# Normalized solutions of $L^{2}$-supercritical NLS equations on noncompact
metric graphs with localized nonlinearities

Jack<EMAIL_ADDRESS>Université de Franche-Comté,
Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray,
25000 Besançon, France  Xiaojun Chang<EMAIL_ADDRESS>School of
Mathematics and Statistics & Center for Mathematics and Interdisciplinary
Sciences, Northeast Normal University, Changchun 130024, Jilin, PR China
Louis<EMAIL_ADDRESS>Université de Franche-Comté,
Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray,
25000 Besançon, France  Nicola<EMAIL_ADDRESS>Dipartimento di
Matematica “Giuseppe Peano”, Università degli Studi di Torino, 10123 Torino,
Italy

###### Abstract

In this paper we are concerned with the existence of normalized solutions for
nonlinear Schrödinger equations on noncompact metric graphs with localized
nonlinearities. In a $L^{2}$-supercritical regime, we obtain the existence of
solutions for any prescribed mass. This result is obtained through an approach
which could prove successful to treat more general equations on noncompact
graphs.

Keywords: Nonlinear Schrödinger equations; $L^{2}$-supercritical; Noncompact
metric graphs; Localized nonlinearities; Variational methods.
Mathematics Subject Classification: 35J60, 47J30

Acknowledgements: X. J. Chang is partially supported by NSFC (11971095). N.
Soave is partially supported by the INDAM-GNAMPA group.
J. Borthwick gratefully acknowledges that part of this work was supported by
the French Investissements d’Avenir program, project ISITE-BFC (contract ANR-
15-IDEX-0003).

##  1 Introduction and main results

This paper is devoted to the existence of bound states of prescribed mass for
the $L^{2}$_-supercritical_ nonlinear Schrödinger equation (NLS) on
$\mathcal{G}$

$-u^{\prime\prime}+\lambda u=\kappa(x)|u|^{p-2}u,$ (1.1)

coupled with Kirchhoff conditions at the vertices, see (1.4) below. Here
$\lambda$ appears as a Lagrange multiplier, $p>6$, $\mathcal{G}$ is a
noncompact connected metric graph, and $\kappa$ is the characteristic function
of the compact core $\mathcal{K}$ of $\mathcal{G}$. We recall, see [4], that
if $\mathcal{G}$ is a metric graph with a finite number of vertices its
compact core $\mathcal{K}$ is defined as the metric subgraph of $\mathcal{G}$
consisting of all the bounded edges of $\mathcal{G}$.

A solution with prescribed mass is often referred to as a _normalized
solution_. For Equation (1.1) they correspond to critical points of the NLS
energy functional $E(\cdot\,,\mathcal{G}):H^{1}(\mathcal{G})\to\mathbb{R}$
defined by

$\displaystyle
E(u,\mathcal{G})=\frac{1}{2}\int_{\mathcal{G}}|u^{\prime}|^{2}\,dx-\frac{1}{p}\int_{\mathcal{K}}|u|^{p}\,dx$
(1.2)

under a mass constraint

$\int_{\mathcal{G}}|u|^{2}\,dx=\mu>0.$ (1.3)

The study of NLS on metric graphs has attracted much attention over the last
few decades. Among the physical motivations to consider Schrödinger equations
on a metric graph, the most prominent one comes from the study of quantum
graphs, see for example [6, 11, 29, 32] and the references therein. In
particular, the investigation of NLS with localized nonlinearities appeared in
optical fibers and Bose-Einstein condensates in non-regular traps (see [24,
32]).

Recently, much effort has been devoted to the existence of normalized
solutions of NLS on metric graphs, in the $L^{2}$_-subcritical_ (i.e.,
$p\in(2,6)$) or $L^{2}$_-critical regimes_ (i.e., $p=6$). In these two
regimes, the energy functional $E(\cdot,\mathcal{G})$ is bounded from below on
the mass constraint and one may develop minimization methods. A relevant
notion is indeed the one of ground states, namely of solutions which minimize
the energy functional on the constraint. For the existence of ground states
solutions, see [1, 2, 3, 4, 5, 33, 35] for noncompact $\mathcal{G}$, and [14,
16] for the compact case; some studies are also conducted on the existence of
local minimizers, see [36]. Regarding problems with a localized nonlinearity
as in Equation (1.1), existence or non-existence of ground states solutions
was discussed in [39, 42] for the $L^{2}$-subcritical case, and in [18, 19]
for the $L^{2}$-critical case. Moreover, in the $L^{2}$-subcritical regime,
genus theory was applied in [38] to obtain the existence of multiple bound
states solutions at negative energy levels.

In the $L^{2}$-supercritical regime on general metric graphs, i.e., when
$p>6$, the energy functional $E(\cdot,\mathcal{G})$ is always unbounded from
below. Moreover, due to the fact that graphs are not scale invariant, the
techniques based on scalings, usually employed in the Euclidean setting and
related to the validity of a Pohozaev identity (see [26] or [8, 25, 40, 41]),
do not work. These two features make the search for normalized solutions in
the $L^{2}$-supercritical regime delicate. Recently, in [15], this issue was
considered on compact metric graphs. The existence of solutions was proved for
small values of $\mu>0$, see [15, Theorem 1.1] for a precise statement. In the
present paper we pursue this study by allowing the graph to be noncompact.
Although in our proofs we make use of the fact that the nonlinearity in
Equation (1.1) only acts on a compact part of the graph, we believe that our
approach could prove fruitful to study more general situations.

### Basic notations and main result

Throughout this paper, we consider a connected noncompact metric graph
$\mathcal{G}=(\mathcal{E},\mathcal{V})$, where $\mathcal{E}$ is the set of
edges and $\mathcal{V}$ is the set of vertices. We assume that $\mathcal{G}$
has a finite number of edges and the compact part $\mathcal{K}$ is non-empty.
Without loss of generality, we may assume that $\\{{\rm e}_{1},\cdots,{\rm
e}_{m_{1}}\\}$ are the bounded edges, $\\{l_{1},\cdots,l_{m_{2}}\\}$ are the
half-lines, and $\\{{\rm{v}}_{1},\cdots,{\rm{v}}_{m_{3}}\\}$ are the finite
vertices of $\mathcal{G}$. Here $m_{1},m_{2},m_{3}$ are positive integers.
Moreover, any bounded edge ${\rm e}$ is identified with a closed bounded
interval $I_{\rm e}$, typically $[0,\ell_{\rm e}]$ (where $\ell_{\rm e}$ is
the length of ${\rm e}$), while each unbounded edge is identified with a
closed half-line $I_{\rm e}=[0,+\infty)$. For details, one can refer to [3, 4,
11]. A function on metric graph $u:\mathcal{G}\to\mathbb{R}$ is identified
with a vector of functions $\\{u_{\mathrm{e}}\\}$, where each $u_{\mathrm{e}}$
is defined on the corresponding interval $I_{\rm e}$ such that
$u|_{\mathrm{e}}=u_{\mathrm{e}}$. Endowing each edge with Lebesgue measure,
one can define the space $L^{p}(\mathcal{G})$ in a natural way, with norm

$\|u\|_{L^{p}(\mathcal{G})}^{p}=\sum_{\mathrm{e}}\|u_{\mathrm{e}}\|_{L^{p}(\mathrm{e})}^{p}.$

The Sobolev space $H^{1}(\mathcal{G})$ consists of the set of continuous
functions $u:\mathcal{G}\to\mathbb{R}$ such that $u_{\mathrm{e}}\in
H^{1}([0,\ell_{\mathrm{e}}])$ for every edge $\mathrm{e}$ (note that
$\ell_{\mathrm{e}}$ can be $+\infty$); the norm in $H^{1}(\mathcal{G})$ is
defined as

$\|u\|_{H^{1}(\mathcal{G})}^{2}=\sum_{\mathrm{e}}\|u_{\mathrm{e}}^{\prime}\|_{L^{2}(\mathrm{e})}^{2}+\|u_{\mathrm{e}}\|_{L^{2}(\mathrm{e})}^{2}.$

We shall study the existence of critical points of the functional
$E(u,\mathcal{G})$ constrained on the $L^{2}$-sphere

$H_{\mu}^{1}(\mathcal{G}):=\left\\{u\in
H^{1}(\mathcal{G}):\int_{\mathcal{G}}|u|^{2}\,dx=\mu\right\\}.$

If $u\in H^{1}_{\mu}(\mathcal{G})$ is such a critical point, then there exists
a Lagrange multiplier $\lambda\in\mathbb{R}$ such that $u$ satisfies the
following problem:

$\begin{cases}-u^{\prime\prime}+\lambda u=\kappa(x)|u|^{p-2}u~{}~{}&\mbox{on
every edge}~{}{\rm e}~{}\mbox{of}~{}\mathcal{G},\\\ \sum\limits_{{\rm
e}\succ{\rm v}}u_{{\rm e}}^{\prime}({\rm v})=0~{}~{}&\mbox{at every
vertex}~{}{\rm v}\in\mathcal{K},\end{cases}$ (1.4)

where ${\rm e}\succ{\rm v}$ means that the edge ${\rm e}$ is incident at ${\rm
v}$, and the notation $u^{\prime}_{{\rm e}}({\rm v})$ stands for
$u^{\prime}_{{\rm e}}(0)$ or $-u^{\prime}_{{\rm e}}(\ell_{{\rm e}})$,
according to whether the vertex ${\rm v}$ is identified with $0$ or
$\ell_{\mathrm{e}}$ (namely, the sum involves the outer derivatives at ${\rm
v}$). The second equation is the so-called _Kirchhoff condition_.

Our main result can be stated as follows.

######  Theorem 1.1.

Let $\mathcal{G}$ be any noncompact metric graph having a non-empty compact
core $\mathcal{K}$, and let $p>6$. Then, for any $\mu>0$, problem (1.4) with
the mass constraint (1.3) has a positive solution $u$ associated to a Lagrange
multiplier $\lambda>0$. This solution corresponds to a critical point of
$E(\cdot\,,\mathcal{G})$ on $H_{\mu}^{1}(\mathcal{G})$ at a strictly positive
energy level.

Let us present the main ideas to prove Theorem 1.1.

