Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs
Simone Dovetta Louis Jeanjean Enrico Serra


 
      Extending Cislunar Communication Network Reach Using Reconfigurable Intelligent Surfaces
     Aamer Mohamed Huroon*§,
    Baris Donmez§,
    Yu-Chih Huang†,
     Gunes Karabulut Kurt§,
   and Li-Chun Wang‡
   

* 
            EECS International Graduate Program, National Yang Ming Chiao Tung University  

    † 
            Institute of Communications Engineering, National Yang Ming Chiao Tung University

         ‡ Department of Electrical and Computer Engineering, National Yang Ming Chiao Tung University 

       § 
           Dept. of Electrical Engineering, Polytechnique Montréal, Montréal, Canada

       E-mail: *†‡{aamer.ee08, jerryhuang, wang}@nycu.edu.tw §{aamer.mohamed-huroon-mohamed, baris.donmez, gunes.kurt}@polymtl.ca 
    April 29, 2024
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We consider the existence of normalized solutions to nonlinear Schrödinger equations on noncompact metric graphs in the L^2 supercritical regime. For sufficiently small prescribed mass (L^2 norm), we prove existence of  positive solutions on two classes of graphs: periodic graphs and noncompact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques.  
A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an L^2 supercritical nonlinearity extended on the whole graph and unraveling the role of topology in the existence of solutions.


^*simone.dovetta@polito.it  ^†louis.jeanjean@univ-fcomte.fr  ^enrico.serra@polito.it



Statements and Declarations: the authors have no relevant financial or non-financial interests to disclose. 

Data availability: data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.




§ INTRODUCTION



Throughout the paper we consider metric graphs =(_, _) such that

	
  *  is connected and has at most countably many edges;
	
  * deg()<∞ for every ∈_, where deg() is the total number of edges incident at the vertex ;
	
  * e_0:=inf_e∈_|e|>0, where |e| is the length of the edge e.

For simplicity, if a graph  satisfies these properties, we say that  is in the class  𝐆, and we write ∈𝐆. The notion of metric graph is detailed in <cit.>.
A graph in 𝐆 is noncompact if at least one of its edges is unbounded (that is, the graph has at least one half-line) or if the total number of edges in the graph is infinite. It is evident that the set of all noncompact graphs in 𝐆 is extremely wide and varied, as it contains structures that may exhibit sensibly different behaviours, making thus impossible to handle them all at once. For this reason, in the present paper we will focus on two specific families of noncompact graphs: graphs with finitely many edges (and thus with at least one half-line, see e.g. Figure <ref>) and periodic graphs (see e.g. Figure <ref>), i.e. graphs with infinitely many edges, whose length is uniformly bounded from above, arranged in a periodic pattern (for a rigorous definition of periodic graphs see e.g. <cit.>). 





Given a graph ∈𝐆 and p>2, we look for solutions (u, λ) of the problem

    u”+u^p-1=λ u    on every e∈_
    
    		u∈ H^1(),   u>0    on 
    ∑_e≻du/dx_e()=0    for every ∈_ ,

where e≻ indicates that the edge e is incident at the vertex , and du/dx_e() denotes the derivative of u at  along the edge e identified with the interval [0,|e|] in such a way that the vertex  corresponds to 0.  

Our focus here is devoted to normalized solutions of (<ref>), i.e. solutions u in the mass constrained space

    H_μ^1():={v∈ H^1() : v_L^2()^2=μ},

in the L^2-supercritical regime p>6.  To this end, we look for critical points of the energy functional E:H^1()→

    E(u,):=1/2u'_L^2()^2-1/pu_L^p()^p

constrained to H_μ^1(). It is well-known that these critical points
satisfy (<ref>) (except possibly for the sign condition), for a suitable λ that arises as a Lagrange multiplier associated with the mass constraint.


Recently, much effort has been devoted to establish the
existence of normalized solutions of nonlinear Schrödinger equations (<ref>) on metric graphs in the
L^2-subcritical (i.e., p∈(2,6)) or L^2-critical
  regimes (i.e., p=6). In these two cases, the energy functional
E(·, ) is bounded from below and coercive on the mass
constraint (one needs to require μ smaller than a threshold when p=6).  Various results of existence, non-existence and multiplicity have been obtained (see e.g. <cit.>). 


Conversely, in the L^2-supercritical regime on general metric graphs,
i.e. when p>6, the energy functional E(·, ) is always
unbounded from below on H_μ^1() and the approaches developed when p ≤ 6 are not directly applicable anymore. As a matter of fact, the existence of normalized solutions in the case p>6 only started to be considered very recently in <cit.> when the graph is compact or when the nonlinearity is restricted to a compact subset of the graph.


The main existence results of the paper are stated in the next two theorems.
Here, by pendant we mean a bounded edge with a vertex of degree 1, whereas by a signpost we mean the union of a bounded edge and a loop, glued together at a common vertex. 

	
	Let ∈𝐆 be a noncompact metric graph with finitely many edges and at least one pendant or one signpost. Then, for every p>6 there exists μ_p,>0, depending on p and , such that, for every μ∈(0,μ_p,], problem (<ref>) admits a solution u∈ H_μ^1() satisfying E(u,) >0 and λ >0.




	
	Let ∈𝐆 be a periodic graph. Then, for every p>6 there exists μ_p,>0, depending on p and , such that, for every μ∈(0,μ_p,], problem (<ref>) admits a  solution u∈ H_μ^1() satisfying E(u,) >0 and λ >0.

Theorems <ref>–<ref>  are the first results on the existence of normalized solutions to (<ref>) when p is L^2-supercritical, the graph  is noncompact, and the nonlinearity acts on the whole of  (whereas without the mass constraint some results can be found e.g. in <cit.>). The only results for supercritical p obtained so far <cit.> either require  to be compact or the nonlinearity to be localized on the compact core of the graph. The solutions found in those papers are critical points of mountain pass type for the energy E(·,) in the mass constrained space H_μ^1(), and the assumptions on the compactness of the graph or on the localization of the nonlinearity are essentially used to obtain a minimum of compactness needed to prove that Palais–Smale sequences converge. Such an issue, together with the very boundedness of these sequences, is one of the major problems when dealing with supercritical problems. We recall that arguments based on scaling, employed for example for the NLS equation on ^N, are not available on metric graphs since these are not scale invariant.

The solutions found in Theorems <ref>–<ref> are again of mountain pass type for E(·,) in H_μ^1(), in the same spirit of <cit.>. In our results, however, we manage to avoid all compactness assumptions on  and on the nonlinearity, replacing them with the requirement that the prescribed mass of the solutions we seek be small and exploiting specific topological features of the graphs under exam. 


Let us give some elements of our strategy.  As in <cit.>, we shall obtain the solutions of Theorems <ref>–<ref> as a limit of a sequence of solutions of approximating problems.  To this aim, we first consider the family of functionals E_ρ:H^1()→  given by

    E_ρ(u,):=1/2u'_L^2()^2-ρ/pu_L^p()^p   where  ρ∈ [1/2,1].

The functionals E_ρ(·,) have a mountain pass geometry and adapting <cit.> to our context we show that, for almost every ρ∈ [1/2,1], E_ρ(·,) admits a bounded Palais-Smale sequence at the mountain pass level with certain Morse index-type properties.
We prove that this sequence converges and that its limit is a critical point u_ρ of E_ρ(·, ) constrained to H_μ^1(). We then consider a sequence ρ_n → 1 and show that the sequence of corresponding solutions u_ρ_n converges to a critical point u ∈ H_μ^1() of E(·, ), which is the solution we are looking for in Theorems <ref>–<ref>. 

Whether it is to derive the existence of u_ρ, for almost every ρ∈ [1/2,1], or later in the analysis of the sequence (u_ρ_n)_n, we are led to study the convergence of such Palais-Smale sequences. 
The noncompactness of the domain and the information on the Morse index guarantee that the corresponding Lagrange multipliers are nonnegative. Then, in order to recover the necessary compactness, we rule out the possible partial loss of mass at infinity proving that, on every noncompact graph, all solutions of (<ref>) have positive energy. This result, which is perhaps of independent interest, requires the small mass assumption and is given in Proposition <ref>. To conclude, we are  left to exclude that the whole mass runs away at infinity. This is accomplished either exploiting the invariance under translations on periodic graphs, or  suitable topological assumptions on graphs with finitely many edges. Noteworthy, for this final step to be completed, the previous conditions must be combined again with the small mass assumption.

We wish to notice that both the requirement of small mass and on the topology of the graph are far from arbitrary or technical, but reflect instead important features of the NLS equation on graphs. Indeed, it seems to us that existence results for mountain pass type solutions with the nonlinearity acting on the whole structure are likely not to hold on general noncompact graphs when even just one of our assumptions is dropped.

Consider first the topological assumptions we exploited: in Theorem <ref> the presence of a pendant or a signpost, and in Theorem <ref> the periodicity of . As already said, these kind of assumptions have been deeply investigated in the subcritical and critical cases (see e.g. <cit.>) to obtain existence or nonexistence of ground states and of more general solutions. This is, on the contrary, the first time that topological properties of the graphs are shown to play a role in the supercritical regime. Here they are crucial to provide suitable estimates on the mountain pass level needed to restore compactness.  The relevance of these conditions is highlighted by the fact that for nonperiodic graphs that do not satisfy the assumptions of Theorem  <ref> one cannot hope, generally, to prove existence of solutions as in the preceding results. In fact, in Proposition <ref> below we show that for graphs satisfying the so–called assumption (H) (a topological condition on graphs with half-lines originally introduced in <cit.>) the mountain pass level we construct coincides with the level of the corresponding problem (with the same prescribed mass) on . However, a prototypical example of graphs fulfilling assumption (H) is the star graph (Figure <ref>), for which it is well-known (see e.g. <cit.>) that, for any mass μ, no solution of (<ref>) exists with energy equal to that of the mountain pass level on . We conjecture that the same is true for all graphs satisfying assumption (H) but, at the present time, we are not able to prove this statement in its full generality. If a proof of this result were available it would be extremely interesting, since it would reveal a deep property of the NLS equation on graphs: in <cit.>, it is shown that in the subcritical and critical regimes, assumption (H) prevents the existence of ground states; the validity of our conjecture would then provide a similar statement for mountain pass solutions in  the supercritical case.



