82) Γ n u (cid:48) n dx (cid:107) u n (cid:107) ∞ , G n . (39)Now, let us show that lim n (cid:82) Γ n u (cid:48) n dx (cid:107) u n (cid:107) ∞ , G n = 0 . (40)By Hölder’s inequality (cid:90) Γ n | u (cid:48) n | dx ≤ | Γ n | / (cid:107) u (cid:48) n (cid:107) , Γ n ≤ diam ( K n ) / (cid:107) u (cid:48) n (cid:107) , G n , so that (cid:82) Γ n u (cid:48) n dx (cid:107) u n (cid:107) ∞ , G n ≤ diam ( K n ) / (cid:107) u (cid:48) n (cid:107) , G n (cid:107) u n (cid:107) ∞ , G n . (41)Moreover, by (34), (cid:107) u (cid:48) n (cid:107) , G n = 13 (cid:107) u n (cid:107) , K n ≤ (cid:107) u n (cid:107) ∞ , G n |K n | , which yields at (cid:107) u (cid:48) n (cid:107) , G n (cid:107) u n (cid:107) ∞ , G n ≤ |K n | / √ (cid:107) u n (cid:107) ∞ , G n . (42)By (15) and the fact that the optimal constant C ∞ = √ , for every G n (see Section 2) |K n | / √ (cid:107) u n (cid:107) ∞ , G n ≤ |K n | / √ µ / K n (cid:107) u (cid:48) n (cid:107) , G n ≤ C |K n | / −→ as n → ∞ , making use of (37). Plugging into (42), we get lim n (cid:107) u (cid:48) n (cid:107) , G n (cid:107) u n (cid:107) ∞ , G n = 0 , which, combined with (41) and diam ( K n ) < C , implies (40). Hence, passing to the limit in (39), lim n m n (cid:107) u n (cid:107) ∞ , G n = 1 and, recalling (38) and the fact that C K n > , (33) follows. (cid:3) The proof of Theorem 1.2 is therefore a straightforward application of Propositions 4.1 and 4.2.
Proof of Theorem 1.2.
Fix ε > . The existence of G ε is immediately guaranteed by Proposition 4.1. Atthe same time, considering graphs as in Proposition 4.2, we easily obtain G ε . (cid:3) -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 17 B (cid:96) H H Γ Figure 6. the signpost graph.4.2.
Graphs without terminal edges and cycle-coverings, with at least two half-lines.
In thislast section, we discuss graphs without terminal edges and cycle-coverings, with at least two half-lines (forwhich it is true that µ R + < µ K ≤ µ R , i.e. C R ≤ C K < C R + , and that ground states exist if and only if µ ∈ [ µ K , µ R ] , provided that µ K (cid:54) = µ R ).Recall that, in a graph having at least two half-lines and neither a terminal edge nor a cycle-covering,there exists of at least one cut edge in the compact core, that is an edge that provides the only connectionbetween two disjoint subgraphs. For the sake of simplicity, in what follows we develop our analysis in thecase of a signpost graph, as in Figure 6, with exactly one cut edge and a circle attached to it.For every (cid:96) > , let G (cid:96) be the graph depicted in Figure 6, with compact core K (cid:96) given by the cut edge B (cid:96) ,of length (cid:96) , and the circle Γ . Moreover, denote by H , H the left and the right half-line of G (cid:96) , respectively,both identified with [ (cid:96), + ∞ ) . In addition, identify the cut edge B (cid:96) with the interval [0 , (cid:96) ] , in such a waythat x B (cid:96) = (cid:96) ( x B (cid:96) being the coordinate on B (cid:96) ) denotes the vertex v such that { v } = H ∩ H . Let also C K (cid:96) be the optimal constant (13) when G = G (cid:96) .We then have the first asymptotic result. Proposition 4.3.
Let G (cid:96) be the graph depicted in Figure 6. Then, the following asymptotics holds: C K (cid:96) −→ C R + as (cid:96) → ∞ . (43) Proof.