It is simple to check that the functional $E(\cdot\,,\mathcal{G})$ admits a
mountain pass geometry in $H_{\mu}^{1}(\mathcal{G})$ for any $\mu>0$. However,
the existence of a bounded Palais-Smale sequence at the mountain pass level, a
key preliminary step to obtain a critical point, is not directly available. To
overcome this issue, following the strategy presented in [15], we first
consider the family of functionals
$E_{\rho}(\cdot,\mathcal{G}):H^{1}(\mathcal{G})\to\mathbb{R}$ given by

$\displaystyle
E_{\rho}(u,\mathcal{G})=\frac{1}{2}\int_{\mathcal{G}}|u^{\prime}|^{2}dx-\frac{\rho}{p}\int_{\mathcal{K}}|u|^{p}dx,\qquad\forall
u\in H^{1}(\mathcal{G}),\,\forall\rho\in\left[\frac{1}{2},1\right].$ (1.5)

As in [15] we shall rely on a parametrized version of a mountain-pass theorem
on a constraint established in [12]. Roughly, this theorem, recalled here as
Theorem 2.4, guarantees, for almost every value of a parameter, the existence
of a bounded Palais-Smale sequence with second-order information. Applying
Theorem 2.4 on the family $E_{\rho}(\cdot,\mathcal{G})$ we deduce that, for
almost every $\rho\in\left[\frac{1}{2},1\right]$, these functionals admit a
bounded Palais-Smale sequence $\\{u_{\rho,n}\\}$ of non negative functions
with “approximate Morse index” information.

The next step is to show that, for such $\rho\in\left[\frac{1}{2},1\right]$,
the sequence $\\{u_{\rho,n}\\}$ converges to some $u_{\rho}$. If this is the
case, as a consequence of the Morse type information on $\\{u_{\rho,n}\\}$ one
can directly deduce that $u_{\rho}$ has a Morse index $m(u_{\rho})$ at most 2,
see Definition 3.4. In [15] the convergence of $\\{u_{\rho,n}\\}$ was
immediate due to the compactness of $\mathcal{G}$. In our setting, despite the
fact that the nonlinearity acts only on the compact core, the non-compactness
of the graph prevents us from directly concluding that bounded Palais-Smale
sequences converge. This kind of difficulty is regularly encountered in the
search of normalized solutions in $L^{2}$-supercritical problems when the
underlying domain is noncompact, see for example [28, 31]. To overcome it one
needs to benefit from information on the sign of an associated sequence of
almost Lagrange multipliers. In our problem we derive this information by
using the “approximate Morse index” information carried by the sequence
$\\{u_{\rho,n}\\}$. This argument seems new and we suspect that it could be
used in other problems. In order to emphasize its generality we present it, in
Section 2, in a general framework.

Having been able to show, for almost every
$\rho\in\left[\frac{1}{2},1\right]$, the convergence of $\\{u_{\rho,n}\\}$ to
some $u_{\rho}$ with Morse index at most 2 (see Proposition 3.5 for a precise
statement), our proof follows the lines of [15]. Namely, taking a sequence
$\rho_{n}\to 1^{-}$, we adapt the blow-up analysis, developed in [15], to show
that $\\{u_{n}\\}:=\\{u_{\rho_{n}}\\}$ is bounded. Then, we use some arguments
previously developed in the proof of Proposition 3.5 to deduce that
$\\{u_{n}\\}$ converges and to end the proof of Theorem 1.1.

The rest of the paper is mainly devoted to the proof of Theorem 1.1. Before
proceeding, we would like to discuss some open problems and possible further
directions of research.

######  Remark 1.2.

Since they are obtained through a mountain pass procedure on a codimension $1$
manifold, it is natural to expect that the solutions we found are orbitally
unstable. This is what happens in mass-superlinear problems when the
underlying equation is autonomous and set on all the space. See, for example,
[39] in that direction. However, here things are more complex since we have
little information on the variational characterizations of the solutions that
we obtain. In comparison to [39], we do not know if our solutions can be
characterized as a ground state or as a global minimizer on some additional
constraint. More globally, the issue of orbital stability (and the related
issue of the monotonicity of the mass with respect to the Lagrange multiplier)
appears complex _on a general non-compact graph_. So far, orbital
stability/instability of bound states on metric graphs has only been
characterized for some specific examples, see [2, 14, 34] and references
therein.

######  Remark 1.3.

Another challenging issue would consist in characterizing precisely the
properties of the solutions in terms of the number and the location of local
and global maximum points, at least in the limit of large/small mass
parameter. This is also related to multiplicity issues. We refer to [7, 13]
for some recent results in the subcritical case, however it seems unlikely
that they are directly adaptable to the supercritical case (the former
exploits a doubly constrained minimization argument; the latter uses some
techniques specifically suited to deal with the cubic NLS).

On the other hand, it is not difficult to prove the following basic result,
which actually holds for _any_ one-signed critical point of
$E(\cdot,\mathcal{G})$ on $S_{\mu}$.

######  Proposition 1.4.

For $p>6$ and $\mu>0$, let $u\in H^{1}_{\mu}(\mathcal{G})$ be a critical point
of $E(\cdot,\mathcal{G})$ defined in (1.2) under the mass constraint (1.3),
and suppose that $u>0$ and $\lambda>0$. If $\bar{x}\in\mathcal{G}$ is a local
maximum point of $u$, then $\bar{x}\in\mathcal{K}$, the compact core.
Furthermore, on any half-line $\ell_{i}\simeq[0,+\infty)$ one has
$u(x)=c_{i}e^{-\sqrt{\lambda}x}$ for some $c_{i}>0$.

The proof follows directly by integrating the equation on each half-line, and
using the fact that $\lambda>0$.

######  Remark 1.5.

Finally, we make a short remark on extending our methods to study the
existence of bound states in cases where the nonlinearity acts on the full
graph, and not only its compact core. In this direction, our min-max geometric
argument still works, but some issues arise in studying the compactness of
Palais-Smale sequences. Considering what happens in the subcritical and
critical cases (see e.g. [3, 4, 5]), it is expected that topological and
metric properties of the underlying graph play a role.

The paper is organized as follows. In Section 2 we recall in Theorem 2.4 the
main result of [15] and explore some of its consequences. In particular, we
show that the second order information on the Palais-Smale sequences can be
used to get uniform bounds from below on the associated sequences of almost
Lagrange parameter, see Lemma 2.5. In Section 3 we show that
$E_{\rho}(\cdot\,,\mathcal{G})$ has a mountain pass critical point for almost
every $\rho\in\left[\frac{1}{2},1\right]$ and we prove Proposition 3.5.
Section 4 presents the proof of Theorem 1.1.

##  2 Some preliminary definitions and results

In this section, we recall [12, Theorem 1] and explore some of its
consequences.

Let $\left(E,\langle\cdot,\cdot\rangle\right)$ and
$\left(H,(\cdot,\cdot)\right)$ be two _infinite-dimensional_ Hilbert spaces
and assume that $E\hookrightarrow H\hookrightarrow E^{\prime}$, with
continuous injections. For simplicity, we assume that the continuous injection
$E\hookrightarrow H$ has norm at most $1$ and identify $E$ with its image in
$H$. Set

$\begin{cases}\|u\|^{2}=\langle u,u\rangle,\\\ |u|^{2}=(u,u),\end{cases}\quad
u\in E,$

and we define for $\mu>0$:

$S_{\mu}=\\{u\in E\big{|}\,|u|^{2}=\mu\\}.$

In the context of this paper, we shall have $E=H^{1}(\mathcal{G})$ and
$H=L^{2}(\mathcal{G})$. Clearly, $S_{\mu}$ is a smooth submanifold of $E$ of
codimension $1$. Its tangent space at a given point $u\in S_{\mu}$ can be
considered as the closed codimension $1$ subspace of $E$ given by:

$T_{u}S_{\mu}=\\{v\in E\big{|}\,(u,v)=0\\}.$

In the following definition, we denote by $\|\cdot\|_{*}$ and
$\|\cdot\|_{**}$, respectively, the operator norm of
$\mathcal{L}(E,\mathbb{R})$ and of $\mathcal{L}(E,\mathcal{L}(E,\mathbb{R}))$.

######  Definition 2.1.

Let $\phi:E\rightarrow\mathbb{R}$ be a $C^{2}$-functional on $E$ and
$\alpha\in(0,1]$. We say that $\phi^{\prime}$ and $\phi^{\prime\prime}$ are
$\alpha$-Hölder continuous on bounded sets if for any $R>0$ one can find
$M=M(R)>0$ such that, for any $u_{1},u_{2}\in B(0,R)$:

$||\phi^{\prime}(u_{1})-\phi^{\prime}(u_{2})||_{*}\leq
M||u_{1}-u_{2}||^{\alpha},\quad||\phi^{\prime\prime}(u_{1})-\phi^{\prime\prime}(u_{2})||_{**}\leq
M||u_{1}-u_{2}||^{\alpha}.$ (2.1)

######  Definition 2.2.

Let $\phi$ be a $C^{2}$-functional on $E$. For any $u\in E$, we define the
continuous bilinear map:

$D^{2}\phi(u)=\phi^{\prime\prime}(u)-\frac{\phi^{\prime}(u)\cdot
u}{|u|^{2}}(\cdot,\cdot).$

######  Definition 2.3.

Let $\phi$ be a $C^{2}$-functional on $E$. For any $u\in S_{\mu}$ and
$\theta>0$, we define the _approximate Morse index_ by

$\tilde{m}_{\theta}(u)=\sup\left\\{\dim\,L\left|\begin{array}[]{l}\ L\text{ is
a subspace of $T_{u}S_{\mu}$ such that:
}D^{2}\phi(u)[\varphi,\varphi]<-\theta\|\varphi\|^{2},\quad\forall\varphi\in
L\setminus\\{0\\}\end{array}\right.\right\\}.$

If $u$ is a critical point for the constrained functional $\phi|_{S_{\mu}}$
and $\theta=0$, we say that this is the _Morse index of $u$ as constrained
critical point_.

The following abstract theorem was established in [12]. Its derivation is
based on a combination of ideas from [22, 23, 27] implemented in a convenient
geometric setting. Note that related theorems, but dealing with unconstrained
functionals, were recently developed in [10, 30].