The importance of the smallness assumption on the mass in Theorems <ref>–<ref> is analogous. As indicated, a key step in our argument is to prove that, for small masses, all solutions of (<ref>) have a nonnegative energy. On the contrary, the next two results show that, in general, for large values of μ there do exist solutions with negative energy. 

	
	Let ∈𝐆 be a noncompact graph with finitely many edges, no pendant and exactly one half-line. Then, there exists δ>0 such that, for every p∈[6,6+δ], there exists μ>0, depending on p and , for which problem (<ref>) admits a positive solution u∈ H_μ^1() with E(u,)<0.


	
	Let ∈𝐆 be a noncompact graph with finitely many edges, at least one of which bounded, and such that every vertex of  is attached to an even number of half-lines (possibly zero). For every p>6 there exists μ > 0, depending on  p and , such that, for all μ≥μ, problem (<ref>) admits a positive solution u∈ H_μ^1() with E(u,)<0.

As its proof shows, the solutions of Theorem <ref> are obtained as a perturbation of negative energy ground states of E at the critical exponent p=6, that on graphs satisfying the hypotheses of Theorem <ref> are known to exist for a full interval of masses (see <cit.>). On the contrary, the result of Theorem <ref> is not perturbative and it was communicated to us by D. Galant <cit.> (to whom we are sincerely grateful). Since their energy is negative, such solutions cannot be found by our mountain pass approach, which is designed to work at strictly positive levels. 
Observe also that the simplest graph fulfilling Theorem <ref> is perhaps the tadpole graph (Figure <ref>), that contains a signpost, whereas the simplest one to which Theorem <ref> applies is the so-called  𝒯-graph (Figure <ref>), that was extensively studied in <cit.> and that contains a pendant. This shows that, without the small mass assumption, there is no chance to recover our mountain pass argument, even in presence of the topological features introduced above. 

We finally note that the advantage given by the small mass to control from below the energy of solutions was already observed and exploited in some related problems posed on ^N (see for example <cit.>).




	
The remainder of the paper is organized as follows. Section <ref> collects some preliminary results on the mountain pass approach we will use and on the NLS equation on the real line. Section <ref> provides a general lower bound on the energy of positive solutions of (<ref>) with small mass. In Section <ref> we introduce the mountain pass problem and we derive some a priori estimates on the mountain pass level depending on the topology of the graphs. Finally, in Sections <ref>–<ref>–<ref> we  prove Theorem <ref>, Theorem <ref> and Theorems <ref>–<ref> respectively. 

Notation. In what follows, for u∈ L^q() we will write u_q in place of u_L^q(), omitting the explicit reference to the domain of integration when it coincides with the whole space. Conversely, the full notation for norms will always be used whenever the domain of integration is a proper subset of the full space.  




§ PRELIMINARIES



This section collects various preliminary results that will be largely used in the rest of the paper.



 §.§ A general existence result for bounded Palais-Smale sequences


The main aim of this subsection is to rewrite in the context of the present paper the content of <cit.>. To this end, for any ρ>0, let E_ρ:H^1()→ be given by

    E_ρ(u,):=1/2u'_2^2-ρ/pu_p^p .


Throughout the paper, the notation E_ρ will be used when ρ≠1 only. When ρ=1, we will always write E as in the Introduction, avoiding the symbol E_1.

We recall the following definition.

	Let ∈𝐆, p>6, μ>0 and ρ>0. For every u∈ H_μ^1() and θ≥ 0, the quantity
	
    m_θ(u):=sup{ dim L: L   subspace of T_uH_μ^1() such that D^2E_ρ(u,)[w,w]<-θw_H^1()^2, ∀ w∈ L∖{0}}

	is called the approximate Morse index of u with respect to θ, where T_uH_μ^1() is the tangent space to H_μ^1() at u and 
	
    D^2E_ρ(u,)[w,w]:=E_ρ”(u,)[w,w]-E_ρ'(u,)[u]/u_2^2 w_L^2()    ∀ w∈ T_uH_μ^1() .

	If u is a critical point of E_ρ constrained to H_μ^1() and θ=0, we say that m_0(u) is the Morse index of u as a constrained critical point.


The next theorem, that is no more than <cit.> applied to the family of functionals E_ρ above, provides a general existence result of bounded Palais-Smale sequences for these functionals constrained to H_μ^1(), with further information on their approximate Morse index.

	
	Let ∈𝐆, p>6 and μ>0 be given. Let I⊂(0,+∞) be a given interval and assume that there exist A,B⊂ H_μ^1() independent of ρ for which
	
    c_ρ:=inf_γ∈Γmax_t∈[0,1]E_ρ(γ(t),)>max{sup_u∈ AE_ρ(u,),sup_u∈ BE_ρ(u,)}    ∀ρ∈ I ,

	where 
	
    Γ:={γ∈ C([0,1],H_μ^1()) : γ(0)∈ A,γ(1)∈ B} .

	Then, for almost every ρ∈ I, there exist sequences (u_n)_n⊂ H_μ^1(), such that, as n→+∞, ζ_n→0^+ and
	
		
  (i) E_ρ(u_n,)→ c_ρ;
		
  (ii) E_ρ'(u_n,)-E_ρ'(u_n,)[u_n]/μu_n→0 in the dual of H_μ^1();
		
  (iii) (u_n)_n is bounded in H_μ^1();
		
  (iv) m_ζ_n(u_n)≤ 1.
	
 
Theorem <ref> will be used as a key tool in the proof of Theorems <ref>–<ref>. The information on the Morse index will play a relevant role combined with the following lemma, that simply rephrases in our setting <cit.>.

	
	Let ∈𝐆, p>6,μ>0 and ρ>0. Assume that (u_n)_n⊂ H_μ^1(), (λ_n)_n⊂ and (ζ_n)_n⊂^+ are such that,  as n→+∞, ζ_n→0^+ and
	
		
  (i) if a subspace W⊂ H^1() satisfies
		
    E_ρ”(u_n,)[w,w]+λ_nw_2^2<-ζ_nw_H^1()^2    ∀ w∈ W∖{0}

		for sufficiently large n, then dim W≤2, and
		
		
  (ii) there exist λ∈, a subspace Y⊂ H^1() with dim Y≥3, and a>0 such that, for every large n,
		
    E_ρ”(u_n,)[w,w]+λw_2^2≤-aw_H^1()^2    ∀ w∈ Y.

	
	Then λ_n>λ for every n large enough.




 §.§ NLS equations on the real line

For every μ,ρ>0, it is well-known that there exists a unique solution (ϕ_μ,ρ,λ_μ,ρ)∈ H_μ^1()×^+ of the problem

    u”+ρ u^p-1=λ u    on 
    
    u>0,   u(0)=max_x∈u(x) .

Setting 

    α=2/6-p ,    β=p-2/6-p ,

this solution can be written explicitly as

    ϕ_μ,ρ(x)=ρ^α/2ϕ_μ,1(ρ^α x)    ∀ x∈,

where

    ϕ_μ,1(x)=μ^αϕ_1,1(μ^β x)    ∀ x∈

and

    ϕ_1,1(x)=C_p sech^α/β(c_p x) ,

with C_p,c_p>0 suitable constants depending only on p. In particular, the previous relations show that, for every  p>6 and x∈, as μ→ 0 it holds

    ϕ_μ,ρ(x)    = C_p'ρ^α/2μ^α e^-2c_p/p-2ρ^αμ^β x+o(μ^α e^-2c_p/p-2ρ^αμ^β x)
    ϕ_μ,ρ'(x)    =C_p”ρ^3α/2μ^α+β e^-2c_p/p-2ρ^αμ^β x+o(μ^α+β e^-2c_p/p-2ρ^αμ^β x)

for suitable constants C_p',C_p”>0 depending only on p. Furthermore, direct computations yield the identity

    ϕ_μ,ρ'_2^2=p-2/2pρϕ_μ,ρ_p^p ,

so that

    E_ρ(ϕ_μ,ρ,)=p-6/4pρϕ_μ,ρ_p^p .

Recalling that E(ϕ_μ,1,)=μ^2β+1E(ϕ_1,1,)>0, it then follows that

    E_ρ(ϕ_μ,ρ)=θ_pρ^4/6-pμ^2β+1    θ_p:=p-6/4pϕ_1,1_p^p>0.

In particular (<ref>) implies that E_ρ(ϕ_μ,ρ) is a strictly decreasing function of μ >0.



 §.§ Gagliardo-Nirenberg inequalities on graphs

For every noncompact graph ∈𝐆, we recall here the following well-known Gagliardo-Nirenberg inequalities

    u_p^p≤ K_p,u_2^p/2+1u'_2^p/2-1    ∀ u∈ H^1(), p>2 ,


    u_∞≤√(2)u_2^1/2u'_2^1/2    ∀ u∈ H^1() ,

with K_p,>0 depending on p and  only (for a proof of these inequalities see e.g. <cit.>).




 §.§ Properties of the Lagrange multiplier

For ∈𝐆, let 

    λ_:=inf_u∈ H^1()∖{0}u'_2^2/u_2^2

be the bottom of the spectrum of the operator -d^2/dx^2 on  endowed with Kirchhoff conditions at the vertices.
Introducing the shorthand notation

    S_μ ={ u∈ H_μ^1() | u  solves (<ref>) for some λ∈},

we have the following lemma.

	
	Let ∈𝐆. There results
	
		
  (a) if u ∈ S_μ, then λ≥  - λ_;
		
  (b) if  is a graph with at least one half-line or a periodic graph,  then λ_=0; 
		
  (c) if  is a noncompact graph with finitely many edges and u  ∈ S_μ for some μ >0, then λ >0; 
		
  (d) if  is a graph with at least one half-line or a periodic graph, then for any m >0 there exists μ_m  >0 such that, if u ∈ S_μ with μ∈ (0, μ_m], then λ≥ m.
	


To prove Lemma <ref>(d), we shall need the next preliminary lemma.

	
	Let u ∈ S_μ. Then 
	
    u_∞≤(p/2λ + pπ^2/2e_0^2)^1/p-2.