As in the proof of Proposition 4.1, let { φ λ } be the sequence of the half-solitons on R + , i.e. thesequence of the restrictions to R + of the functions given by (7). Then, set λ := (cid:96) − / , for every (cid:96) > , anddefine the sequence { u λ } ⊂ H ( G (cid:96) ) such that u λ ( x ) := (cid:40) φ λ (0) if x ∈ Γ φ λ ( x ) if x ∈ G (cid:96) / Γ . Note that, as both H and H are identified with the interval [ (cid:96), ∞ ) , then u λ |H ≡ u λ |H . Hence, directcomputations (recalling φ (0) = 1 ) yield (cid:107) u λ (cid:107) , G (cid:96) = λ | Γ | + (cid:90) B (cid:96) ∪H λ | φ ( λx ) | dx + (cid:90) H λ | φ ( λx ) | dx = (cid:96) − / | Γ | + (cid:107) φ (cid:107) , R + + (cid:107) φ (cid:107) L ( (cid:96) / , ∞ ) (cid:107) u (cid:48) λ (cid:107) , G (cid:96) = (cid:90) B (cid:96) ∪H λ | φ (cid:48) ( λx ) | dx + (cid:90) H λ | φ (cid:48) ( λx ) | dx = (cid:96) − (cid:16) (cid:107) φ (cid:48) (cid:107) , R + + (cid:107) φ (cid:48) (cid:107) L ( (cid:96) / , ∞ ) (cid:17) (cid:107) u λ (cid:107) , K (cid:96) = λ | Γ | + (cid:90) B (cid:96) λ | φ ( λx ) | dx = (cid:96) − (cid:16) (cid:96) − / | Γ | + (cid:107) φ (cid:107) L (0 ,(cid:96) / ) (cid:17) and, since as (cid:96) → ∞ , (cid:96) − / | Γ | , (cid:107) φ (cid:107) L ( (cid:96) / , ∞ ) and (cid:107) φ (cid:48) (cid:107) L ( (cid:96) / , ∞ ) tend to , and (cid:107) φ (cid:107) L (0 ,(cid:96) / ) → (cid:107) φ (cid:107) , R + , itfollows that Q ( u λ ) −→ (cid:107) φ (cid:107) , R + (cid:107) φ (cid:107) , R + (cid:107) φ (cid:48) (cid:107) , R + = C R + . Combining with the fact that Q ( u λ ) ≤ C K < C R + for every λ , we obtain (43). (cid:3) Remark . Clearly, the proof of Proposition 4.3 provides an example of graph such that C K (cid:54) = C R , showing,as anticipated in the Introduction, that the class of graphs of case (iv) in Theorem 1.1 fulfilling µ K (cid:54) = µ R isnot empty. Remark . It is readily seen that the proof of Proposition 4.3 can be straightforwardly generalized to thecase of several cut edges joining components (possibly) different than circles, simply repeating the previousconstruction for any fixed cut edge.The existence of a sequence of graphs whose optimal constants converge to C R , on the contrary, requiresmore efforts. We aim at obtaining such a result by discussing the behaviour of G (cid:96) , as before, but now in theregime (cid:96) → .Some preliminary steps are required: a characterization of the behavior of the (possible) optimizers of C K (cid:96) (Lemmas 4.4–4.5) and a monotonicity property of C K (cid:96) with respect to (cid:96) (Lemma 4.6). For the sake ofsimplicity, we denote in the following by H the union of the half-lines of G (cid:96) , i.e. H := H ∪ H . Lemma 4.4.
Let G (cid:96) be the graph depicted in Figure 6. If u ∈ H ( G (cid:96) ) , u > , is an optimizer of C K (cid:96) , then M B (cid:96) := max x ∈B (cid:96) u ( x ) > max x ∈H u ( x ) =: M H . Proof.
Suppose, by contradiction, that M B (cid:96) ≤ M H . Then, for a.e. t in the image of u , { x ∈ G (cid:96) : u ( x ) = t } ≥ . As a consequence, with (cid:98) u being the symmetric rearrangement of u , C K (cid:96) = Q ( u ) ≤ (cid:107) u (cid:107) , G (cid:96) (cid:107) u (cid:107) , G (cid:96) (cid:107) u (cid:48) (cid:107) , G (cid:96) ≤ (cid:107) (cid:98) u (cid:107) , R (cid:107) (cid:98) u (cid:107) , R (cid:107) (cid:98) u (cid:48) (cid:107) , R ≤ C R . (44)Now, if C K (cid:96) (cid:54) = C R , then it directly entails a contradiction, in view of Proposition 3.1. On the other hand, if C K (cid:96) = C R , then (44) implies that (cid:98) u is an optimizer of C R on the real line and, by the properties of symmetricrearrangements, that (cid:107) (cid:98) u (cid:48) (cid:107) , R = (cid:107) u (cid:48) (cid:107) , G (cid:96) . (45)However, since u runs through a vertex of degree and M B (cid:96) ≤ M H , there exists a subregion of G (cid:96) of positivemeasure where all the values attained by u possess at least three pre-images ,which contradicts (45). (cid:3) Lemma 4.5.