######  Theorem 2.4 (Theorem 1 in [12]).

Let $I\subset(0,+\infty)$ be an interval and consider a family of $C^{2}$
functionals $\Phi_{\rho}:E\to\mathbb{R}$ of the form

$\Phi_{\rho}(u)=A(u)-\rho B(u),\qquad\rho\in I,$

where $B(u)\geq 0$ for every $u\in E$, and

either $A(u)\to+\infty$ or $B(u)\to+\infty$ as $u\in E$ and $\|u\|\to+\infty$.
(2.2)

Suppose moreover that $\Phi_{\rho}^{\prime}$ and $\Phi_{\rho}^{\prime\prime}$
are $\alpha$-Hölder continuous on bounded sets for some $\alpha\in(0,1]$.
Finally, suppose that there exist $w_{1},w_{2}\in S_{\mu}$ (independent of
$\rho$) such that, setting

$\Gamma=\left\\{\gamma\in C([0,1],S_{\mu})\big{|}\
\gamma(0)=w_{1},\quad\gamma(1)=w_{2}\right\\},$

we have

$c_{\rho}:=\inf_{\gamma\in\Gamma}\
\max_{t\in[0,1]}\Phi_{\rho}(\gamma(t))>\max\\{\Phi_{\rho}(w_{1}),\Phi_{\rho}(w_{2})\\},\quad\rho\in
I.$ (2.3)

Then, for almost every $\rho\in I$, there exist sequences $\\{u_{n}\\}\subset
S_{\mu}$ and $\zeta_{n}\to 0^{+}$ such that, as $n\to+\infty$,

  * (i)

$\Phi_{\rho}(u_{n})\to c_{\rho}$;

  * (ii)

$||\Phi^{\prime}_{\rho}|_{S_{\mu}}(u_{n})||_{*}\to 0$;

  * (iii)

$\\{u_{n}\\}$ is bounded in $E$;

  * (iv)

$\tilde{m}_{\zeta_{n}}(u_{n})\leq 1$.

It is easily observed, see [12, Remarks 1.3], from Theorem 2.4 (ii)-(iii) that

$\Phi^{\prime}_{\rho}(u_{n})+\displaystyle\lambda_{n}(u_{n},\cdot)\to
0\,\mbox{ in }E^{\prime}\mbox{ as }n\to+\infty$ (2.4)

where we have set

$\lambda_{n}:=-\displaystyle\frac{1}{\mu}(\Phi^{\prime}_{\rho}(u_{n})\cdot
u_{n}).$ (2.5)

Also, Theorem 2.4 (iv) directly implies that if there exists a subspace
$W_{n}\subset T_{u_{n}}S_{\mu}$ such that

$D^{2}\Phi_{\rho}(u_{n})[w,w]=\Phi^{\prime\prime}_{\rho}(u_{n})[w,w]+\displaystyle\lambda_{n}(w,w)<-\zeta_{n}||w||^{2},\quad\mbox{for
all }w\in W_{n}\setminus\\{0\\},$ (2.6)

then necessarily $\dim W_{n}\leq 1$. We called the sequence
$\\{\lambda_{n}\\}\subset\mathbb{R}$ defined in (2.5) the sequence of _almost
Lagrange multipliers_.

We shall see in Section 3 that if $\\{u_{n}\\}\subset S_{\mu}$ converges to
some $u\in S_{\mu}$ then information on the Morse index of $u$ as constrained
critical point can be obtained. First, let us show how information on the
sequence $\\{\lambda_{n}\\}\subset\mathbb{R}$ can be derived.

######  Lemma 2.5.

Let $\\{u_{n}\\}\subset S_{\mu}$, $\\{\lambda_{n}\\}\subset\mathbb{R}$ and
$\\{\zeta_{n}\\}\subset\mathbb{R}^{+}$ with $\zeta_{n}\to 0^{+}$. Assume that
the following conditions hold:

  * (i)

For large enough $n\in\mathbb{N}$, all subspaces $W_{n}\subset E$ with the
property

$\Phi^{\prime\prime}_{\rho}(u_{n})[\varphi,\varphi]+\lambda_{n}|\varphi|^{2}<-\zeta_{n}\|\varphi\|^{2},\qquad\mbox{for
all }\,\varphi\in W_{n}\backslash\\{0\\},$ (2.7)

satisfy: $\dim W_{n}\leq 2$.

  * (ii)

There exist $\lambda\in\mathbb{R}$, a subspace $Y$ of $E$ with $\dim Y\geq 3$
and $a>0$ such that, for large enough $n\in\mathbb{N}$,

$\Phi^{\prime\prime}_{\rho}(u_{n})[\varphi,\varphi]+\lambda|\varphi|^{2}\leq-a||\varphi||^{2},\qquad\mbox{for
all }\,\varphi\in Y.$ (2.8)

Then $\lambda_{n}>\lambda$ for all large enough $n\in\mathbb{N}$. In
particular, if (2.8) holds for any $\lambda<0$, then
$\liminf_{n\to\infty}\lambda_{n}\geq 0$.

###### Proof.

Suppose by contradiction that $\lambda_{n}\leq\lambda$ along a subsequence.
From (2.8) we have, for all $\varphi\in Y\backslash\\{0\\},$

$\begin{split}\Phi^{\prime\prime}_{\rho}(u_{n})[\varphi,\varphi]+\lambda_{n}|\varphi|^{2}&=\Phi^{\prime\prime}_{\rho}(u_{n})[\varphi,\varphi]+\lambda|\varphi|^{2}+(\lambda_{n}-\lambda)|\varphi|^{2}\\\
&\leq-a||\varphi||^{2}+(\lambda_{n}-\lambda)|\varphi|^{2}.\end{split}$

Now, since $\zeta_{n}\to 0^{+}$, there exists $n_{0}\in\mathbb{N}$ such that:
$\forall n\geq n_{0}$, $\zeta_{n}<a$. Thus, for an arbitrary $n\geq n_{0}$, we
obtain

$\Phi^{\prime\prime}_{\rho}(u_{n})[\varphi,\varphi]+\lambda_{n}|\varphi|^{2}\leq-\zeta_{n}\|\varphi\|^{2},\qquad\mbox{for
all }\,\varphi\in Y\backslash\\{0\\},$

in contradiction with (2.7) since dim $Y>2$. ∎

##  3 Mountain pass solutions for approximating problems

Theorem 2.4 will be a key ingredient in the proof of Theorem 1.1. To show that
the family $E_{\rho}(\cdot\,\mathcal{G})$ enters into the framework of Theorem
2.4 we first need to show that it has a mountain pass geometry on
$H^{1}_{\mu}(\mathcal{G})$ uniformly with respect to
$\rho\in\left[\frac{1}{2},1\right]$.

######  Lemma 3.1.

For every $\mu>0$, there exist $w_{1},w_{2}\in S_{\mu}$ independent of
$\rho\in\left[\frac{1}{2},1\right]$ such that

$\displaystyle
c_{\rho}:=\inf\limits_{\gamma\in\Gamma}\,\max\limits_{t\in[0,1]}E_{\rho}(\gamma(t),\mathcal{G})>\max\\{E_{\rho}(w_{1},\mathcal{G}),E_{\rho}(w_{2},\mathcal{G})\\},\qquad\forall\rho\in\left[\frac{1}{2},1\right],$

where $\Gamma:=\left\\{\gamma\in
C([0,1],H^{1}_{\mu}(\mathcal{G}))\big{|}~{}\gamma~{}\mbox{is
continuous},~{}\gamma(0)=w_{1},\gamma(1)=w_{2}\right\\}.$

###### Proof.

For any $\mu,k>0$, denote

$\displaystyle A_{\mu,k}:=\\{u\in
H_{\mu}^{1}(\mathcal{G})\big{|}\int_{\mathcal{G}}|u^{\prime}|^{2}dx<k\\},$
$\displaystyle\partial A_{\mu,k}:=\\{u\in
H_{\mu}^{1}(\mathcal{G})\big{|}\int_{\mathcal{G}}|u^{\prime}|^{2}dx=k\\}.$

First note that, since the graph is noncompact, for any $\mu,k>0$, we have
$A_{\mu,k}\neq\emptyset$ (and, similarly, $A_{\mu,k}\neq\emptyset$). Indeed,
take any function $v\in C^{\infty}_{c}(\mathbb{R})$ with
$\|v\|_{L^{2}(\mathbb{R})}^{2}=\mu$, and consider
$v_{t}(x)=t^{\frac{1}{2}}v(tx)$, for $t>0$. We have that
$\|v_{t}\|_{L^{2}(\mathbb{R})}^{2}=\mu$ and
$\|v_{t}^{\prime}\|_{L^{2}(\mathbb{R})}^{2}=t^{2}\|v^{\prime}\|_{L^{2}(\mathbb{R})}^{2}$
for every $t>0$. Since $\mathcal{G}$ contains at least a half-line, it
contains arbitrarily large intervals. Therefore, any function $v_{t}$ can be
regarded as a function in $H^{1}_{\mu}(\mathcal{G})$, still denoted by
$v_{t}$, with the support entirely located on the half-line $l_{1}$; and, in
particular, $v_{t}\in A_{\mu,k}$ for $t$ small enough.

Now, by the Gagliardo-Nirenberg inequality on metric graphs (see Proposition
2.1 in [4]), we obtain

$\displaystyle E_{\rho}(u,\mathcal{G})$
$\displaystyle\geq\frac{1}{2}\|u^{\prime}\|_{L^{2}(\mathcal{G})}^{2}-\rho\frac{C_{p}}{p}\mu^{\frac{p}{4}+\frac{1}{2}}\|u^{\prime}\|_{L^{2}(\mathcal{G})}^{\frac{p}{2}-1},\qquad\forall
u\in H^{1}_{\mu}(\mathcal{G}).$ (3.1)

Then, for any $\mu>0$ and $u\in\partial A_{\mu,k_{0}}$ with
$k_{0}=\frac{1}{2}(\frac{p}{2C_{p}})^{\frac{4}{p-6}}\mu^{-\frac{p+2}{p-6}}$,
we have

$\displaystyle\inf\limits_{u\in\partial
A_{\mu,k_{0}}}E_{\rho}(u,\mathcal{G})\geq
k_{0}\left(\frac{1}{2}-\frac{C_{p}}{p}\mu^{\frac{p+2}{4}}k_{0}^{\frac{p-6}{4}}\right)=:\alpha>0,$
(3.2)

for every $\rho\in[1/2,1]$.