	This can be checked by standard phase plane considerations as follows. Let M := u_∞ >0. If M^p-2≤ pλ/2, there is nothing to prove, so we only deal with the case M^p-2 > pλ/2. Let then M be achieved in some edge e at a point called q. Reversing, if necessary, the orientation of the coordinate along e, we can assume u to be nonincreasing on an interval [q,r] of length a ≥ e_0/2. Indeed, if no such interval existed, then u would attain a local minimum in the interior of e, which is impossible since u>0 on , while the condition  M^p-2 > pλ/2 forces all local minima of u to be strictly negative.
	
	Since u solves (<ref>), for every x∈[q,r] there results
	
    1/2 |u'(x)|^2 + 1/p |u(x)|^p - λ/2 |u(x)|^2 =  1/p |u(q)|^p - λ/2 |u(q)|^2 =  1/p M^p - λ/2 M^2>0.

	Then
	
    a = ∫_q^r dx = ∫_u(r)^M du/√(2/p(M^p -u^p)- λ(M^2-u^2))

	or, setting t = u/M,
	
    a =  ∫_u(r)/M^1 dt/√(2/p M^p-2(1 -t^p)- λ(1-t^2)).

	Now, as t ∈ (0,1], we have 1-t^p ≥ 1-t^2, so that 
	
    a ≤1/√(2/p M^p-2 -λ)∫_u(r)/M^1 dt/√(1-t^2)≤1/√(2/p M^p-2 -λ)∫_0^1 dt/√(1-t^2)
    	= π/2√(2/p M^p-2-λ) ,

	namely
	
    M^p-2≤p/2λ + pπ^2/8a^2≤p/2λ + pπ^2/2e_0^2,

	since a ≥ e_0/2.




Point (a) is somehow standard, but we prove it for completeness. Let  be a given connected, bounded subset of , and let φ solve

    -φ”=λ_1() φ   on every edge of 
    φ >0    on 
    ∑_e≻d φ/dx_e()=0    for every vertex  in \∂
    φ(x) = 0    for x ∈∂,

with λ_1()  the first eigenvalue of -d^2/dx^2 on  with the above boundary conditions. Let u ∈ H^1() solve

    u” + u^p-1 =λ u     on every  e∈_
    
    		u >0    on 
    ∑_e≻d u/dx_e()=0    for every ∈_,

for some λ∈. Multiplying the equation of u by φ and the equation of φ by u  we get

    - ∫_u' φ '  dx+ ∫_ u^p-1φ dx = λ∫_ u φ   dx   and  
    -  φ' u_| ∂ + ∫_φ' u' dx = λ_1() ∫_ u φ dx.

Note that - φ' u_| ∂ >0, so that - ∫_φ'u' dx > - λ_1() ∫_u φ dx. Coupling with the other equation yields

    ∫_ u^p-1φ dx  < (λ + λ_1()) ∫_ u φ dx.

Since the integrals are positive, λ > - λ_1(). By the arbitrariness of , this gives λ≥ - λ_.
The proof of (b)  is evident for graphs with half-lines, whereas for periodic graphs we refer e.g. to <cit.>. To prove (c) it is enough to observe that, if u∈ S_μ, then it is strictly positive everywhere on  by definition. This is impossible if λ=0, because in this case all nonzero solutions on a half-line are restrictions of periodic functions (and thus not in L^2()). 
To prove (d), let  m>0 be arbitrary and assume by contradiction that there exist sequences μ_n → 0 and (u_n)_n ⊂ S_μ_n solving (<ref>) with λ_n such that 0 ≤λ_n <m.
	  Since u_n ∈ S_μ_n,
	
    u'_n_2^2 ≤u'_n_2^2 +λ_n u_n_2^2 = u_n_p^p ,

	which coupled with (<ref>) yields
	
    u'_n_2^2 ≤u_n_p^p ≤  K_p,μ_n^p+2/4u_n'_2^p-2/2,

	namely
	
    1 ≤ K_p,μ_n^p+2/4u_n'_2^p-6/2.

	Since μ_n → 0, this shows that u_n'_2→ +∞, and then the same for u_n_p, by (<ref>). Hence, by u_n_p^p ≤u_n_∞^p-2μ_n, we see that u_n_∞→∞, and the contradiction is achieved  invoking (<ref>).




§ A GENERAL ENERGY ESTIMATE FOR POSITIVE SOLUTIONS



The aim of this section is to prove an asymptotic estimate on the energy of small mass positive solutions of (<ref>). 
The main result of the section is the following.

	 Let ∈𝐆 be either a noncompact graph with finitely many edges or a periodic graph. If  has no pendant, then
	
    lim inf_μ→ 0inf_u ∈ S_μE(u,)/E(ϕ_μ,1,)≥ 1.

	If ∈𝐆 has at least one pendant, then
	
    lim inf_μ→ 0inf_u ∈ S_μE(u,)/E(ϕ_2μ,1,^+)≥ 1.





	We start proving (<ref>). If it is false, there exist δ∈ (0,1), a sequence μ_n → 0 and a sequence (u_n) ⊂ S_μ_n such that, for every n, 
	
    E(u_n,)/E(ϕ_μ_n,1,)≤ 1-δ.

	Each u_n solves (<ref>) for some λ_n ≥ 0. In view of Lemma <ref>(d) we have that λ_n → +∞ as n →∞.
	To proceed, consider a point q_n in some edge e_n where u_n attains a local maximum. Since u_n solves (<ref>), it must be u_n(q_n) ≥λ_n^1/p-2. If equality holds,  u_n is constant on e_n by uniqueness in the Cauchy problem, and e_n is a bounded edge. But then
	
    μ_n = ∫_ |u_n|^2≥∫_e_n |u_n|^2 = λ_n^2/p-2|e_n| ≥λ_n^2/p-2|e_0|,

	which is impossible since λ_n → +∞ and μ_n → 0. Therefore, u_n is strictly larger than λ_n^1/p-2 at all its local maximum points, which are therefore nondegenerate. Since u_n ∈ H_μ_n^1(), outside a compact set K_n ⊂, u_n is smaller than some number, say 1 (certainly smaller than λ_n^1/p-2), and thus all local maximum points of u_n are in K_n. Since by the preceding argument they are also isolated, it then follows that each u_n can have at most a finite number of maximum points.
	
	Since μ_n → 0, we can also assume that min_e u_n < 1 for every e ∈_ and every n.
	We then set
	
    A_n = {x ∈| u_n(x) >1}

	and we note that no local minimum point of u_n can be in A_n.
	Each A_n  is the disjoint union of a finite number of connected open subsets B_1^n,…, B_k_n^n of  such that 
	
		
  (i) |B_i^n| → 0 for every i=1,…,k_n as n→∞;
		
  (ii) each B_i^n contains at least one local maximum point of u_n;
		
  (iii) each B_i^n contains at most one vertex of ;
		
  (iv) if B_i^n contains more than one local maximum point of u_n, then it contains exactly one vertex of .
	
	Indeed, (i) is obvious as u_n ≥ 1 on each B_i^n and μ_n → 0. Moreover, 
	since each B_i^n is non-empty, bounded, contained in A_n and u = 1 on ∂ B_i^n, it contains at least one local maximum point of u_n, i.e. property (ii) holds.
	Property (iii) follows by the fact that, if some B_i^n contains at least two vertices of , then, being connected, it contains at least a whole edge of , which is impossible because the minimum of u_n on each edge of  is strictly smaller than 1.
	Finally, to prove (iv),  observe that, if two local maximum points of u_n belong to the same edge of , then the fact that u_n solves (<ref>) implies that u_n has a local minimum point halfway between the two local maximum points. Moreover, by phase plane analysis it is easy to see that the value attained by u_n at this local minimum point coincides with the minimum value of u_n on the whole edge. Hence, u_n is smaller than or equal to 1 at least at one point between two consecutive local maximum points belonging to the same edge, in turn implying that these local maximum points do not belong to the same B_i^n. Therefore if B_i^n contains at least two local maximum points, these must belong to different edges. By connectedness, then, B_i^n must contain a vertex, and by (iii) this vertex is unique.
	
	Now, since 0<u_n≤ 1 on ∖ A_n, it follows that 
	
    E(u_n,∖ A_n) ≥  -1/pu_n_L^p(∖ A_n)^p ≥  - 1/pu_n_L^2(∖ A_n)^2 ≥ -1/pμ_n = o(1)

	as n →∞.
	
	For every n, we denote shortly by B_n a set B_i_n^n such that
	
    E(u_n, B_i_n^n) = min_i E(u_n, B_i^n),

	which exists since for every n the number of B_i^n's is finite. Up to subsequences, we can assume that either every B_n contains exactly one vertex _n, or that no B_n contains a vertex, and we treat the two cases separately.
	
	Let us first suppose that B_n contains no vertex of , for every n. In this case, for each n, B_i^n is an interval (-a_n,a_n) contained in an edge e_n and the restriction of u_n to [-a_n,a_n] is strictly positive,  symmetric with respect to 0 and decreasing, say,  on [0,a_n].  It satisfies
	
    u_n”+u_n^p-1=λ_n u_n    on [-a_n,a_n] ,
    
    	u_n(0)= u_n_L^∞(-a_n,a_n) > λ_n^1/p-2,   
     
    	u_n(± a_n) =1

	and 
	
    1/2 |u_n'(x)|^2 + 1/p |u_n(x)|^p - λ_n/2 |u_n(x)|^2 = C_n      for every  x ∈ [-a_n,a_n]

	for some constant C_n. By Lemma <ref>,  we see that, as n →∞,
	
    C_n  = 1/2  |u_n(0)|^2(2/p  |u_n(0)|^p-2 -λ _n )
    	≤1/2  |u_n(0)|^2 (λ_n + π^2/e_o^2 -λ _n ) = 
    	O(λ_n^2/p-2).

	We now set 
	
    v_n(x):=λ_n^-1/p-2u_n(x/√(λ_n)),

	so that v_n is a strictly positive function defined on the interval [-√(λ_n)a_n,√(λ_n)a_n], symmetric with respect to 0 and  decreasing on [0,√(λ_n)a_n]. It satisfies
	
    v_n”+v_n^p-1=v_n    on [-√(λ_n)a_n,√(λ_n)a_n] ,
    
    	v_n(0)= v_n_L^∞(-√(λ_n) a_n,√(λ_n) a_n)≥ 1,    
     
    	v_n(±√(λ_n) a_n) =λ_n^-1/p-2

	and
	
    1/2 |v_n'(x)|^2 + 1/p |v_n(x)|^p - 1/2 |v_n(x)|^2 = λ_n^-p/p-2 C_n      for every  x ∈[-√(λ_n)a_n,√(λ_n)a_n].