Let G (cid:96) be the graph depicted in Figure 6. If u ∈ H ( G (cid:96) ) , u > , is an optimizer of C K (cid:96) , then u | H is symmetric with respect to v (recall { v } := H ∩ H ) and non-increasing both on H and on H .Proof. Let us prove first that u |H attains the maximum M H at the vertex v and that it is non-increasingboth on H and on H . To this aim, assume by contradiction that u |H does not posses any of the previousfeatures.Let x be the closest point to the circle Γ in B (cid:96) such that u ( x ) = M B (cid:96) (see, e.g., Figure 7(a)). Denote,also, by G the subgraph of G (cid:96) obtained by removing the circle Γ and the segment that joins x and thevertex v (cid:48) given by the intersection between Γ and B (cid:96) . Then, let A := { x ∈ G : u ( x ) > M H } B := { x ∈ G : u ( x ) ≤ M H } . and let u A , u B be the restrictions of u to A and B , respectively. Note that A and B do not need to beconnected sets, but the images of u A and u B are nevertheless both connected.By Lemma 4.4 M H < M B (cid:96) = max x ∈ A u ( x ) , and hence A ⊂ B (cid:96) and x + | A | ≤ (cid:96) . Therefore, letting u ∗ A bethe decreasing rearrangement of u A on [0 , | A | ] (see, e.g., Figure 7(b)), we get u ∗ A (0) = M B (cid:96) , u ∗ A ( | A | ) = M H and (cid:107) ( u ∗ A ) (cid:48) (cid:107) L (0 , | A | ) ≤ (cid:107) u (cid:48) A (cid:107) L ( A ) . On the other hand, since M H = max x ∈ B u ( x ) , it follows that { x ∈ B : u ( x ) = t } ≥ for almost every value t ∈ (0 , M H ] , and rearranging symmetrically u B on R (see again Figure7(b)) we get (cid:98) u B (0) = M H and (cid:107) (cid:98) u (cid:48) B (cid:107) L ( R ) ≤ (cid:107) u (cid:48) B (cid:107) L ( B ) . In addition, one can see that if u |H does not attain -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 19 M B (cid:96) M H x M H A B (a) the function u on G (cid:96) | A | M B (cid:96) M H M H − (cid:96) − x −| A | (cid:96) − x −| A | (b) the decreasing rearrangement of u A on [0 , | A | ] and the symmetric rearrangement of u B on R M B (cid:96) M H x x + | A | (c) the function ˜ u on G (cid:96) Figure 7. the steps of the construction of ˜ u starting from u as in the proof of Lemma 4.5. M H at v or is non-monotone either on H or on H , then there exists a subregion of B with positive measuresuch that all the values attained there by u | B are actually realized at least three times each, leading to (cid:107) (cid:98) u (cid:48) B (cid:107) L ( R ) < (cid:107) u (cid:48) B (cid:107) L ( B ) (46)Then, we can use u ∗ A , (cid:98) u B in order to construct a new function (cid:101) u ∈ H ( G (cid:96) ) (see, e.g., Figure 7(c)). First,on G (cid:96) /G , set (cid:101) u ≡ u , and on B (cid:96) ∩ [ x , x + | A | ] , set (cid:101) u ( x ) = u ∗ A ( x − x ) . In addition, consider the restriction of (cid:98) u