Next, observe that for any $u\in A_{\mu,k}$

$\displaystyle
E_{\rho}(u,\mathcal{G})\leq\frac{1}{2}\int_{\mathcal{G}}|u^{\prime}|^{2}\,dx\leq\frac{1}{2}k.$

Thus, recalling that $A_{k,\mu}\neq\emptyset$ for all $\mu,k>0$, it is
possible to choose a $w_{1}\in H^{1}_{\mu}(\mathcal{G})$ such that

$\|w_{1}^{\prime}\|_{L^{2}(\mathcal{G})}^{2}<k_{0}\quad\text{and}\quad
E_{\rho}(w_{1},\mathcal{G})\leq\frac{\alpha}{2}\quad\forall\rho\in\left[\frac{1}{2},1\right].$
(3.3)

Moreover, we also observe that we can identify any bounded edge, say
${\mathrm{e}_{1}}$, with the interval $[-\ell_{1}/2,\ell_{1}/2]$. It follows
that any compactly supported $H^{1}$ function $w$ on such interval, with mass
$\mu$, can be seen as a function in $H^{1}_{\mu}(\mathcal{G})$. Defining
$w_{t}(x):=t^{1/2}w(tx)$, with $t>1$, it is not difficult to check that
$w_{t}\in H^{1}_{\mu}(\mathcal{G})$ (notice in particular that the support of
$w_{t}$ is shrinking on ${\rm{e}_{1}}$ as $t$ becomes larger), and that

$\displaystyle E_{\rho}(w_{t},\mathcal{G})$
$\displaystyle=\frac{t^{2}}{2}\int_{{\mathrm{e}_{1}}}|w^{\prime}|^{2}\,dx-\frac{\rho
t^{\frac{p-2}{2}}}{p}\int_{{\mathrm{e}_{1}}}|w|^{p}\,dx\leq\frac{t^{2}}{2}\left(\int_{{\mathrm{e}_{1}}}|w^{\prime}|^{2}\,dx-\frac{t^{\frac{p-6}{2}}}{p}\int_{{\mathrm{e}_{1}}}|w|^{p}\,dx\right),$

for every $\rho\in[\frac{1}{2},1]$. Since $p>6$, then the right-hand side
tends to $-\infty$ as $t\to+\infty$, and in particular there exists $t_{2}>0$
large enough such that
$\|w^{\prime}_{t}\|^{2}_{L^{2}(\mathcal{G})}=t^{2}\|w^{\prime}\|_{L^{2}(\mathcal{G})}^{2}>2k_{0}$
and $E_{\rho}(w_{t},\mathcal{G})<0$ for all $t>t_{2}$ and
$\rho\in[\frac{1}{2},1]$. We set $w_{2}:=w_{2t_{2}}$, and we point out that

$\|w_{2}^{\prime}\|_{L^{2}(\mathcal{G})}^{2}>2k_{0}\quad\text{and}\quad
E_{\rho}(w_{2},\mathcal{G})<0\quad\forall\rho\in\left[\frac{1}{2},1\right].$
(3.4)

At this point the thesis follows easily. Let $\Gamma$ and $c_{\rho}$ be
defined as in the statement of the lemma for our choice of $w_{1}$ and
$w_{2}$; the fact that $\Gamma\neq 0$ is straightforward, since

$\gamma_{0}(t):=\frac{\mu^{1/2}}{\|(1-t)w_{1}+tw_{2}\|_{L^{2}(\mathcal{G})}}\left[(1-t)w_{1}+tw_{2}\right]\qquad
t\in[0,1]$

belongs to $\Gamma$. By (3.3) and (3.4), we note that for every
$\gamma\in\Gamma$ there exists $t_{\gamma}\in[0,1]$ such that
$\gamma(t_{\gamma})\in\partial A_{\mu,k_{0}}$, by continuity. Therefore, for
any $\rho\in[1/2,1]$, we have for every $\gamma\in\Gamma$

$\max_{t\in[0,1]}E_{\rho}(\gamma(t),\mathcal{G})\geq
E_{\rho}(\gamma(t_{\gamma}),\mathcal{G})\geq\inf_{u\in\partial
A_{\mu,k_{0}}}E_{\rho}(u,\mathcal{G})\geq\alpha$

(see (3.2)) and thus $c_{\rho}\geq\alpha$ while

$\max\\{E_{\rho}(w_{1},\mathcal{G}),E_{\rho}(w_{2},\mathcal{G})\\}=E_{\rho}(w_{1},\mathcal{G})<\frac{\alpha}{2}.\qed$

The following lemma, together with Lemma 2.5, will be central to establish the
convergence of the Palais-Smale sequences provided by the application of
Theorem 2.4.

######  Lemma 3.2.

For any $\lambda<0$, there exists a subspace $Y$ of $H^{1}(\mathcal{G})$ with
dim $Y=3$ such that

$\int_{\mathcal{G}}|w^{\prime}|^{2}\,dx+\lambda\int_{\mathcal{G}}|w|^{2}\,dx\leq\frac{\lambda}{2}\|w\|_{H^{1}(\mathcal{G})}^{2},\qquad\forall\,w\in
Y.$ (3.5)

###### Proof.

Take $\phi\in C_{0}^{\infty}(\mathbb{R}^{+})$ with
${\rm{supp}}\,\phi\subset[\frac{3}{4},\frac{5}{4}]$ such that
$\int_{0}^{+\infty}|\phi|^{2}dx=1$. Recalling that $l_{1}$ is an arbitrary
half-line of $\mathcal{G}$, we define the following functions for $\sigma>0$:

$\displaystyle\psi_{0,\sigma}(x)=\left\\{\begin{array}[]{ll}0{}{}&\mbox{for}~{}x\in\mathcal{\mathcal{G}}\setminus
l_{1}\\\ \sigma^{\frac{1}{2}}\phi(\sigma x){}{}&~{}\mbox{for}~{}x\in
l_{1},\end{array}\right.$
$\displaystyle\psi_{1,\sigma}(x)=\left\\{\begin{array}[]{ll}0{}{}&\mbox{for}~{}x\in\mathcal{\mathcal{G}}\setminus
l_{1}\\\ \sigma^{\frac{1}{2}}\phi(\sigma x-1){}{}&~{}\mbox{for}~{}x\in
l_{1},\end{array}\right.$

and

$\displaystyle\psi_{2,\sigma}(x)=\left\\{\begin{array}[]{ll}0{}{}&\mbox{for}~{}x\in\mathcal{\mathcal{G}}\setminus
l_{1}\\\ \sigma^{\frac{1}{2}}\phi(\sigma x-2){}{}&~{}\mbox{for}~{}x\in
l_{1}.\end{array}\right.$

Clearly, for any $\sigma>0$, $\int_{\mathcal{G}}|\psi_{j,\sigma}|^{2}\,dx=1$
for all $j=0,1,2$, and the three functions are mutually orthogonal in
$H^{1}(\mathcal{G})$, having disjoints supports. Assume that
$w=\sum\limits_{j=0}^{2}\theta_{j}\psi_{j,\sigma}$. Then

$\begin{split}\int_{\mathcal{G}}|w^{\prime}|^{2}\,dx+\lambda\int_{\mathcal{G}}|w|^{2}\,dx&=\sigma^{2}\Big{(}\sum_{j=0}^{2}\theta_{j}^{2}\int_{0}^{+\infty}|\phi^{\prime}|^{2}\,dx\Big{)}+\lambda\Big{(}\sum_{j=0}^{2}\theta_{j}^{2}\int_{0}^{+\infty}|\phi|^{2}\,dx\Big{)}\\\
&=(\sigma^{2}\alpha+\lambda)\sum_{j=0}^{2}\theta_{j}^{2},\end{split}$

where $\alpha:=\int_{0}^{+\infty}|\phi^{\prime}|^{2}\,dx>0$. Similarly,
$\|w\|_{H^{1}(\mathcal{G})}^{2}=(\sigma^{2}\alpha+1)\sum_{j=0}^{2}\theta_{j}^{2}$.
Therefore, provided that $w\neq 0$,

$\frac{\int_{\mathcal{G}}|w^{\prime}|^{2}\,dx+\lambda\int_{\mathcal{G}}|w|^{2}\,dx}{\|w\|_{H^{1}(\mathcal{G})}^{2}}=\frac{\sigma^{2}\alpha+\lambda}{\sigma^{2}\alpha+1}\leq\frac{\lambda}{2}$

for every $\sigma>0$ sufficiently small, and for every
$\theta_{0},\theta_{1},\theta_{2}\in\mathbb{R}$. ∎

######  Remark 3.3.

The validity of the previous lemmata relies on the existence of at least one
half-line, thus on the non-compact character of the graph. Results like Lemma
3.2 are expected to hold broadly in problems set on a non-compact domain.

Finally, we recall a definition. In what follows, $\chi_{A}$ denotes the
characteristic function of $A$.

######  Definition 3.4.

For any graph $\mathcal{F}$, any subgraph
$\mathcal{F^{\prime}}\subset\mathcal{F}$, and any solution $U\in
C(\mathcal{F})\cap H^{1}_{{\rm loc}}(\mathcal{F})$, not necessarily in
$H^{1}(\mathcal{F})$, of

$\begin{cases}-U^{\prime\prime}+\lambda
U=\rho|U|^{p-2}U\chi_{\mathcal{F}^{\prime}}&\text{in $\mathcal{F}$},\\\
\sum_{{\rm e}\succ{\rm v}}U^{\prime}({\rm v})=0&\text{for any vertex ${\rm v}$
of $\mathcal{F}$},\end{cases}$ (3.9)

with $\lambda,\rho\in\mathbb{R}$, we consider

$Q(\varphi;U,\mathcal{F}):=\int_{\mathcal{F}}\left(|\varphi^{\prime}|^{2}+(\lambda-(p-1)\rho|U|^{p-2}\chi_{\mathcal{F^{\prime}}})\varphi^{2}\right)\,dx,\quad\forall\varphi\in
H^{1}(\mathcal{F})\cap C_{c}(\mathcal{F}).$ (3.10)

The _Morse index of $U$_, denoted by $m(U)$, is the maximal dimension of a
subspace $W\subset H^{1}(\mathcal{F})\cap C_{c}(\mathcal{F})$ such that
$Q(\varphi;U,\mathcal{F})<0$ for all $\varphi\in W\setminus\\{0\\}$.

The main aim of this section is to prove the following result.

######  Proposition 3.5.

For any fixed $\mu>0$ and almost every $\rho\in I_{\rho}$, there exists
$(u_{\rho},\lambda_{\rho})\in H^{1}_{\mu}(\mathcal{G})\times\mathbb{R}^{+}$
which solves

$\begin{cases}-u_{\rho}^{\prime\prime}+\lambda_{\rho}u_{\rho}=\rho\kappa(x)u_{\rho}^{p-1},\quad
u_{\rho}>0&\text{in $\mathcal{\mathcal{G}}$},\\\ \sum_{{\rm e}\succ{\rm
v}}u_{\rho}^{\prime}({\rm v})=0&\text{for any vertex ${\rm v}$}.\end{cases}$
(3.11)

Moreover, $E_{\rho}(u_{\rho},\mathcal{G})=c_{\rho}$ and $m(u_{\rho})\leq 2$.