	Then, recalling (<ref>), we deduce that, as n→∞,
	
    |v'(±√(λ_n)a_n)|^2 =  2λ_n^-p/p-2 C_n + |v(±√(λ_n)a_n)|^2 - 2/p|v(±√(λ_n)a_n)|^p = O(λ_n^-1) + o(1) = o(1).

	Summing up, v_n satisfies (<ref>) and  (v_n(±(√(λ_n)a_n)),v_n'(±√(λ_n)a_n)) → (0,0)  as n→+∞. By continuity in the phase plane, it then follows that  √(λ_n)a_n →∞ and that
	
    E(v_n,[-√(λ_n)a_n,√(λ_n)a_n])=E(ϕ,)+o(1)    as n→+∞ ,

	where ϕ is the unique solution in H^1() of 
	
    ϕ”+ϕ^p-1=ϕ   on 
    ϕ(0)=ϕ_L^∞()≥ 1.

	Note that ϕ=ϕ_ν,1 for ν:=lim_n→+∞v_n_L^2(-√(λ_n) a_n,√(λ_n) a_n)^2>0. Since by direct computations 
	
    E(u_n,B_n)=λ_n^p+2/2(p-2)E(v_n,[-√(λ_n)a_n,√(λ_n)a_n]),

	by (<ref>) it follows
	
    E(u_n,B_n) ≥λ_n^p+2/2(p-2)E(ϕ_ν,1, ) + o( λ_n^p+2/2(p-2)) =     E(ϕ_ν_n,1,) + o( E(ϕ_ν_n,1,)) 
    ≥     E(ϕ_μ_n,1,) + o( E(ϕ_μ_n,1,)),

	the last inequality coming by the fact that ν_n:=λ_n^6-p/2(p-2)ν≤μ_n by definition of ν and 
	
    v_n_L^2(-√(λ_n) a_n,√(λ_n) a_n)^2=λ_n^p-6/2(p-2)u_n_L^2(B_n)^2≤λ_n^p-6/2(p-2)μ_n    ∀ n .
 
	Hence, by (<ref>),
	
    E(u_n, ) =  E(u_n, A_n) + E(u_n, ∖ A_n)  ≥    E(u_n, B_n) +o(1) ≥  E(ϕ_μ_n,1,) + o( E(ϕ_μ_n,1,)),

	so that
	
    E(u_n, ) / E(ϕ_μ_n,1,) ≥ 1 + o(1),

	which violates (<ref>) when n→∞ and completes the proof in case B_n contains no vertex of .
	
	We now assume (keeping in mind (<ref>)), that every B_n contains a (unique) vertex _n. Thus, each B_n is the union of 
	k_n ≥ 3 line segments glued together at this vertex.
	
	Hence, letting k_n:=deg(_n), there exist a_1^n,…,a_k_n^n>0 such that B_n can be identified with the intervals [0,a_1^n),…,[0,a_k_n^n) glued together at the origin (identified with _n). Since by construction B_n contains at least one local maximum point of u_n on , there are i_n ≥ 0 intervals [0,a_1^n),…,[0,a_i_n^n) on which u_n is increasing from 0 to a local maximum point and then decreasing from this maximum point to the end of the interval, whereas on the intervals [0,a_i_n+1^n),…,[0,a_k_n^n) u_n is decreasing from 0 to the end of the interval. Note that i_n<k_n, because otherwise u_n would have a local minimum point at _n, and by the Kirchhoff condition this would imply u_n(_n)≤ 1, contradicting _n∈ B_n.
	
	Observe that, if
	
    lim sup_n→+∞λ_n^-1/p-2u_n(_n)=0,

	arguing as in the previous part of the proof on each interval (if any) [0,a_i^n], i = 1, …, i_n,  yields, as n→+∞,
	
    E(u_n,[0,a_i^n]) = λ_n^p+2/2(p-2)E(ϕ_ν,1,)+o(λ_n^p+2/2(p-2))
 
	for ν≤λ_n^p-6/2(p-2)μ_n. On the other hand, on each interval [0,a_i^n] with i = i_n+1, …, k_n, if a_i^n = o(1/√(λ_n)), then
	
    E(u_n,[0,a_i^n]) ≥ -∫_0^a_n|u_n|^p≥ o (λ_n^p/p-2)a_i^n = o (λ_n^p+2/2(p-2)).

	If, on the contrary, √(λ_n)a_i^n is bounded away from 0, then E(u_n, [0,a_i^n])>0 (this can be seen e.g. combining the scaling procedure used in the previous part of the proof with <cit.>). All in all, this yields, as above, 
	
    E(u_n, B_n)   ≥  E(ϕ_μ_n,1,) + o( E(ϕ_μ_n,1,)).

	Conversely, if (<ref>) does not hold, there exists K>0 
	such that along a subsequence (not relabeled) we have
	
    λ_n^-1/p-2u_n(_n) ≥ K    ∀ n .

	Considering the restriction of u_n to B_n, set then v_n(x):=λ_n^-1/p-2u_n(x/√(λ_n)). The function v_n is defined on a star graph with k_n bounded edges, identified with the intervals [0,√(λ_n)a_i^n], i= 1, …, k_n. Moreover, on each edge, v_n is either increasing from 0 to a local maximum point and then decreasing to the end of the edge or decreasing, and it satisfies
	
    v_n”+v_n^p-1=v_n    on each [0,√(λ_n)a_i^n]
    
    	v_n(0) ≥ K     
    ∑_i=1^k_ndv_n/dx_i(0)=0,

	where dv_n/dx_i(0) denotes the outgoing derivative of v_n at 0 along [0,√(λ_n)a_i^n]. Furthermore, v_n(√(λ_n)a_i^n)→0 as n→+∞ for every i. 
	
	
	Arguing exactly as in the previous part of the proof, we obtain again that  |v_n'(√(λ_n)a_j^n)|→0 and √(λ_n)a_i^n→+∞ as n→+∞ for every i. By continuity in the phase plane, this implies that the restriction of v_n to each interval [0,a_i^n) converges to the restriction of the function ϕ of (<ref>) to a suitable interval of the form [b_i,+∞). Together with the fact that v_n satisfies the Kirchhoff condition at the vertex of the star graph, this implies that 
	
    E(v_n,√(λ_n)B_n)=E(w_n,S_k_n)+o(1)    as n→+∞ ,

	where S_k_n is the infinite star graph with k_n half-lines and w_n is a H^1(S_k_n) positive solution of 
	
    w_n”+w_n^p-1=w_n    on S_k_n
    
    	w_n(0) ≥ K    
    ∑_i=1^k_ndw_n/dx_i(0)=0.

	Since k_n≥ 3 for every n, it is well known that E(w_n,S_k_n) ≥ E(ϕ_ν_n,1,) (see e.g. <cit.>), where ν_n ≤μ_n is the mass of w_n.
	Hence, recalling that E(u_n,B_n)=λ_n^p+2/2(p-2)E(v_n,√(λ_n)B_n), we conclude as in the previous case. This completes the proof of (<ref>).
	
	The proof of (<ref>) is completely analogous to that of (<ref>). The only difference is that if u_n “concentrates” on a pendant, then one must use the half-soliton ϕ_2μ_n,1 and its level E(ϕ_2μ_n,1, ^+) <E(ϕ_μ_n,1,) in the asymptotic estimates.


	
	If  has no pendant, it is clear that the proof of Proposition <ref>  can be easily modified to check that
	
    lim inf_μ→0inf_u∈ S_μ,ρE_ρ(u,)/E_ρ(ϕ_μ,ρ,)≥1    ∀ρ∈[1/2,1],

	where S_μ,ρ is the set of all solutions of the problem
	
    u”+ρ u^p-1=λ u    on every e∈_
    
    	u∈ H_μ^1(),   u>0    on 
    ∑_e≻du/dx_e()=0    for every ∈_ .

	Observe that, since u∈ S_μ if and only if ρ^-1/p-2u∈ S_ρ^-2/p-2μ,ρ, this implies that, given  with no pendant, p>6 and ε>0, there exists μ_,p,ε>0 independent of ρ∈[1/2,1] for which 
	
    inf_u∈ S_μ,ρE_ρ(u,)/E_ρ(ϕ_μ,ρ,)≥ 1-ε    ∀μ∈(0,μ_,p,ε], ∀ρ∈[1/2,1] .

	Analogously to Proposition <ref>, the same is true for graphs with a pendant, replacing E_ρ(ϕ_μ,ρ,) with E_ρ(ϕ_2μ,ρ,^+).



Recalling (see (<ref>)) that both E(ϕ_μ,1,) → + ∞ and E(ϕ_2μ,1,^+) → + ∞ as μ→ 0, the following is a direct consequence of Proposition <ref>.


Let ∈𝐆 be either a noncompact graph with finitely many edges or a periodic graph.  There exists μ̃ = μ̃(, p)>0 such that, if μ∈ (0, μ̃], then every solution u ∈ H_μ^1() to (<ref>) (or (<ref>) in which u^p-1 is replaced by ρ u^p-1) satisfies 
E(u, ) > 0 (respectively E_ρ(u, ) > 0).






§ MOUNTAIN PASS STRUCTURE AND A PRIORI ESTIMATES



In this section we introduce the mountain pass geometry that will be exploited to prove Theorems <ref>–<ref> and we derive topology-driven a priori estimates on the mountain pass level.

We start with the next lemma, analogous to those in <cit.>, that provides the precise description of the mountain pass structure we will consider.

	
	Let ∈𝐆 be either a noncompact graph with finitely many edges or a periodic graph. Then, for every p>6 and every μ>0, there exists δ>0, depending on ,p,μ but not on ρ∈[1/2,1], such that
	
    c_ρ():=inf_γ∈Γmax_t∈[0,1]E_ρ(γ(t),)>max{sup_u∈ A_δE_ρ(u,),sup_u∈ BE_ρ(u,)}    ∀ρ∈[1/2,1] ,

	where Γ:={γ∈ C([0,1],H_μ^1()) : γ(0)∈ A_δ, γ(1)∈ B
	} and  
	
    A_δ:={u∈ H_μ^1() : u'_2^2≤δ} ,     B:={u∈ H_μ^1() :  E_1/2(u,)<0} .