B to the interval (cid:2) − (cid:96) − x −| A | , (cid:96) − x −| A | (cid:3) (see again Figure 7(b)), rearrange it decreasingly on [0 , (cid:96) − x −| A | ] ,denoting by (cid:98) u ∗ B such a rearrangement, and then set (cid:101) u ( x ) = (cid:98) u ∗ B ( x − x − | A | ) on B (cid:96) \ [0 , x + | A | ] (assumingthat on B (cid:96) the vertex v (cid:48) is represented by x = 0 ). Finally, define (cid:101) u on H as the restriction of (cid:98) u B to R / (cid:2) − (cid:96) − x −| A | , (cid:96) − x −| A | (cid:3) , glued together at .As a consequence, we have (cid:107) (cid:101) u (cid:107) , G (cid:96) = (cid:107) u (cid:107) , G (cid:96) (cid:107) (cid:101) u (cid:48) (cid:107) , G (cid:96) < (cid:107) u (cid:48) (cid:107) , G (cid:96) (cid:107) (cid:101) u (cid:107) , K (cid:96) ≥ (cid:107) u (cid:107) , K (cid:96) , the strict inequality being given by (46), and thus C K (cid:96) = Q ( u ) < Q ( (cid:101) u ) ≤ C K (cid:96) , i.e. a contradiction. Hence, u |H has a maximum at v and is non-increasing both on H and on H .Moreover, if u H is non-symmetric with respect to v , then a symmetric rearrangement on H would providea better competitor (it is well known, indeed, that on the real line a symmetric rearrangement of a non-symmetric function has a strictly smaller kinetic energy), serving again a contradiction. (cid:3) Remark . Note that neither Lemma 4.4 nor Lemma 4.5 discusses the existence of optimizers for C K (cid:96) ingeneral. They only state some a priori conditions to be satisfied by possible minimizers. Lemma 4.6.
Let G (cid:96) be the graph depicted in Figure 6. Then, for every < (cid:96) ≤ (cid:96) , it is true that C K (cid:96) ≤ C K (cid:96) . Proof. If C K (cid:96) = C R , then the claim is trivial by (20). Let thus C K (cid:96) > C R . In view of Proposition 3.3, thereexists u ∈ H ( G (cid:96) ) such that Q ( u ) = C K ( G (cid:96) ) , and, by Lemma 4.5, it is symmetric on H and non-increasingboth on H and on H .Setting Λ := (cid:96) − (cid:96) , let I ⊂ H be a symmetric interval of measure | I | = Λ centered at v , u | I the restrictionof u to I and u ∗| I its decreasing rearrangement of [0 , Λ] . Subsequently, consider the function v ∈ H ( G (cid:96) ) defined by v ( x ) := u ( x ) if x ∈ Γ ∪ ( B (cid:96) ∩ [0 , (cid:96) ]) u ∗| I ( x − (cid:96) ) if x ∈ B (cid:96) ∩ [ (cid:96) , (cid:96) ] u ( | x | + Λ / if x ∈ H . As a consequence (cid:107) v (cid:107) , G (cid:96) = (cid:107) u (cid:107) , G (cid:96) , (cid:107) v (cid:48) (cid:107) , G (cid:96) < (cid:107) u (cid:107) , G (cid:96) and (cid:107) v (cid:107) , K (cid:96) > (cid:107) u (cid:107) , K (cid:96) . yielding C K (cid:96) = Q ( u ) < Q ( v ) ≤ C K (cid:96) . (cid:3) Now, we can prove the existence of a sequence of graphs whose optimal constants converge to C R . Proposition 4.7.