###### Proof.

For simplicity, we omit the dependence of the functionals
$E_{\rho}(\cdot\,\mathcal{G})$ on $\mathcal{G}$. We apply Theorem 2.4 to the
family of functionals $E_{\rho}$, with $E=H^{1}(\mathcal{G})$,
$H=L^{2}(\mathcal{G})$, $S_{\mu}=H^{1}_{\mu}(\mathcal{G})$, and $\Gamma$
defined in Lemma 3.1. Setting

$A(u)=\frac{1}{2}\int_{\mathcal{G}}|u^{\prime}|^{2}\,dx\quad\text{and}\quad
B(u)=\frac{\rho}{p}\int_{\mathcal{G}}|u|^{p}\,dx,$

assumption (2.2) holds, since we have that

$u\in H^{1}_{\mu}(\mathcal{G}),\ \|u\|\to+\infty\quad\implies\quad
A(u)\to+\infty.$

Let $E^{\prime}_{\rho}$ and $E^{\prime\prime}_{\rho}$ denote respectively the
free first and second derivatives of $E_{\rho}$. Clearly, $E^{\prime}_{\rho}$
and $E^{\prime\prime}_{\rho}$ are both of class $C^{1}$, and hence locally
Hölder continuous, on $H_{\mu}^{1}(\mathcal{G})$, which implies that
assumption (2.1) holds.

Thus, taking into account Lemma 3.1, by Theorem 2.4 and the considerations
just after it, for almost every $\rho\in[1/2,1]$, there exist a bounded
sequence $\\{u_{n,\rho}\\}\subset H_{\mu}^{1}(\mathcal{G})$, that we shall
denote simply by $\\{u_{n}\\}$ from now on, and a sequence
$\\{\zeta_{n}\\}\subset\mathbb{R}^{+}$ with $\zeta_{n}\to 0^{+}$, such that

$E^{\prime}_{\rho}(u_{n})+\lambda_{n}u_{n}\to 0\quad\mbox{in the dual of}\quad
H^{1}_{\mu}(\mathcal{G}),$ (3.12)

where

$\lambda_{n}:=-\frac{1}{\mu}E^{\prime}_{\rho}(u_{n})u_{n}.$ (3.13)

Moreover, if the inequality

$\displaystyle\int_{\mathcal{G}}\left[|\varphi^{\prime}|^{2}+\left(\lambda_{n}-(p-1)\rho\kappa(x)|u_{n}|^{p-2}\right)\varphi^{2}\right]\,dx=E^{\prime\prime}_{\rho}(u_{n})[\varphi,\varphi]+\lambda_{n}||\varphi||_{L^{2}(\mathcal{G})}^{2}<-\zeta_{n}\|\varphi\|_{H^{1}(\mathcal{G})}^{2}$
(3.14)

holds for any $\varphi\in W_{n}\setminus\\{0\\}$ in a subspace $W_{n}$ of
$T_{u_{n}}S_{\mu}$, then the dimension of $W_{n}$ is at most $1$. In addition,
as explained in [12, Remark 1.4], since $u\in H^{1}_{\mu}(\mathcal{G})\
\Longrightarrow|u|\in H^{1}_{\mu}(\mathcal{G})$, $w_{1},w_{2}\geq 0$, the map
$u\mapsto|u|$ is continuous, and $E_{\rho}(u)=E_{\rho}(|u|)$, it is possible
to choose $\\{u_{n}\\}$ with the property that $u_{n}\geq 0$ on $\mathcal{G}$.

The sequence $\\{u_{n}\\}$ being bounded, it follows by (3.13) that
$\\{\lambda_{n}\\}$ is bounded. Then, passing to a subsequence, there exists
$\lambda_{\rho}\in\mathbb{R}$ such that
$\lim\limits_{n\to+\infty}\lambda_{n}=\lambda_{\rho}$. Furthermore, there
exists $u_{\rho}\in H^{1}(\mathcal{G})$ such that

$\displaystyle u_{n}\rightharpoonup
u_{\rho}~{}~{}\mbox{in}~{}~{}H^{1}(\mathcal{G}),$ (3.15) $\displaystyle
u_{n}\to u_{\rho}~{}~{}\mbox{in}~{}~{}L_{{\rm{loc}}}^{r}(\mathcal{G}),r>2,$
(3.16) $\displaystyle u_{n}(x)\to u_{\rho}(x)~{}~{}\mbox{for
a.e.}~{}x\in\mathcal{G},$ (3.17)

which implies that $u_{\rho}\geq 0$. Also, recalling (3.12) and that
$\lim\limits_{n\to+\infty}\lambda_{n}=\lambda_{\rho}$, we see that $u_{\rho}$
satisfies

$-u_{\rho}^{\prime\prime}+\lambda_{\rho}u_{\rho}=\rho\kappa(x)u_{\rho}^{p-1}$
(3.18)

with the Kirchhoff condition at the vertices. In order to show that
$u_{\rho}\not\equiv 0$ we shall first prove that $\lambda_{\rho}\geq 0$. Here
the Morse type information (3.14) proves decisive. Firstly, since the
codimension of $T_{u_{n}}S_{\mu}$ is $1$, we infer that if the inequality
(3.14) holds for every $\varphi\in V_{n}\setminus\\{0\\}$ for a subspace
$V_{n}$ of $H^{1}(\mathcal{G})$, then the dimension of $V_{n}$ is at most $2$.
Secondly, by Lemma 3.2, we have that for every $\lambda<0$, there exist a
subspace $Y$ of $E$ with $\dim Y\geq 3$ and $a>0$ such that, for
$n\in\mathbb{N}$ large,

$\begin{split}E^{\prime\prime}_{\rho}(u_{n})[\varphi,\varphi]+\lambda|\varphi|^{2}&\leq\int_{\mathcal{G}}|\varphi^{\prime}|^{2}\,dx+\lambda\int_{\mathcal{G}}|\varphi|^{2}\leq-a\|\varphi\|^{2},\qquad\mbox{for
all }\,\varphi\in Y\backslash\\{0\\}.\end{split}$

Therefore, Lemma 2.5 implies that $\lambda_{\rho}\geq 0$.

Now from (3.12) and the fact that $\lambda_{n}\to\lambda_{\rho}$, we deduce
that

$\int_{\mathcal{G}}\left(u_{n}^{\prime}\varphi^{\prime}+\lambda_{\rho}u_{n}\varphi\right)\,dx-\rho\int_{\mathcal{K}}u_{n}^{p-1}\varphi\,dx=o(1)\|\varphi\|_{H^{1}(\mathcal{G})}.$

Moreover, by (3.18),

$\int_{\mathcal{G}}\left(u_{\rho}^{\prime}\varphi^{\prime}+\lambda_{\rho}u_{\rho}\varphi\right)\,dx-\rho\int_{\mathcal{K}}u_{\rho}^{p-1}\varphi\,dx=0$

Therefore, taking the difference and exploiting (3.15)-(3.17) we infer that

$\int_{\mathcal{G}}|(u_{n}-u_{\rho})^{\prime}|^{2}\,dx+\lambda_{\rho}\int_{\mathcal{G}}|u_{n}-u_{\rho}|^{2}\,dx\to
0$ (3.19)

as $n\to\infty$. If we assume that $u_{\rho}\equiv 0$ then we deduce that

$\int_{\mathcal{G}}|u_{n}^{\prime}|^{2}\,dx+\lambda_{\rho}\int_{\mathcal{G}}|u_{n}|^{2}\,dx\to
0.$ (3.20)

If $\lambda_{\rho}>0$, this is not possible since
$\|u_{n}\|_{L^{2}(\mathcal{G})}^{2}=\mu>0$. If $\lambda_{\rho}=0$, then (3.20)
contradicts the fact that the mountain-pass level satisfies $c_{\rho}>0$.
Hence, $u_{\rho}\not\equiv 0$. Appealing to the Kirchhoff condition and the
uniqueness theorem for ODEs, we have in fact that $u_{\rho}>0$ in
$\mathcal{G}$: indeed, assume by contradiction that there exists
$x_{0}\in\mathcal{G}$ such that $u_{\rho}(x_{0})=0$. If $x_{0}$ stays in the
interior of some edge, then by $u_{\rho}\geq 0$ on $\mathcal{G}$ it follows
that $u^{\prime}_{\rho}(x_{0})=0$. If instead $x_{0}$ is a vertex, then by
$u_{\rho}(x)\geq 0$ and by the Kirchhoff condition we also get
$u^{\prime}_{\rho}(x_{0})=0$. Hence, by uniqueness, $u_{\rho}\equiv 0$ on any
edge containing $x_{0}$; but then, by repeating this argument a finite number
of times (since $\mathcal{G}$ has finitely many vertices and edges), we deduce
that $u_{\rho}\equiv 0$ on $\mathcal{G}$, which is the desired contradiction.

Now we claim that $\lambda_{\rho}>0$. If not, then $\lambda_{\rho}=0$.
Identifying $l_{1}$ with $[0,+\infty)$, we have that $u_{\rho}$ is a $C^{2}$
function on $[0,+\infty)$ such that $u_{\rho}^{\prime\prime}(x)=0$ for all
$x\in(0,+\infty)$. It follows that $u_{\rho}>0$ on $[0,+\infty)$,

$u_{\rho}(x)=u_{\rho}(0)+u_{\rho}^{\prime}(0)x,\quad\text{and }\quad
u_{\rho}\in L^{2}(\mathcal{G}),$

which are incompatible. Hence the claim holds.

Having proved that $\lambda_{\rho}>0$, we get from (3.19) that $u_{n}\to
u_{\rho}$ strongly in $H^{1}(\mathcal{G})$.