	The proof works exactly as that of <cit.>, additionally recalling here that, for every μ,k>0, the set
	
    A_k:={u∈ H_μ^1() : u'_2^2≤ k}

	is nonempty whenever ∈𝐆 is a noncompact graph with finitely many edges or a periodic graph because for all such graphs we already pointed out in (<ref>) that λ_=0.



	
	On the real line it is well-known that 
	
    c_ρ()=E_ρ(ϕ_μ,ρ,)    ∀μ>0, ∀ρ∈[1/2,1],

	where ϕ_μ,ρ is as in Section <ref>. Furthermore, setting a_p:=(p-2/4)^2/p-6, exploiting (<ref>) it is easy to see that there exists a (small) ε_p>0 such that the path γ:=(γ_t)_t∈[0,1]⊂ H_μ^1() given by
	
    γ_t(x):=√(a_p t+ε_p) ϕ_μ,ρ((a_p t+ε_p)x)

	satisfies γ_0∈ A_δ, γ_1∈ B and c_ρ()=max_t∈[0,1]E_ρ(γ_t,)=E_ρ(γ_1-ε_p/a_p,). Observe also that, by (<ref>), (<ref>), (<ref>), 
	
    γ_t'_2^2=Cμ^2β+1,    γ_t_p^p=C'μ^2β+1

	with C,C'>0 depending on p,ρ,t only, and being bounded both from above and away from zero, uniformly for ρ∈[1/2,1] and t∈[0,1].  Clearly, since for every μ and ρ there results
	
    c_ρ(^+)= 2^2 βc_ρ(),

	an analogous path can be constructed at mass μ on the half-line simply taking the restriction to ^+ of the path γ⊂ H_2μ^1().


The main result of this section is the following proposition, that establishes suitable a priori estimates on the level c_ρ().


	
	For every ∈𝐆 and every p>6, when μ→ 0
	
    c_ρ()≤ c_ρ()+o(c_ρ()).

	Moreover, if  has a pendant, 
	
    c_ρ()≤ c_ρ(^+)+o(c_ρ(^+)),

	whereas if  has a signpost, then 
	
    c_ρ()< c_ρ().

	All relations are uniform for ρ∈[1/2,1].


	We argue in three steps.
	
	Step 1: proof of c_ρ()≤ c_ρ()+o(c_ρ()). Let ∈𝐆 and p>6 be fixed. Let e∈_ be any given edge of , and let 4ℓ:=|e| be its length. To prove the first part of Proposition <ref>, we will now show that, as soon as μ→ 0, there exists a path γ=(γ_t)_t∈[0,1]⊂ H_μ^1() such that γ_t is compactly supported on e for every t∈[0,1] and max_t∈[0,1]E_ρ(γ_t,)≤ c_ρ()+o(c_ρ()) for every ρ∈[1/2,1]. 
	
	To this end, let γ⊂ H_μ^1() be the path given in Remark <ref> and define v_t:e→ as
	
    v_t(x):=γ_t(x)    if x∈[-ℓ,ℓ]
    γ_t(ℓ)(2-x/ℓ)    if x∈(ℓ,2ℓ]
    γ_t(-ℓ)(2+x/ℓ)    if x∈[-2ℓ,-ℓ),

	where we tacitly identified the edge e with the interval [-2ℓ,2ℓ]. Extending then v_t to be identically zero on ∖ e, we define γ_t∈ H_μ^1() as
	
    γ_t:=√(μ)/v_t_2 v_t .

	Note that, by definition of v_t, Remark <ref> and (<ref>), 
	
    v_t_2^2=γ_t_L^2(-ℓ,ℓ)^2+2ℓ/3γ_t(ℓ)^2    =μ+2ℓ/3γ_t(ℓ)^2-2γ_t_L^2(ℓ,+∞)^2
        =μ+Cμ^2αe^-2τ cμ^βℓ+o(μ^2αe^-2τ cμ^βℓ)    as μ→0 ,

	for constants C,c>0 depending on p,ρ,t only and both bounded from above and away from zero uniformly for ρ∈[1/2,1] and t∈[0,1], and τ:=α/β (with α,β as in (<ref>)). Analogous computations lead to
	
    v_t'_2^2    =γ_t'_2^2+C'μ^2α+βe^-2τ cμ^βℓ+o(μ^2α+βe^-2τ cμ^βℓ)=γ_t'_2^2+o(γ_t'_2^2)
    v_t_p^p    =γ_t_p^p+C”μ^α pe^-pτ cμ^βℓ+o(μ^α pe^-pτ cμ^βℓ)=γ_t_p^p++o(γ_t_p^p)    as μ→0 ,

	again with C',C”>0 depending on p,ρ,t only and both bounded from above and away from zero uniformly for ρ∈[1/2,1] and t∈[0,1] by Remark <ref>. All in all, as soon as μ is small enough we obtain γ_0∈ A_δ, γ_1∈ B and
	
    E_ρ(γ_t,)=μ/v_t_2^21/2v_t'_2^2-(μ/v_t_2^2)^p/21/pv_t_p^p= E_ρ(γ_t,)+o(E_ρ(γ_t,))

	uniformly for ρ∈[1/2,1] and t∈[0,1], in turn yielding
	
    c_ρ()≤max_t∈[0,1]E_ρ(γ_t,)=max_t∈[0,1](E_ρ(γ_t,)+o(E_ρ(γ_t,)))=c_ρ()+o(c_ρ())

	for sufficiently small μ.
	
	Step 2: proof of c_ρ() ≤ c_ρ(^+)+o(c_ρ(^+)) for graphs with a pendant. This step works exactly as Step 1, with the only difference that we now construct a path γ⊂ H_μ^1() such that γ_t is compactly supported on the pendant of  for every t∈ [0,1] and max_t∈[0,1]E_ρ(γ_t)≤ c_ρ(^+)+o(c_ρ(^+))<c_ρ(). By Remark <ref>, such a path γ can be obtained repeating verbatim the construction of Step 1 starting with the restriction to ^+ of γ⊂ H_2μ^1().
	
	Step 3: proof of c_ρ()<c_ρ() for graphs with a signpost. The argument is again analogous to that of Step 1, suitably adapted to functions defined on the tadpole graph, i.e. the graph 𝒯_ℓ given by a loop ℒ of length 2ℓ>0 and a single half-line . Note first that, for every μ, ρ, it holds
	
    c_ρ(𝒯_ℓ)<c_ρ() .

	To see this, take again the path γ⊂ H_μ^1() introduced in Remark <ref> and define γ⊂ H_μ^1(𝒯_ℓ) as
	
    γ_t(x):=γ_t(x)    if x∈[-ℓ,ℓ]
    γ_t(x+2ℓ/2)    if x∈^+

	where, with a slight abuse of notation, we identified the loop ℒ with the interval [-ℓ,ℓ] and the half-line  with ^+. Straightforward computations show that, for every t∈[0,1],
	
    γ_t_2^2=μ ,        γ_t_L^p(𝒯_ℓ)^p=γ_t_L^p()^p ,
       γ_t'_L^2(𝒯_ℓ)^2=γ_t'_L^2(-ℓ,ℓ)^2+1/4γ_t'_L^2(∖[-ℓ,ℓ])^2=γ_t'_2^2-3/4γ_t'_L^2(∖[-ℓ,ℓ])^2 ,

	so that γ_0∈ A_δ, γ_1∈ B and E_ρ(γ_t,𝒯_ℓ)<E_ρ(γ_t,), in turn yielding (<ref>).
	
	To show then that the same inequality holds on  when it has a signpost, it is enough to adapt the construction in Step 1 to the path γ. Denote by 𝒫 the signpost of  and let 2ℓ be the length of its loop and 2a be that of its bounded edge. For every t∈[0,1], we define v_t:𝒫→ as
	
    v_t(x):=γ_t(x)    if x∈ℒ
    γ_t(x)    if x∈[0,a]∩
    γ_t(a)(2-x/a)    if x∈[a,2a] ,

	again with the slight abuse of notation to identify the loop of 𝒫 with the loop ℒ of the tadpole 𝒯_ℓ and its bounded edge with the interval [0,2a] on the half-line  of 𝒯_ℓ. Extending v_t to zero on the rest of , we then define γ⊂ H_μ^1() as γ_t:=√(μ)/v_t_2 v_t, for every t∈[0,1]. Arguing as in Step 1, by (<ref>) there results
	
    v_t_2^2=    γ_t_L^2(ℒ∪(∩[0,a]))^2	+γ_t(a+2ℓ/2)^2a/3=μ-γ_t_L^2(∩(a,+∞))^2+γ_t(a+2ℓ/2)^2a/3
    
    	=    μ-∫_a^+∞|γ_t(x+2ℓ/2)|^2 dx+γ_t(a+2ℓ/2)^2a/3=μ-2∫_a+2ℓ/2^∞|γ_t(y)|^2 dy+γ_t(a+2ℓ/2)^2a/3
    
    	=    μ+Kμ^2αe^-2τ cμ^β(a+2ℓ/2)+o(μ^2αe^-2τ cμ^β(a+2ℓ/2))    as μ→0 ,

	where c,τ are the same as in (<ref>) and K>0 is a constant depending on p,ρ,t only, bounded from above and away from zero uniformly for ρ∈[1/2,1] and t∈[0,1]. Similarly, recalling also (<ref>), we have
	
    v_t'_2^2=    γ_t'_L^2(ℒ∪(∩[0,a]))^2+γ_t(a+2ℓ/2)^2/a=γ_t_L^2(𝒯_ℓ)^2-γ_t'_L^2(∩(a,+∞))^2+γ_t(a+2ℓ/2)^2/a
    