Let G (cid:96) be the graph depicted in Figure 6. Then, the following asymptotics holds: C K (cid:96) −→ C R as (cid:96) → . Proof. If C K (cid:96) = C R , for some (cid:96) > , then the statement follows by Lemma 4.6. Assume, then, C K (cid:96) > C R ,for every (cid:96) > . By Proposition 3.3, there exists a sequence { v (cid:96) } (cid:96)> such that v (cid:96) ∈ H ( G (cid:96) ) , v (cid:96) > and Q ( v (cid:96) ) = C K ( G (cid:96) ) . Also, by homogeneity of Q , we can set (cid:107) v (cid:96) (cid:107) , G (cid:96) = µ K (cid:96) , so that (cid:107) v (cid:96) (cid:107) , K (cid:96) (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) = 3 ∀ (cid:96) > . (47)Combining (47) with the modified Gagliardo-Nirenberg inequality (16) leads to (cid:107) v (cid:96) (cid:107) , K (cid:96) (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) ≤ (cid:107) v (cid:96) (cid:107) , G (cid:96) (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) ≤ (cid:16) µ K (cid:96) µ R (cid:17) (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) + C √ µ R (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) = 3 (cid:18) µ K (cid:96) µ R (cid:19) + C √ µ R (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) and, rearranging terms, we get (cid:32) − µ K (cid:96) µ R (cid:33) (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) ≤ C √ µ R . (48)As a consequence, if lim sup (cid:96) → (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) = ∞ , then (48) immediately implies µ K (cid:96) → µ R , that is C K (cid:96) → C R .On the other hand, assume that (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) is bounded. Define a new graph (cid:101) G (cid:96) with compact core (cid:101) K (cid:96) asfollows: for every (cid:96) > , replace the cut edge B (cid:96) of G (cid:96) with two edges, say B (cid:96) , B (cid:96) , joining the same vertices,each of length (cid:96) . Furthermore, let (cid:101) v (cid:96) be the function in H ( (cid:101) G (cid:96) ) defined by (cid:101) v (cid:96) ( x ) := v (cid:96) ( x ) if x / ∈ B (cid:96) ∪ B (cid:96) v (cid:96) ( x ) if x ∈ B i(cid:96) , i = 1 , . -CRITICAL NLS ON GRAPHS WITH LOCALIZED NONLINEARITY 21 Now, (cid:107) (cid:101) v (cid:96) (cid:107) pp, (cid:101) K (cid:96) = (cid:90) Γ (cid:96) | (cid:101) v (cid:96) | p dx + (cid:88) i =1 (cid:90) B i(cid:96) | (cid:101) v (cid:96) | p dx = (cid:90) Γ (cid:96) | v (cid:96) | p dx + 2 (cid:90) (cid:96) | (cid:101) v (cid:96) ( x ) | p dx = (cid:90) Γ (cid:96) | v (cid:96) | p dx + 2 (cid:90) (cid:96) | v (cid:96) ( x/ | p dx = (cid:107) v (cid:96) (cid:107) pp, K (cid:96) + 3 (cid:90) B (cid:96) | v (cid:96) | p dx , (49)and (cid:107) (cid:101) v (cid:48) (cid:96) (cid:107) , (cid:101) G (cid:96) = (cid:90) (cid:101) G (cid:96) \B (cid:96) ∪B (cid:96) | (cid:101) v (cid:48) (cid:96) | dx + (cid:88) i =1 (cid:90) B i(cid:96) | (cid:101) v (cid:48) (cid:96) | dx = (cid:90) G (cid:96) \B (cid:96) | v (cid:48) (cid:96) | dx + 2 (cid:90) (cid:96) | (cid:101) v (cid:48) (cid:96) ( x ) | dx = (cid:90) G (cid:96) \B (cid:96) | v (cid:48) (cid:96) | dx + 12 (cid:90) (cid:96) | v (cid:48) (cid:96) ( x/ | dx = (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) . (50)Since by construction (cid:101) G (cid:96) admits a cycle-covering, then C (cid:101) K (cid:96) = C R by Proposition 3.3, which, combined with(49) and (50), entails that C R ≥ Q (˜ v (cid:96) ) = (cid:107) v (cid:96) (cid:107) , G (cid:96) + 3 (cid:82) B (cid:96) | v (cid:96) | dx (cid:16) (cid:107) v (cid:96) (cid:107) , G (cid:96) + 3 (cid:82) B (cid:96) | v (cid:96) | dx (cid:17) (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) . (51)In addition, given that (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) is bounded by assumption, (15) and (10) ensure that (cid:107) v (cid:96) (cid:107) ∞ , G (cid:96) ≤ √ (cid:107) v (cid:96) (cid:107) / , G (cid:96) (cid:107) v (cid:48) (cid:96) (cid:107) / , G (cid:96) ≤ C , ∀ (cid:96) > , so that, as (cid:96) → , (cid:90) B (cid:96) | v (cid:96) | dx ≤ (cid:107) v (cid:96) (cid:107) ∞ , G (cid:96) (cid:96) → (cid:90) B (cid:96) | v (cid:96) | dx ≤ (cid:107) v (cid:96) (cid:107) ∞ , G (cid:96) (cid:96) → . Hence, passing to the limit in (51), C R ≥ lim sup (cid:96) → (cid:107) v (cid:96) (cid:107) , K (cid:96) (cid:107) v (cid:96) (cid:107) , G (cid:96) (cid:107) v (cid:48) (cid:96) (cid:107) , G (cid:96) = lim sup (cid:96) → C K (cid:96) ≥ lim inf (cid:96) → C K (cid:96) ≥ C R thus concluding the proof. (cid:3) Remark . Proposition 4.7 clearly holds as well replacing the circle Γ with a generic compact graphadmitting a cycle-covering. Proof of Theorem 1.3.