It remains to show that the Morse index $m(u_{\rho})$ is at most $2$. Since
the tangent space $T_{u}H^{1}_{\mu}(\mathcal{G})$ has codimension 1, it
suffices to show that $u_{\rho}\in S_{\mu}$ has Morse index at most $1$ as a
constrained critical point. If not, in view of Definition 2.3 we may assume by
contradiction that there exists a $W_{0}\subset T_{u}S_{\mu}$ with $\dim
W_{0}=2$ such that

$D^{2}E_{\rho}(u_{\rho})[w,w]<0\quad\mbox{for all }w\in
W_{0}\backslash\\{0\\}.$ (3.21)

Then, since $W_{0}$ is of finite dimension, there exists $\beta>0$ such that

$D^{2}E_{\rho}(u_{\rho})[w,w]<-\beta\quad\mbox{for all }w\in
W_{0}\backslash\\{0\\}~{}~{}\mbox{with}~{}~{}\|w\|_{H^{1}(\mathcal{G})}=1,$

using the homogeneity of $D^{2}E_{\rho}(u_{\rho})$, we deduce that

$D^{2}E_{\rho}(u_{\rho})[w,w]<-\beta||w||_{H^{1}(\mathcal{G})}^{2}\quad\mbox{for
all }w\in W_{0}\backslash\\{0\\}.$

Now, from [12, Corollary 1] or using directly that $E_{\rho}^{\prime}$ and
$E_{\rho}^{\prime\prime}$ are $\alpha$-Hölder continuous on bounded sets for
some $\alpha\in(0,1]$, it follows that there exists $\delta_{1}>0$ small
enough such that, for any $v\in S_{\mu}$ satisfying $||v-u||\leq\delta_{1}$,

$D^{2}E_{\rho}(v)[w,w]<-\frac{\beta}{2}||w||_{H^{1}(\mathcal{G})}^{2}\quad\mbox{for
all }w\in W_{0}\backslash\\{0\\}.$ (3.22)

Hence, noting that $||u_{n}-u_{\rho}||_{H^{1}(\mathcal{G})}\leq\delta_{1}$ for
$n\in\mathbb{N}$ large enough, by (3.21)-(3.22) and $\zeta_{n}\to 0^{+}$ we
get

$D^{2}E_{\rho}(u_{n})[w,w]<-\frac{\beta}{2}||w||_{H^{1}(\mathcal{G})}^{2}<-\zeta_{n}||w||_{H^{1}(\mathcal{G})}^{2}\quad\mbox{for
all }w\in W_{0}\backslash\\{0\\}$ (3.23)

for any such large $n$. Therefore, since $\dim W_{0}>1$ and recalling that

$E^{\prime\prime}_{\rho}(u_{n})[w,w]+\lambda_{\rho}||w||_{L^{2}(\mathcal{G})}^{2}=D^{2}E_{\rho}(u_{n})[w,w],$

Equation (3.23) provides a contradiction with Equation (3.14). Thus we infer
that $\tilde{m}_{0}(u_{\rho})\leq 1$. ∎

##  4 Proof of Theorem 1.1

From Proposition 3.5 we know that there exists a sequence $\rho_{n}\to 1^{-}$,
and a corresponding sequence of critical points $u_{\rho_{n}}\in
H^{1}_{\mu}(\mathcal{G})$ of $E_{\rho_{n}}(\cdot\,,\mathcal{G})$ constrained
to $H^{1}_{\mu}(\mathcal{G})$, lying at the mountain-pass level
$c_{\rho_{n}}$, with a Morse index $m(u_{\rho_{n}})$ lesser or equal to $2$.
Additionally, the associated Lagrange multipliers satisfy $\lambda_{n}>0$.
To obtain the existence of solutions for the original problem, at a strictly
positive energy level, it clearly suffices to show that $\\{u_{\rho_{n}}\\}$
converges. In this direction the key point is to show that
$\\{u_{\rho_{n}}\\}$ is bounded in $H^{1}(\mathcal{G})$.

By Lemma 3.1 and the monotonicity of $c_{\rho}$,

$E_{1}(w_{1},\mathcal{G})\leq E_{\rho}(w_{1},\mathcal{G})\leq c_{\rho}\leq
c_{1/2},\qquad\forall\rho\in\left[\frac{1}{2},1\right],$

which implies that $c_{\rho_{n}}$ is bounded. In addition, thanks to the
Kirchhoff condition

$\int_{\mathcal{G}}\left(|u_{n}^{\prime}|^{2}+\lambda_{n}u_{n}^{2}\right)dx=\int_{\mathcal{K}}|u_{n}|^{p}\,dx,$

whence it follows that

$\left(\frac{1}{2}-\frac{1}{p}\right)\int_{\mathcal{G}}|u_{n}^{\prime}|^{2}\,dx=c_{n}+\frac{\lambda_{n}\mu}{p}.$

Therefore, if $\\{\lambda_{n}\\}$ is bounded, then $\\{u_{\rho_{n}}\\}$ is
bounded as well. We shall thus assume that $\lambda_{n}\to+\infty$, up to
extraction of a subsequence and, in order to reach a contradiction, develop a
blow-up analysis for $\\{u_{\rho_{n}}\\}$ as in [15] (see also [20, 21, 37]).

Precisely, writing for simplicity $u_{n}:=u_{\rho_{n}}$ for $\rho_{n}\to
1^{-}$, we consider the behavior of a sequence of solutions $\\{u_{n}\\}\in
H^{1}(\mathcal{G})$ of the following problem:

$\begin{cases}-u_{n}^{\prime\prime}+\lambda_{n}u_{n}=\rho_{n}\kappa(x)u_{n}^{p-1}&\text{on
$\mathcal{G}$},\\\ u_{n}>0&\text{on $\mathcal{G}$},\\\
\sum_{\rm{e}\succ\rm{v}}u_{e,n}^{\prime}(\rm{v})=0&\forall\rm{v}\in\mathcal{V},\end{cases}$
(4.1)

where $m(u_{n})\leq 2$ for all $n\in\mathbb{N}$, and it is assumed that
$\lambda_{n}\to+\infty$.

Firstly, we note that the local maximum points of $u_{n}$ stay on the compact
part $\mathcal{K}$ of $\mathcal{G}$.

######  Lemma 4.1.

Let $x_{n}\in\mathcal{G}$ be a local maximum point of $u_{n}$. Then
$x_{n}\in\mathcal{K}$, and

$u_{n}(x_{n})\geq\lambda_{n}^{\frac{1}{p-2}}.$

###### Proof.

Let $x_{n}$ be in the interior of some half line $l_{i}\in\mathcal{E}$. In
coordinates, $l_{i}\simeq[0,+\infty)$ and $x_{n}\in(0,+\infty)$, and the
restriction of $u_{n}$ on $l_{i}$ is a $C^{2}$ function on $[0,+\infty)$, by
regularity. Since $x_{n}$ is a local maximum point, we have
$u^{\prime\prime}_{n}(x_{n})\leq 0$, which implies that
$\lambda_{n}u_{n}(x_{n})\leq 0$. A contradiction. Thus $x_{n}\in\mathcal{K}$,
and at this point the estimate can be proved as in [15, Lemma 4.1]. ∎

In the following, we denote by $B_{r}(x_{0})=\\{x\in\mathcal{G}:\ {\rm
dist}(x,x_{0})<r\\}$. Moreover, we denote by $\mathcal{G}_{m}$ the star-graph
with $m\geq 1$ half-lines glued together at their common origin $0$. In
particular, $\mathcal{G}_{1}=\mathbb{R}^{+}$, and $\mathcal{G}_{2}$ is
isometric to $\mathbb{R}$.

Since $m(u_{n})\leq 2$, we can describe the asymptotic behavior of the
sequence $\\{u_{n}\\}$ near a local maximum point as $\lambda_{n}\to+\infty$.
From now on, we denote by $\chi_{A}$ the characteristic function of a set $A$.

######  Proposition 4.2.

Let $x_{n}\in\mathcal{G}$ be such that, for some $R_{n}\to\infty$,

$u_{n}(x_{n})=\max_{B_{R_{n}\tilde{\varepsilon}_{n}}(x_{n})}u_{n}\quad\text{where
}\tilde{\varepsilon}_{n}=(u_{n}(x_{n}))^{-\frac{p-2}{2}}\to 0.$

Suppose moreover that

$\limsup_{n\to\infty}\frac{{\rm{dist}}(x_{n},\mathcal{V})}{\tilde{\varepsilon}_{n}}=+\infty.$
(4.2)

Then, up to extraction of a subsequence, points ($i$)-($iv$) in [15, Theorem
4.2] holds. If, instead of (4.2), we suppose that

$\limsup_{n\to\infty}\frac{{\rm{dist}}(x_{n},\mathcal{V})}{\tilde{\varepsilon}_{n}}<+\infty;$
(4.3)

then, up to extraction of a subsequence :

  * ($i^{\prime}$)

$x_{n}\to{\rm{v}}\in\mathcal{V}$, and all the $x_{n}$ lie on the same edge
${\rm{e}}_{1}\simeq[0,\rho_{1}]$, where ${\rm{e}}_{1}\subset\mathcal{K}$ and
the vertex ${\rm{v}}$ is identified by the coordinate $0$ on ${\rm{e}}_{1}$.

  * ($ii^{\prime}$)

Let ${\rm{e}}_{2}\simeq[0,r_{2}]$, …, ${\rm{e}}_{j}\simeq[0,r_{j}]$ be the
other bounded edges of $\mathcal{G}$ having ${\rm{v}}$ as a vertex (if any),
and let $l_{j+1}\simeq[0,+\infty)$, …, $l_{m}\simeq[0,+\infty)$ be the half-
lines of $\mathcal{G}$ having $\mathrm{v}$ as a vertex (if any), where
${\rm{v}}$ is identified by the coordinate $0$ on each ${\rm{e}}_{i}$ and
$l_{i}$. Setting $\varepsilon_{n}=\lambda_{n}^{-\frac{1}{2}}$, we have that

$\begin{split}\frac{\tilde{\varepsilon}_{n}}{\varepsilon_{n}}&\to(0,1],\\\
\limsup_{n\to\infty}&\frac{{\rm{dist}}(x_{n},\mathcal{V})}{\varepsilon_{n}}<+\infty,\end{split}$
(4.4)

and the scaled sequence defined by

$v_{n}(y):=\varepsilon_{n}^{\frac{2}{p-2}}u_{n}(\varepsilon_{n}y)\quad\text{for
}y\in\frac{{\rm{e}}_{i}}{\varepsilon_{n}},\text{ with
$i=1,\dots,j$},\quad\text{or for }y\in\frac{l_{i}}{\varepsilon_{n}},\text{
with $i=j+1,\dots,m$}$

converges to a limit $V$ in $C^{0}_{\rm{loc}}(\mathcal{G}_{m})$ as
$n\to\infty$. Denoting by $V_{i}$ the restriction of $V$ to the $i$-th half-
line $\ell_{i}$ of $\mathcal{G}_{m}$, and by $v_{i,n}$ the restriction of
$v_{n}$ to ${\rm e}_{i}/\varepsilon_{n}$, we have moreover that $v_{i,n}\to
V_{i}$ in $C^{2}_{\rm{loc}}([0,+\infty))$. Finally, $V\in
H^{1}(\mathcal{G}_{m})$ is a positive solution to the NLS-type equation on the
star-graph