    	=    γ_t'_2^2-3/4γ_t'_L^2(∖[-ℓ,ℓ])^2-γ_t'_L^2(∩(a,+∞))^2+γ_t(a+2ℓ/2)^2/a
    
    	=    γ_t'_2^2-K'μ^2α+βe^-2τ cμ^βℓ+o(μ^2α+βe^-2τ cμ^βℓ)

	and
	
    v_t_p^p=    γ_t_L^p(ℒ∪(∩[0,a]))^p+|γ_t(a+2ℓ/2)|^pa/p+1= γ_t_L^p(𝒯_ℓ)^p-γ_t_L^p(∩(a,+∞))^p+|γ_t(a+2ℓ/2)|^pa/p+1
    
    	=    γ_t_L^p()^p+K”μ^α pe^-pτ cμ^β(a+2ℓ/2)+o(e^-pτ cμ^β(a+2ℓ/2)) ,

	for constants K',K”>0 depending on p,ρ,t only and bounded from above and away from zero uniformly for ρ∈[1/2,1] and t∈[0,1].
	Coupling (<ref>) and (<ref>) yields, since p>2 and a>0,
	
    E_ρ(v_t,)=E_ρ(γ_t,)-K'μ^2α+βe^-2τ cμ^βℓ+o(μ^2α+βe^-2τ cμ^βℓ)

	as μ→0, which together with (<ref>) (and recalling again Remark <ref> and (<ref>)) gives
	
    E_ρ(γ_t,)    =μ/v_t_2^21/2v_t'_2^2-(μ/v_t_2^2)^p/21/pv_t_p^p=(1-O(μ^2αe^-2τ cμ^β(a+2ℓ/2)))E_ρ(v_t,)
        = E_ρ(γ_t,)-K'μ^2α+βe^-2τ cμ^βℓE_ρ(γ_t,)+o(μ^2α+βe^-2τ cμ^βℓE_ρ(γ_t,))

	as μ→ 0. Since K' is bounded away from zero uniformly for ρ∈[1/2,1] and t∈[0,1], by Remark <ref> the previous identity entails c_ρ()≤max_t∈[0,1]E_ρ(γ_t,)<max_t∈[0,1]E_ρ(γ_t,)=c_ρ() as soon as μ is small enough, and we conclude.

We end this section with a more precise characterization of the mountain pass level c_ρ() under further topological conditions on the graph. To this end, we recall that a noncompact graph ∈𝐆 with finitely many edges is said to satisfy assumption (H) if every x∈ lies on a trail (i.e. a path of adjacent edges of , each one run through exactly once) containing two half-lines. Assumption (H) has been introduced for the first time in <cit.>, but the formulation we reported here is taken from <cit.>.

	
	Let ∈𝐆 be a noncompact graphs with finitely many edges. Then 
	
    c_ρ()≤ c_ρ()    ∀μ>0, ∀ρ∈[1/2,1].

	Furthermore, if  satisfies assumption (H), then equality holds in (<ref>).


	The validity of (<ref>) is evident as soon as  has at least one half-line. To show that c_ρ()=c_ρ() whenever  satisfies assumption (H) recall that, by standard properties of rearrangements on graphs (see e.g. <cit.>), if u∈ H^1() is a positive function on  satisfying assumption (H), then its symmetric decreasing rearrangement on the line u∈ H^1() is such that
	
    u_r=u_r,  ∀ r≥1,    u'_2≥u'_2 ,

	By (<ref>) it follows that, if γ∈ C([0,1],H_μ^1()) is a path in H_μ^1() such that γ_0∈ A_α and γ_1∈ B on , then the symmetric decreasing rearrangement γ_t of γ_t provides a path γ∈ C([0,1], H_μ^1())  (the strong continuity in H^1() of the map t↦γ_t can be proved e.g. as in <cit.>) such that γ_0∈ A_α and γ_1∈ B on , and E_ρ(γ_t,)≤ E_ρ(γ_t,) for every t. As this ensures that c_ρ()≥ c_ρ(), coupling with (<ref>) yields the desired identity.




§ GRAPHS WITH FINITELY MANY EDGES: PROOF OF THEOREM <REF>



In this section we provide the proof of the first main existence result of the paper, i.e. the existence of positive solutions of mountain pass type on noncompact graphs with finitely many edges stated as in Theorem <ref>. 

Before proving Theorem <ref>, we establish a useful alternative for bounded Palais-Smale sequences of E_ρ with small mass.

	
	Let ∈𝐆 be a noncompact graph with finitely many edges and p>6. There exists μ=μ(p,)>0 such that, for every μ∈(0,μ] and every ρ∈[1/2,1], if (u_n)_n⊂ H_μ^1() satisfies u_n≥0 on  and, as n→+∞,
	
		
  (a) E_ρ(u_n,)→ c_ρ(),
		
  (b) E_ρ'(u_n,)+λ_n u_n→0 in the dual of H^1(), with λ_n:=-E_ρ'(u_n,)[u_n]/μ=ρu_n_p^p-u_n'_2^2/μ for every n, and
		
  (c) u_n⇀ u in H^1(),
	
	then either u≡ 0 on  or u is a critical point of E_ρ constrained to H_μ^1() at level c_ρ().


	Since (u_n)_n is bounded in H^1() by (c), so is (λ_n)_n ⊂. Hence, up to subsequences, λ_n→λ as n→+∞, for some λ∈.
	
	By weak lower semicontinuity, m:=u_2^2∈[0,μ]. If m=0, then u≡0. Conversely, if m=μ, then u∈ H_μ^1(), u_n→ u in L^2() and thus in L^p(), so that by (b) and λ_n→λ it follows
	
    u_n'-u'_2^2+λu_n-u_2^2=o(1)    as n→+∞ ,

	that is u_n→ u in H^1(), and u is a critical point of E_ρ constrained to H_μ^1() at level c_ρ(). Hence, to prove the lemma it is enough to rule out the case m∈(0,μ).
	
	To this end, we argue by contradiction. Assume thus 0<m<μ, so that u≢0 on  satisfies
	
    u”+ρ u^p-1=λ u    on every e∈_
    
    	u≥ 0    on 
    ∑_e≻du/dx_e()=0    for every ∈_ .
 
	By standard arguments, u>0 on , that is u∈ S_m,ρ (where S_m,ρ is the set defined in Remark <ref>). Thus we have from Lemma <ref>(c)  that λ >0.  By Remark <ref>, see also Corollary <ref>,
	there exists μ>0, depending on p and  only, such that for every μ∈(0,μ) and every ρ∈[1/2,1], there results
	
    E_ρ(u,) > 0 .

	Set then v_n:=u_n-u. By weak convergence of u_n to u in H^1(), as n →∞ we have
	
    v_n_2^2=μ-m+o(1),    v_n'_2^2=u_n'_2^2-u'_2^2+o(1) ,

	so that by Brezis-Lieb lemma <cit.> and (a)
	
    E_ρ(v_n,)=E_ρ(u_n,)-E_ρ(u,)+o(1)=c_ρ()-E_ρ(u,)+o(1)    as n→+∞ .

	Moreover, by (b), (c), (<ref>) and λ_n→λ, it is easy to see that v_n→0 in H^1(), where  denotes as usual the compact core of . This can be easily seen for instance by taking the difference between (b) and (<ref>) both tested with φ_n:=(u_n-u)η, where η is compactly supported on a fixed neighbourhood of  and η≡1 on . Hence, up to negligible corrections in H^1() that do not affect the validity of (<ref>) and (<ref>), we can take v_n to be identically zero on  for every n. Since ∖=⋃_i=1^N_i, where _i, i=1,…,N, are all the half-lines of , this means that we can think of v_n as the sum of its restrictions v_n,i to the half-lines _i, i=1,…,N. As v_n,i vanishes at the vertex of _i, we can embed _i in  and extend each v_n,i by 0. Up to translations, there is no loss of generality in further assuming that each v_n,i satisfies v_n,i_∞=v_n,i(0). Moreover, standard arguments (see e.g. <cit.>) show that, for every i=1,…, N, as n→+∞,
	
    ∫_ v_n,i'φ' dx-ρ∫_ v_n,i^p-1φ dx+λ∫_ v_n,iφ dx=o(1)φ_H^1()    ∀φ∈ H^1() .

	Observe that, together with (<ref>) and λ >0, this entails that (up to subsequences) v_n,i_∞↛0 as n→+∞ at least for one value of i. Indeed, otherwise v_n,i_p^p →0 as n→+∞ contradicting (<ref>).
	
	All in all, (<ref>) and (<ref>) imply that there exists k∈, k≥1, such that
	
    E_ρ(v_n,)=∑_i=1^NE_ρ(v_n,i,)=k E_ρ(ϕ_μ-m/k,ρ,)+o(1)

	as soon as n is large enough. Coupling with (<ref>) and (<ref>) then yields
	
    c_ρ() ≥ k E_ρ(ϕ_μ-m/k,ρ,)+o(1) ≥ E_ρ(ϕ_μ-m/k,ρ,)+o(1)  .

	However, by Proposition <ref>, for every >0 there exists  μ>0  such that c_ρ()≤ (1-)c_ρ()= (1-)E_ρ(ϕ_μ,ρ,) for every μ∈(0,μ] and every ρ∈[1/2,1]. Since μ-m<μ by assumption, coupling with (<ref>) and recalling (<ref>) (and possibly lowering μ) provides the contradiction we seek and completes the proof.
 


	The proof is divided in two steps: in the first one, we prove existence of positive critical points of E_ρ constrained to H_μ^1() for almost every ρ∈[1/2,1], in the second one we prove that (a subsequence of) these critical points converges, as ρ→1^-, to a positive critical point of E in H_μ^1().
	
	Step 1. For every fixed μ>0, by Lemma <ref> and Theorem <ref> it follows that, for almost every ρ∈[1/2,1], there exists a bounded sequence (u_n)_n⊂ H_μ^1() satisfying properties (i)–(iv) of Theorem <ref>. Furthermore, as pointed out in <cit.>, it is not restrictive to assume u_n≥0 on  for every n. Set
	
    λ_n:=-E_ρ'(u_n,)[u_n]/u_n_2^2=ρu_n_p^p-u_n'_2^2/μ .

	Since (u_n)_n is bounded in H^1() by property (iii) of Theorem <ref>, (λ_n)_n is bounded too, so that as n→+∞ up to subsequences λ_n→λ_ρ, for some λ_ρ∈, and u_n⇀ u_ρ in H^1(), for some u_ρ∈ H^1() satisfying
	
    u_ρ”+u_ρ^p-1=λ_ρ u_ρ   on every e∈_
    
    	u_ρ≥0    on 
    ∑_e≻du_ρ/dx_e()=0    for every ∈_ .