Given ε > , the existence of G ε and G ε is a direct consequence of Propositions 4.3and 4.7. (cid:3) References [1] R. Adami, S.Dovetta, E. Serra, P. Tilli, Dimensional crossover with a continuum of critical exponents for NLS on doublyperiodic metric graphs,
Analysis and PDEs , to appear.[2] R. Adami, E. Serra, P. Tilli, NLS ground states on graphs,
Calc. Var. Partial Differential Equations (2015), no. 1,743–761.[3] R. Adami, E. Serra, P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs, J.Funct. Anal. (2016), no. 1, 201–223.[4] R. Adami, E. Serra, P. Tilli, Negative energy ground states for the L2-critical NLSE on metric graphs,
Comm. Math.Phys. (2017), no. 1, 387–406.[5] G. Berkolaiko, P. Kuchment,
Introduction to quantum graphs , Mathematical Surveys and Monographs 186, AmericanMathematical Society, Providence, RI, 2013.[6] W. Borrelli, R. Carlone, L. Tentarelli, Nonlinear Dirac equation on graphs with localized nonlinearities: bound states andnonrelativistic limit, arXiv:1807.06937 [math.AP] (2018). [7] C. Cacciapuoti, S. Dovetta, E. Serra, Variational and stability properties of constant solutions to the NLS equation oncompact metric graphs,
Milan Journal of Mathematics , to appear.[8] C. Cacciapuoti, D. Finco, D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graphwith potentials,
Nonlinearity (2017), no. 8, 3271–3303.[9] T. Cazenave, Semilinear Schrödinger equations , Courant Lecture Notes in Mathematics 10, American MathematicalSociety, Providence, RI, 2003.[10] J. Dolbeault, M.J. Esteban, A. Laptev, L. Michael, One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarkson duality and flows,
J. Lond. Math. Soc. (2) (2014), no. 2, 525-550.[11] S. Dovetta, Existence of infinitely many stationary solutions of the L -subcritical and critical NLSE on compact metricgraphs, J. Differential Equations (2018), no. 7, 4806–4821.[12] S. Dovetta, L. Tentarelli, Ground states of the L -critical NLS equation with localized nonlinearity on a tadpole graph,arXiv:1804.11107 [math.AP] (2018).[13] A. Duca, Global exact controllability of the bilinear Schrödinger potential type models on quantum graphs,arXiv:1710.06022 [math.OC] (2017).[14] A. Kairzhan, D.E. Pelinovsky, Nonlinear instability of half-solitons on star graphs, J. Differential Equations (2018),no. 12, 7357–7383.[15] Y. Li, F. Li, J. Shi, Ground states of nonlinear Schrödinger equation on star metric graphs,
J. Math. Anal. Appl. (2018), no. 2, 661–685.[16] J. L. Marzuola, D.E. Pelinovsky, Ground state on the dumbbell graph, arXiv:1509.04721 [math.AP] (2017).[17] D. Mugnolo, D. Noja, C. Seifert, Airy-type evolution equations on star graphs,
Anal. PDE (2018), no. 7, 1625–1652.[18] E. Serra, L. Tentarelli, Bound states of the NLS equation on metric graphs with localized nonlinearities, J. DifferentialEquations (2016), no. 7, 5627–5644.[19] E. Serra, L. Tentarelli, On the lack of bound states for certain NLS equations on metric graphs,
Nonlinear Anal. (2016), 68–82.[20] L. Tentarelli, NLS ground states on metric graphs with localized nonlinearities,
J. Math. Anal. Appl. (2016), no. 1,291–304.
Politecnico di Torino, Dipartimento di Scienze Matematiche “G.L. Lagrange”, Corso Duca degli Abruzzi,24, 10129, Torino, ItalyUniversità degli Studi di Torino, Dipartimento di Matematica “G. Peano”, Via Carlo Alberto, 10, 10123,Torino, Italy
E-mail address : [email protected] Università degli Studi di Roma“La Sapienza”, Dipartimento di Matematica “G. Castelnuovo”, P.le AldoMoro, 5, 00185, Roma, Italy.
E-mail address ::