$\begin{cases}-V^{\prime\prime}+V=V^{p-1}\chi_{\ell_{1}\cup\dots\cup\ell_{j}},\quad
V>0&\text{in $\mathcal{G}_{m}$},\\\ \sum_{i=1}^{m}V_{i}^{\prime}(0^{+})=0,\\\
V(x)\to 0&\text{as }{\rm dist}(x,0)\to\infty\end{cases}$

with a global maximum point $\bar{x}$ located on $\ell_{1}$, whose coordinate
is

$\bar{x}=\lim_{n\to\infty}\bar{x}_{n}\in[0,+\infty),\quad\text{where}\quad\bar{x}_{n}:=\frac{{\rm{dist}}(x_{n},\mathcal{V})}{\varepsilon_{n}}.$

  * ($iii^{\prime}$)

There exists $\phi_{n}\in H^{1}(\mathcal{G}_{m})\cap C_{c}(\mathcal{G}_{m})$,
with ${\rm{supp}}\,\phi_{n}\subset B_{\bar{R}\varepsilon_{n}}(x_{n})$ for some
$\bar{R}>0$, such that

$Q(\phi_{n};u_{n},\mathcal{G})<0.$

  * ($iv^{\prime}$)

For all $R>0$ and $q\geq 1$, we have that

$\lim_{n\to\infty}\lambda_{n}^{\frac{1}{2}-\frac{q}{p-2}}\int_{B_{R\varepsilon_{n}}(x_{n})}u_{n}^{q}\,dx=\lim_{n\to\infty}\int_{B_{R}(\bar{x}_{n})}v_{n}^{q}\,dy=\int_{[0,\bar{x}+R]}V_{1}^{q}\,dy+\sum_{i=2}^{m}\int_{[0,R-\bar{x}]}V_{i}^{q}\,dy=\int_{B_{R}(\bar{x})}V^{q}\,dy$

(where $B_{R}(\bar{x}_{n})$ and $B_{R}(\bar{x})$ denote the balls in the
scaled and in the limit graphs, respectively).

The proof of the proposition can be obtained by making minor modifications to
the one of Theorem 4.2 in [15]. Roughly speaking, with respect to the setting
in [15], we have to take into account the fact that the points $x_{n}$ (which
belong to the compact core of $\mathcal{G}$, by Proposition 4.1) may
accumulate on a vertex ${\rm{v}}$ of $\mathcal{G}$ which belongs to some half-
line. We will need the following two statements, which correspond to Lemmata
4.3 and 4.5 in [15].

######  Lemma 4.3.

Let $U\in H^{1}_{{\rm loc}}(\mathcal{G}_{m})$ be a solution to

$\begin{cases}-U^{\prime\prime}+\lambda U=\rho
U^{p-1}\chi_{\ell_{1}\cup\dots\cup\ell_{j}}&\text{in $\mathcal{G}_{m}$},\\\
U>0&\text{in $\mathcal{G}_{m}$},\\\
\sum_{i=1}^{m}U_{i}^{\prime}(0)=0,\end{cases}$ (4.5)

for some $p>2$, $\rho,\lambda>0$, where $U_{i}$ denotes the restriction of $U$
on the $i$-th half-line $\ell_{i}$ of $\mathcal{G}_{m}$, and $1<j\leq m$.
Suppose that $U$ is bounded, and that $U$ is stable outside a compact set $K$,
in the sense that $Q(\varphi;U,\mathcal{G}_{m})\geq 0$ for all $\varphi\in
H^{1}(\mathcal{G}_{m})\cap C_{c}(\mathcal{G}_{m}\setminus K)$. Then $U(x)\to
0$ as ${\rm{dist}}(x,0)\to+\infty$, and $U\in H^{1}(\mathcal{G})$.

###### Proof.

On $\ell_{1},\dots,\ell_{j}$, one can argue exactly as in [15] (which in turn
follows [21, Theorem 2.3]). On $\ell_{j+1},\dots,\ell_{m}$, we note that $U$
satisfies $U^{\prime\prime}=\lambda U$ for some $\lambda>0$, and, since $U$ is
bounded, we deduce that the restriction of $U$ on each of these half-lines has
the form $U(x)=ce^{-\sqrt{\lambda}x}$, for some $c>0$. ∎

######  Lemma 4.4.

Let $U\in H^{1}(\mathcal{G}_{m})$ be any non-trivial solution of (4.5). Then
its Morse index $m(U)$ is strictly positive.

###### Proof.

Thanks to the Kirchhoff condition

$\int_{\mathcal{G}_{m}}\left(|U^{\prime}|^{2}+\lambda
U^{2}\right)dx=\sum_{i=1}^{j}\int_{\ell_{i}}\rho|U|^{p}\,dx.$

Therefore

$Q(U;U,\mathcal{G}_{m})=(2-p)\sum_{i=1}^{j}\int_{\ell_{i}}|U|^{p}\,dx<0$

(recall that $j>1$), and the thesis follows by density of
$H^{1}(\mathcal{G}_{m})\cap C_{c}(\mathcal{G}_{m})$ in
$H^{1}(\mathcal{G}_{m})$. ∎

###### Proof of Proposition 4.2.

We only emphasize the main differences with respect to [15]. Since
$\\{x_{n}\\}\subset\mathcal{K}$ by Lemma 4.1, and $\mathcal{G}$ has a finite
number of vertices and edges, up to extraction of a subsequence, the maximum
points $x_{n}$ belong to the same bounded edge ${\rm{e}_{1}}\simeq[0,r_{1}]$,
and converge. If (4.2) holds, then the proof proceeds exactly as the one of
[15, Theorem 4.2 under assumption (4.3)]. If instead (4.3) holds, then
$x_{n}\to{\rm{v}}\in\mathcal{V}$, since $\tilde{\varepsilon}_{n}\to 0$.
Contrary to [15], we have to take into account the possibility that ${\rm{v}}$
is on some half-line of $\mathcal{G}$; thus, the proof does not directly
follows from [15], but can be easily adjusted.
By the previous discussion, ($i^{\prime}$) holds, and we can suppose that

$\frac{d_{n}}{\tilde{\varepsilon}_{n}}\to\eta\in[0,+\infty),\quad d_{n}:={\rm
dist}(x_{n},\mathcal{V})=x_{n}.$

Let

$\tilde{u}_{i,n}(y):=\tilde{\varepsilon}_{n}^{\frac{2}{p-2}}u_{n}(\tilde{\varepsilon}_{n}y)\quad\text{for
}y\in\tilde{\rm e}_{i,n}:=\frac{{\rm e}_{i}}{\tilde{\varepsilon}_{n}},\
\text{with $i=1,\dots,j$},\text{ and for
}y\in\tilde{l}_{i,n}:=\frac{l_{i}}{\tilde{\varepsilon}_{n}},\ \text{with
$i=j+1,\dots,m$},$

and $\tilde{u}_{n}=(\tilde{u}_{1,n},\dots,\tilde{u}_{m,n})$. Note that
$\tilde{u}_{n}$ is defined on a graph $\mathcal{G}_{m,n}$ consisting in $j$
expanding edges and $m-j$ half-lines, glued together at their common origin,
which is identified with the coordinate $0$ on each edge or half-line. In the
limit $n\to\infty$, this graph converges to the star-graph $\mathcal{G}_{m}$.
Clearly, for every $a>\eta+1$ and large enough $n$

$\tilde{u}_{1,n}\left(\frac{d_{n}}{\tilde{\varepsilon}_{n}}\right)=1=\max_{B_{a}(0)}\tilde{u}_{n}$
(4.6)

(since $u_{n}(x_{n})=\max_{B_{R_{n}\tilde{\varepsilon}_{n}}(x_{n})}u_{n}$ for
some $R_{n}\to+\infty$),

$-\tilde{u}_{n}^{\prime\prime}+\tilde{\varepsilon}_{n}^{2}\lambda_{n}\tilde{u}_{n}=\rho_{n}\tilde{u}_{n}^{p-1}\chi_{\tilde{{\rm{e}}}_{1,n}\cup\dots\cup\tilde{{\rm{e}}}_{j,n}},\quad\tilde{u}_{n}>0$

on any edge of $\mathcal{G}_{m,n}$, and the Kirchhoff condition at the origin
holds. Also, by Lemma 4.1,

$\tilde{\varepsilon}_{n}^{2}\lambda_{n}\in(0,1]\quad\forall n.$

Thus, by elliptic estimates, for $i=1,\dots,j$ we have that
$\tilde{u}_{i,n}\to\tilde{u}_{i}$ in $C^{2}_{\rm loc}([0,+\infty))$ for every
$i=1,\dots,j$, and the limit $\tilde{u}_{i}$ solves

$-\tilde{u}^{\prime\prime}_{i}+\tilde{\lambda}\tilde{u}_{i}=\tilde{u}_{i}^{p-1},\quad\tilde{u}_{i}\geq
0\quad\text{in $(0,+\infty)$}$ (4.7)

for some $\tilde{\lambda}\in[0,1]$. Analogously, for $i=j+1,\dots,m$ we have
that $\tilde{u}_{i,n}\to\tilde{u}_{i}$ in $C^{2}_{\rm loc}([0,+\infty))$, and
the limit $\tilde{u}_{i}$ solves

$-\tilde{u}^{\prime\prime}_{i}+\tilde{\lambda}\tilde{u}_{i}=0,\quad\tilde{u}_{i}\geq
0\quad\text{in $(0,+\infty).$}$ (4.8)

Moreover, since $\tilde{u}_{n}$ is continuous on $\mathcal{G}_{m,n}$ and by
uniform convergence, $\tilde{u}_{i_{1}}(0)=\tilde{u}_{i_{2}}(0)$ for every
$i_{1}\neq i_{2}$, so that
$\tilde{u}\simeq(\tilde{u}_{1},\dots,\tilde{u}_{m})$ can be regarded as a
function defined on $\mathcal{G}_{m}$. Since the convergence
$\tilde{u}_{i,n}\to\tilde{u}_{i}$ takes place in $C^{2}$ up to the origin, the
Kirchhoff condition also passes to the limit. Now we exclude the case that
$\tilde{u}\equiv 0$ on some half-line of $\mathcal{G}_{m}$. By local uniform
convergence, we have that

$\tilde{u}_{1}(\eta)=\lim_{n\to\infty}\tilde{u}_{1,n}\left(\frac{d_{n}}{\tilde{\varepsilon}_{n}}\right)=1,$

and $\eta$ is a global maximum point of $\tilde{u}$, in view of (4.6). This
implies that $\tilde{u}_{1}>0$ in $(0,+\infty)$, by the uniqueness theorem for
ODEs. In turn, the Kirchhoff condition, and the uniqueness theorem for ODEs
again, ensure that $\tilde{u}_{i}>0$ on $(0,+\infty)$ for every $i$. Finally,
we claim that

$\text{the Morse index of $\tilde{u}$ is at most $2$}.$ (4.9)

The proof of this claim exploits the fact that $m(u_{n})\leq 2$ for every $n$,
is analogue to the one of [15, Eq. (4.10)], and hence is omitted.