	Since  is a graph with at least one half-line and u_n satisfies properties (iv) of Theorem <ref>, by Lemma <ref> and <cit.> we obtain λ_ρ≥0.
	
	We now show that u_ρ≢0 on . Assume by contradiction that this is the case, so that u_n⇀ 0 in H^1() and u_n→0 in L_loc^∞() as n→+∞. Note that the convergence of u_n to 0 cannot be in L^∞(), because then u_n∈ H_μ^1() would imply u_n→0 in L^p(), which together with (<ref>) and lim infλ_n≥0 would yield E_ρ(u_n,)→0, that is impossible since E_ρ(u_n,)→ c_ρ()>0. Hence, u_n→0 in L_loc^∞() but not in L^∞(). Whenever the mass is smaller than a threshold independent of ρ, arguing as in the proof of Lemma <ref> it is then easy to see that, denoting by _i, i=1,…,N, the half-lines of , as n→+∞ there results
	
    u_n_L^2(⋃_i=1^N_i)^2=μ+o(1)
       ∫__iu_n'φ' dx-∫__iu_n^p-1φ dx+λ∫__iu_nφ dx=o(1)φ_H^1(_i),    ∀ i=1,…,N ,

	and, up to negligible corrections in H^1(), u_n can be taken to be identically zero on . Since u_n_∞ is uniformly bounded away from 0, it then follows that there exists k∈, k≥1, such that
	
    E_ρ(u_n,)=k E_ρ(ϕ_μ/k,ρ,)+o(1)≥ E_ρ(ϕ_μ,ρ,)+o(1)=c_ρ()+o(1)    as n→+∞ ,

	the first inequality following by (<ref>). However, since by assumption  contains either a pendant or a signpost, Proposition <ref> ensures that E_ρ(u_n,)=c_ρ()+o(1)<c_ρ() as μ→ 0  (uniformly in ρ), contradicting (<ref>). Hence, u_ρ≢0 on .
	
	Up to possibly lowering the threshold on the mass, by Lemma <ref> we then have that u_ρ is a critical point of E_ρ constrained to H_μ^1() at level c_ρ(). Moreover, since u_ρ satisfies (<ref>) and  is a noncompact graph with finitely many edges, we have λ_ρ>0 and u_ρ>0 on . 
	
	Step 2. Since the previous argument works for every μ∈(0,μ_p,] and almost every ρ∈[1/2,1], for some value μ_p,>0 depending on p and  only, by Step 1 there exists u_ρ>0 critical point of E_ρ constrained to H_μ^1() at level c_ρ(), for every μ∈(0,μ_p,] and for almost every ρ∈[1/2,1]. We then fix any μ in this interval and consider the limit ρ→1^- (along a subsequence of ρ's, not relabebed). Recall first that u_ρ satisfies (<ref>) with λ_ρ>0. Since u_ρ_2^2=μ for every ρ, arguing exactly as in the proof of Proposition <ref> proves that (λ_ρ)_ρ is bounded as ρ→1^-.  By (<ref>), the boundedness of λ_ρ and that of E_ρ(u_ρ,)=c_ρ(), we obtain that (u_ρ)_ρ is bounded in H_μ^1() as ρ→1^-. All in all, this implies that, as ρ→ 1, (u_ρ)_ρ⊂ H_μ^1() satisfies E(u_ρ)→ c()+o(1). Indeed, the continuity of the map ρ↦ c_ρ() as ρ→1^- follows repeating the argument of <cit.>. Furthemore,
	
    E'(u_ρ,)+λ_ρ u_ρ→0    in the dual of H^1()
 
	where, up to subsequences, λ_ρ→λ≥0 as ρ→1^-. Therefore, repeating verbatim the argument of Step 1 yields a positive critical point of E in H_μ^1() at level c(), i.e a solution of (<ref>) as desired.




§ PERIODIC GRAPHS: PROOF OF THEOREM <REF>


In this section we prove our existence result for mountain pass solutions on periodic graphs. The line of the argument is essentially the same as the one exploited in the previous section for graphs with half-lines, but the periodicity of the graph will now be crucial to recover the necessary compactness. The first result in this direction is indeed the analogue of Lemma <ref>.

	
	Let ∈𝐆 be a periodic graph and p>6. There exists μ=μ(p,)>0 such that, for every μ∈(0,μ] and every ρ∈[1/2,1], if (u_n)_n⊂ H_μ^1() satisfies u_n≥0 on  and, as n→+∞,
	
		
  (a) E_ρ(u_n,)→ c_ρ(),
		
  (b) E_ρ'(u_n,)+λ_n u_n→0 in the dual of H^1(), with λ_n:=-E_ρ'(u_n, )[u_n]/μ for every n, and
		
  (c) u_n⇀ u in H^1(),
	
	then either u≡ 0 on  or u is a critical point of E_ρ constrained to H_μ^1() at level c_ρ().


	Since the proof is very similar to that of Lemma <ref>, we only highlight the main differences. 
	
	Let us consider first the case of a periodic graph  with no pendant. If we assume m:=u_2^2∈(0,μ), arguing exactly as above we obtain that u∈ S_m,ρ solves (<ref>) for some  λ∈, where up to subsequences λ_n→λ as n→+∞. Note that by Lemma <ref>(b), λ≥ 0 and also, assuming μ >0 sufficiently small, λ >0 by Lemma <ref>(c).  Moreover, exploiting the periodicity of , we can consider u_n-u and suitably translate it on  to obtain a function v_n such that, as n→+∞,
	
    v_n_2^2=μ-m+o(1)
       E_ρ(v_n,)=c_ρ()-E_ρ(u,)+o(1)
       E_ρ'(v_n,)+λ v_n→0 in the dual of H^1()

	and, for every n, v_n attains its L^∞ norm in a given compact subset of  independent of n. Since (v_n)_n is bounded in H^1() by definition,  the argument used for u_n implies that v_n⇀ w in H^1(), for some w satisfying again (<ref>) with the same λ as u. Iterating this procedure, we eventually end up with finitely many functions w_k, k=1,…, N, each one solving (<ref>) with the same λ>0 (here w_1=u, w_2=w as above) and such that
	
    μ=u_n_2^2=∑_k=1^Nw_k_2^2,     c_ρ()=E_ρ(u_n,)=∑_k=1^NE_ρ(w_k,)+o(1)

	as n → +∞. Note however that N≥2 since by assumption w_1_2^2=m∈(0,μ), so that in particular there exists k∈{1,…,N} for which w_k_2^2≤μ/2. Remark <ref> and Proposition <ref> then ensure the existence of a value μ>0, depending only on p and , such that  for every ρ∈[1/2,1] and μ∈ (0, μ], 
	
    E_ρ(ϕ_μ,ρ,)+o(E_ρ(ϕ_μ,ρ,))≥ c_ρ()>E_ρ(w_k,)≥ E_ρ(ϕ_μ/2,ρ,)+o(E_ρ(ϕ_μ/2,ρ,)) ,

	i.e. a contradiction in view of (<ref>).
	
	If  has a pendant, the proof is exactly the same, making use of the corresponding estimates in Remark <ref> and Proposition <ref>.
 
We are now almost ready to prove Theorem <ref>, for which we only need the following preliminary lemma, that is the counterpart of <cit.> in the context of periodic graphs. 



	
	Let ∈𝐆 be a periodic graph. For every λ<0 there exists a subspace Y⊂ H^1() such that dim Y=3 and
	
    w'_2^2+λw_2^2≤λ/2w_H^1()^2    ∀ w∈ Y.



	It is a direct consequence of the fact that λ_=0 (with λ_ given by (<ref>)) for every periodic graph . Indeed, from this it follows that, for every λ<0, there exists w_0∈ H^1() with compact support such that
	
    w_0'_2^2≤-λ/2-λw_0_2^2 ,

	i.e. w_0 satisfies (<ref>). Exploiting the periodicity of the graph, it is then straightforward to take w_1,w_2∈ H^1() to be two different copies of w_0, suitably translated on  so that w_0, w_1, w_2  have disjoint supports. The proof is then completed simply taking Y= span{w_0,w_1,w_2}.


	The proof follows exactly the same line as that of Theorem <ref>, replacing whenever needed Lemma <ref> with Lemma <ref>, and <cit.> with Lemma <ref>. The only significant difference arises when one shows that the weak limit u_ρ of the Palais-Smale sequence (u_n)_n⊂ H_μ^1() for E_ρ at level c_ρ() is not identically equal to 0 on . In the case of a periodic graph, it is even easier to perform this passage. Indeed, exploiting the periodicity of , there is no loss of generality in taking from the very beginning each u_n to attain its L^∞ norm in a given compact subset of  independent of n. Like this, if we assume by contradiction that u_n⇀0 in H^1() as n→+∞, then u_n→0 in L^p(), in turn entailing (just as in the proof of Theorem <ref>) u_n'_2^2→0 and E_ρ(u_n,)→0 as n→+∞, that is impossible since E_ρ(u_n,)→ c_ρ()>0 by assumption.






§ PROOF OF THEOREMS <REF>–<REF>



This section is devoted to the proof of Theorems <ref>–<ref>, i.e. to show the existence of solutions of (<ref>) with negative energy on noncompact graphs ∈𝐆 with finitely many edges and suitable topological assumptions when the mass is large enough.

We begin with the proof of Theorem <ref>. Since we will prove the theorem with a perturbative argument around p=6, let us briefly recall some previous results at this critical exponent. First, when p=6 it is well-known that the function ϕ_μ,1 as in Section <ref> solves the minimization problem

    inf_u∈ H_μ^1()E(u,)

if and only if μ=μ_:=√(3)π/2. In this case, E(ϕ_μ_,1,)=0.  Moreover, for every λ>0, the function φ_λ(x):=√(λ)ϕ_μ_,1(λ x) satisfies φ_λ∈ H_μ_^1(), E(φ_λ,)=0 and 

    λ/2μ_=J_λ,6(φ_λ,)=inf_v∈_λ,6()J_λ,6(v,)

where 

    J_λ,p(v,):=1/2v'_2^2+λ/2v_2^2-1/pv_p^p

and 

    _λ,p():={v∈ H^1() : v'_2^2+λv_2^2=v_p^p} .

Conversely, when p=6 and  has finitely many edges, no pendant and exactly one half-line, it has been shown in <cit.> that the minimization problem inf_u∈ H_μ^1()E(u,) has a solution if and only if μ∈(μ_/2,μ_]. Furthermore, for every μ∈(μ_/2,μ_], if u∈ H_μ^1() is such that E(u,)= inf_u∈ H_μ^1()E(u,), then E(u,)<0. 

The following preliminary lemma moves from these considerations to establish existence of solutions to the problem

    inf_v∈_λ,p()J_λ,p(v,)

for p slightly above 6 (where J_λ,p(v,), _λ,p() are defined analogously to (<ref>), (<ref>)).

	
	Let ∈𝐆 be a noncompact graph with finitely many edges, no pendant and exactly one half-line. Then, there exist λ>0 and δ>0 (depending on ) such that inf_v∈_λ,p()J_λ,p(v,) is attained, for every p∈[6,6+δ).


	Recall first that, by <cit.>, if ∈𝐆 is a noncompact graph with finitely many edges, for every p>2 and λ>0 the minimization problem inf_v∈_λ,p()J_λ,p(v,) admits a solution if
	
    inf_v∈_λ,p()J_λ,p(v,)<inf_v∈_λ,p()J_λ,p(v,) .

	Hence, to prove the lemma it is enough to show that (<ref>) holds at a certain value of λ for every p sufficiently close to 6, whenever  has finitely many edges, exactly one half-line and no pendant. Moreover, since at fixed λ>0 the map p↦inf_v∈_λ,p()J_λ,p(v,) is easily proved to be continuous, we are left to prove the validity of  (<ref>) at p=6.
	
	Let then p=6. In this case, by the aforementioned result of <cit.>, there exists u∈ H_μ_^1() such that
	
    E(u,)=inf_u∈ H_μ_^1()E(u,)<0 .

	By <cit.>, it then follows that
	
    J_λ,6(u,)=inf_v∈_λ,6()J_λ,6(v,),   with  λ:=u_6^6-u'_2^2/μ_ .
 
	 By (<ref>) and (<ref>), we then have
	
    inf_v∈_λ,6()J_λ,6(v,)=E(u,)+λ/2u_2^2<λ/2μ_=inf_v∈_λ,6()J_λ,6(v,) ,

	and we conclude.




	Since it is a standard fact that solutions of the problem inf_v∈_λ()J_λ,p(v,) satisfy (<ref>), in view of Lemma <ref> to complete the proof of Theorem <ref> it is enough to show that, as soon as p is sufficiently close to 6, functions u∈_λ,p() for which J_λ,p(u,)=inf_v∈_λ,p()J_λ,p(v,) satisfy E(u,)<0 (where λ is the number given by Lemma <ref>). 
	
	To this end, we argue by contradiction. Assume that there exist exponents p_n→ 6^+ and functions u_n∈_λ,p_n() such that J_λ,p_n(u_n,)=inf_v∈_λ,p_n()J_λ,p_n(v,) and E(u_n,)≥0. Since p_n is bounded and the proof of Lemma <ref> shows that inf_v∈_λ,p_n()J_λ,p_n(v,)<inf_v∈_λ,p_n()J_λ,p_n(v,), it follows that (u_n)_n is bounded in H^1(). Hence, up to subsequences  u_n⇀ u in H^1() and u_n→ u in L_loc^∞() as n→+∞, for some u∈ H^1(). 
	
	Note first that u≢0 on . Indeed, if it were u≡0 on , we would have u_n→0 in L^∞(). Therefore, denoting by  the unique half-line of , it would follow
	
    inf_v∈_λ,p_n()J_λ,p_n(v,)=J_λ,p_n(u_n,)=J_λ,p_n(u_n,)+o(1)≥inf_v∈_λ,p_n() J_λ,p_n(v,)+o(1)

	which is impossible by the proof of Lemma <ref>.
	
	Arguing in a similar way, we also observe that u_n converges strongly in H^1() to u. Indeed, u_n being a solution of (<ref>) with λ=λ implies that u_n(x)=φ_λ,p_n(x+a_n) for every x∈, where φ_λ,p_n is the unique H^1 solution of (<ref>) with λ=λ and p=p_n (and ρ=1), and a_n∈ is a suitable number depending on n. It is then easy to see that 
	
    lim inf_n→+∞a_n>-∞ ,

	because if this were not the case we would obtain again a contradiction as in (<ref>). By (<ref>), p_n→6^+ and the properties of φ_λ,p_n, it thus follows that the restriction of u_n to  converges strongly in H^1() to φ_λ,6(·+a), for a suitable a∈∪{+∞} (if a=+∞, then u_n tends to 0 strongly in H^1()). By uniqueness of the limit, this means that u(·)≡φ_λ,6(·+a) on  and, in particular,
	
    u_n_L^2()→u_L^2(),    u_n_L^p_n()→u_L^6()    as n→+∞ .

	Moreover, since u_n→ u in L^∞_loc(), then u_n(x)→ u(x) almost everywhere on the compact core 𝒦 as n→+∞. Since  is a set of finite measure by assumption and u_n is uniformly bounded from above (by the uniform boundedness in H^1()), by dominated convergence  we have
	
    u_n_L^p_n()→u_L^6()    as n→+∞ .

	Together with (<ref>), the strong convergence of u_n to u in L^2() and weak lower semicontinuity, this yields
	
    u_n_2→u_2,    u_n_p_n→u_6    as n→+∞
       lim inf_n→+∞u_n'_2≥u'_2 ,

	that is
	
    σ:=u'_2^2+λu_2^2/u_6^6≤lim inf_n→+∞u_n'_2^2+λu_n_2^2/u_n_p_n^p_n=1 ,

	the last equality being a consequence of u_n∈_λ,p_n() for every n. Noting that σ^1/4u∈_λ,6() by definition, we obtain
	
    inf_v∈_λ,6()J_λ,6(v,)≤ J_λ,6(σ^1/4u)=    (1/2-1/6)σ^3/2u_6^6≤lim inf_n→+∞(1/2-1/p_n)u_n_p_n^p_n
    
    	=    lim inf_n→+∞inf_v∈_λ,p_n()J_λ,p_n(v,)=inf_v∈_λ,6()J_λ,6(v,),

	where we exploited also the continuity of the map p↦inf_v∈_λ,p()J_λ,p(v,). The previous chain of inequalities shows that σ=1, u∈_λ,6(), J_λ,6(u,)=inf_v∈_λ,6()J_λ,6(v,) and that the convergence of u_n to u is strong in H^1(). By Lemma <ref> above, <cit.> and the aforementioned results of <cit.>, this implies that u_2^2=μ_ and E(u,)=inf_v∈ H_μ_^1()E(v,)<0. By strong convergence in H^1(), this entails E(u_n,)<0 for sufficiently large n, providing the contradiction we seek and concluding the proof.


We can now conclude our analysis by proving Theorem <ref>.


Let  be a noncompact graph with finitely many edges, at least one of which bounded, and such that every vertex is attached at an even number (possibly zero) of half-lines. Observe first that, on a graph with these properties, for every λ>0 there exists a positive solution u_λ of (<ref>). Indeed, since for every vertex _i∈_ there exists k_i∈ such that _i is attached to 2k_i half-lines, we can group these half-lines in pairs and think of them as k_i full lines crossing at the same point _i.  Then, identifying _i with the origin of each of these k_i lines and denoting by φ_λ the unique solution of the nonlinear Schrödinger equation (<ref>) on  (with ρ=1), it is readily seen that the function

    u_λ(x):=λ^1/p-2   for x∈
    φ_λ(x+τ_λ)    for x on each of the k_i lines attached to _i, for every _i∈_,

with  the compact core of  and τ_λ>0 such that φ_λ(τ_λ)=λ^1/p-2, solves (<ref>) on . Note that τ_λ exists because
		
    φ_λ_∞
    			= φ_λ(0)
    			= ( λ p/2)^1/p-2
    			> λ^1/p-2.

To complete the proof of Theorem <ref> we are thus left to show that, varying λ, u_λ has negative energy as soon as its mass is sufficiently large. Since u_λ is made of a constant function on the compact core of  and a certain number, say k, of full copies of φ_λ, we have
		
    u_λ_2^2   = || λ^2/p-2 + k φ_λ_2^2
    			= || λ^2/p-2 + k λ^6-p/2(p-2)φ_1 _2^2 ,
     u_λ_p^p   = || λ^p/p-2 + k φ_λ_p^p 
    			= || λ^p/p-2 + k λ^p+2/2(p-2)φ_1 _p^p ,
     u_λ' _2^2   = k φ'_λ_2^2
    			= k λ^p+2/2(p-2)φ'_1 _2^2 ,

		where we used the well-known relation φ_λ(x) = λ^1/p-2φ_1(√(λ)x).
		Therefore,
		
    E(u_λ,)
    			= 1/2 u_λ' _2^2 - 1/p u_λ_p^p
    			= k ( φ'_1 _2^2/2 - φ_1 _p^p/p) λ^p+2/2(p-2)
    			- ||/pλ^p/p-2.

		Since p > 2, one has 2/p-2>0 and p/p-2 > p+2/2(p-2).
		Hence, if λ is large enough (depending on k, || and p), then u_λ_2 is large and E(u_λ,) < 0.





§ ACKNOWLEDGEMENTS

S.D. acknowledges that this study was carried out within the project E53D23005450006 “Nonlinear dispersive equations in presence of singularities” – funded by European Union – Next Generation EU  within the PRIN 2022 program (D.D. 104 - 02/02/2022 Ministero dell'Università e della Ricerca). This manuscript reflects only the author's views and opinions and the Ministry cannot be considered responsible for them. The work has been partially supported by the INdAM GNAMPA project 2023 “Modelli nonlineari
in presenza di interazioni puntuali”. This work has been carried out in the framework of the Project NQG (ANR-23-CE40-0005-01), funded by the French National Research Agency (ANR).
L. Jeanjean thank the ANR for its support. 


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