The case $\tilde{\lambda}=0$ can be ruled out by phase-plane analysis (the
equation $\tilde{u}_{1}^{\prime\prime}+\tilde{u}_{1}^{p-1}=0$ on
$\ell_{1}=[0,+\infty)$ has only periodic sign-changing solutions, save the
trivial one). Therefore,

$0<\liminf_{n\to\infty}\frac{\lambda_{n}}{(u_{n}(x_{n}))^{p-2}}\leq\limsup_{n\to\infty}\frac{\lambda_{n}}{(u_{n}(x_{n}))^{p-2}}\leq
1,$ (4.10)

which proves the first estimate in (4.4). Furthermore, $\tilde{u}$ is bounded
and hence, by Lemma 4.3, $\tilde{u}\to 0$ as $|x|\to+\infty$, and
$\tilde{u}\in H^{1}(\mathcal{G}_{m})$.

At this point it is equivalent, but more convenient, to work with $v_{n}$
defined in point ($ii^{\prime}$) of the proposition, rather than with
$\tilde{u}_{n}$. By (4.3) and (4.10),

$\limsup_{n\to\infty}\frac{{\rm
dist}(x_{n},\mathcal{V})}{\varepsilon_{n}}<+\infty.$

Thus, similarly to what was done before, one can show that $v_{n}$ converges,
in $C^{0}_{\rm loc}(\mathcal{G}_{m})$ and in $C^{2}_{{\rm loc}}([0,+\infty))$
on every half-line, to a limit function $V\simeq(V_{1},\dots,V_{m})$, which
solves

$\begin{cases}-V^{\prime\prime}+V=V^{p-1}\chi_{\ell_{1}\cup\dots\cup\ell_{j}},\quad
V\geq 0&\text{in $\mathcal{G}_{m}$},\\\
\sum_{i=1}^{m}V_{i}^{\prime}(0^{+})=0;\end{cases}$ (4.11)

furthermore, $V$ has a positive global maximum on the half-line $\ell_{1}$,
$V_{1}(\bar{x})\geq 1$ (thus $V>0$ in $\mathcal{G}_{m}$), and has finite Morse
index $m(V)\leq 2$. Moreover, by Lemma 4.3, $V\to 0$ as $|x|\to\infty$. Thus,
($ii^{\prime}$) is proved. Point ($iv^{\prime}$) follows directly by local
uniform convergence. Finally, point ($iii^{\prime}$) is a consequence of Lemma
4.4. This implies that there exists $\phi\in H^{1}(\mathcal{G}_{m})\cap
C_{c}(\mathcal{G}_{m})$ such that $Q(\phi;V,\mathcal{G}_{m})<0$; thus,
defining

$\phi_{i,n}(x)=\varepsilon_{n}^{\frac{1}{2}}\phi_{i}\left(\frac{x-x_{n}}{\varepsilon_{n}}\right),$

it is not difficult to deduce that for sufficiently large $n$ we have
$Q(\phi_{i,n};u_{n},\mathcal{G})<0$, and ${\rm supp}\,\phi_{i,n}\subset
B_{\bar{R}\varepsilon_{n}}(x_{n})$ for some positive $\bar{R}$. ∎

In what follows, we focus on the global behavior of $\\{u_{n}\\}$ and show the
way that $\\{u_{n}\\}$ decays exponentially away from local maximum points. In
fact, by similar arguments to those used to establish Theorem 4.6 in [15] (see
also [17, 20, 21, 37]), we have the following result.

######  Proposition 4.5.

There exist $k\in\\{1,2\\}$, and sequences of points $\\{P_{n}^{i}\\}$,
$i\in\\{1,k\\}$, such that

$\displaystyle\lambda_{n}{\rm
dist}(P_{n}^{1},P_{n}^{2})\to+\infty\quad~{}(\mbox{if}~{}k=2),$ (4.12)
$\displaystyle
u_{n}(P_{n}^{i})=\max_{B_{R_{n}\lambda_{n}^{-1/2}}(P_{n}^{i})}u_{n}\quad\text{for
some $R_{n}\to+\infty$, for every $i$,}$ (4.13)

and constants $C_{1},C_{2}>0$ such that

$u_{n}(x)\leq
C_{1}\lambda_{n}^{\frac{1}{p-2}}\sum_{i=1}^{k}e^{-C_{2}\lambda_{n}^{\frac{1}{2}}{\rm
dist}(x,P_{n}^{i})}+C_{1}\lambda_{n}^{\frac{1}{p-2}}\sum_{j=1}^{m_{3}}e^{-C_{2}\lambda_{n}^{\frac{1}{2}}{\rm
dist}(x,{\rm v}_{j})},\quad\forall
x\in\mathcal{G}\setminus\bigcup_{i=1}^{k}B_{R\lambda_{n}^{-1/2}}(P_{n}^{i}),$

where ${\rm v}_{1},\dots,{\rm v}_{m_{3}}$ are all the vertices of
$\mathcal{G}$.

###### Proof.

Having proved Proposition 4.2, we can adjust the proof of [15, Theorem 4.6]
with minor changes.

_Step 1) There exist $k\in\\{1,2\\}$, and sequences of points
$\\{P_{n}^{i}\\}$, $i\in\\{1,k\\}$, such that (4.12) and (4.13) hold, and
moreover_

$\lim_{R\to+\infty}\left(\limsup_{n\to\infty}\
\lambda_{n}^{-\frac{1}{p-2}}\max_{d_{n}(x)\geq
R\lambda_{n}^{-1/2}}u_{n}(x)\right)=0.$ (4.14)

_where $d_{n}(x)=\min\\{{\rm dist}(x,P_{n}^{i}):\ i=1,k\\}$ is the distance
function from $\\{P_{1}^{n},P_{k}^{n}\\}$._

The proof of this point, as in [15], can be directly adapted from [21, Theorem
3.2].

_Step 2) Conclusion of the proof._ Here we argue as in the second step of the
proof of [15, Theorem 4.6]. Therein, a comparison argument is used based on
the following estimates: on one side, for all small $\varepsilon\in(0,1)$,
there exist $R>0$ and $n_{R}\in\mathbb{N}$ such that

$\max_{d_{n}(x)>R\lambda_{n}^{-1/2}}u_{n}(x)\leq\lambda_{n}^{\frac{1}{p-2}}\varepsilon,\quad\forall
n\geq n_{R}.$ (4.15)

Moreover, in $A_{n}:=\\{d_{n}(x)>R\lambda_{n}^{-1/2}\\}$ it results that

$-u_{n}^{\prime\prime}+\frac{\lambda_{n}}{2}u_{n}\leq 0.$ (4.16)

Once we have checked that these estimates are satisfied also in the present
setting, we can follow [15] verbatim.

Inequality (4.15) is given by the first step of this proof. Regarding (4.16),
its validity on $\mathcal{K}\cap A_{n}$ follows from (4.15), since on this set

$u_{n}^{\prime\prime}=(\lambda_{n}-u_{n}^{p-2})u_{n}$

and $\lambda_{n}-u_{n}^{p-2}\geq\lambda_{n}/2$ provided that $\varepsilon>0$
is small enough. On $\mathcal{G}\setminus\mathcal{K}$, instead, we can
directly observe that

$u_{n}^{\prime\prime}=\lambda_{n}u_{n}\geq\frac{\lambda_{n}}{2}u_{n}$

since $\lambda_{n}>0$ and $u_{n}\geq 0$. Therefore, (4.15) and (4.16) holds,
and the thesis follows. ∎

Now, we can give

###### Proof of Theorem 1.1.

We have already observed that if $\\{\lambda_{n}\\}$ is bounded then also
$\\{u_{\rho_{n}}\\}$ is bounded. We shall thus assume by contradiction that
$\lambda_{n}\to+\infty$, up to extraction of a subsequence. Then, Propositions
4.2 and 4.5 hold for $u_{n}:=u_{\rho_{n}}$ and by similar arguments as in the
proof of Proposition 5.1 in [15], we can get a contradiction.

Since $\\{\lambda_{n}\\}$ is bounded, passing to a subsequence, there exists
$\lambda^{*}\in\mathbb{R}$ such that
$\lim\limits_{n\to+\infty}\lambda_{n}=\lambda^{*}$. Also, since $\\{u_{n}\\}$
is bounded and $u_{n}>0$ on $\mathcal{G}$, there exists $u^{*}\in
H^{1}(\mathcal{G})$ such that

$\displaystyle u_{n}\rightharpoonup
u^{*}~{}~{}\mbox{in}~{}~{}H^{1}(\mathcal{G}),$ $\displaystyle u_{n}\to
u^{*}~{}~{}\mbox{in}~{}~{}L_{{\rm{loc}}}^{r}(\mathcal{G}),r>2,$ $\displaystyle
u_{n}(x)\to u^{*}(x)~{}~{}\mbox{for a.e.}~{}x\in\mathcal{G},$

which implies that $u^{*}\geq 0$ and $u^{*}$ satisfies

$-(u^{*})^{\prime\prime}+\lambda^{*}u^{*}=\kappa(x)(u^{*})^{p-1}$

together with the Kirchhoff condition

$\displaystyle\sum_{{\rm e}_{i}\succ{\rm v}_{j}}(u^{*})^{\prime}({\rm
v}_{j})+\sum_{l_{i}\succ{\rm v}_{j}}(u^{*})^{\prime}({\rm
v}_{j})=0,\qquad\forall j=1,\cdots,m_{3}.$

By similar arguments to those in the proof of Proposition 3.5, we can deduce
that $\lambda^{*}>0$, $u_{n}\to u^{*}$ strongly in $H^{1}(\mathcal{G})$, and
that $u^{*}>0$. This completes the proof. ∎

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