# Phragmèn-Lindelöf type theorems
for elliptic equations
on infinite graphs

Stefano Biagi  Stefano Biagi,
Dipartimento di Matematica,
Politecnico di Milano,
Piazza Leonardo da Vinci 32, 20133, Milano, Italy
_E-mail address:_<EMAIL_ADDRESS>and  Fabio Punzo  Fabio Punzo,
Dipartimento di Matematica,
Politecnico di Milano,
Piazza Leonardo da Vinci 32, 20133, Milano, Italy
_E-mail address:_<EMAIL_ADDRESS>
###### Abstract.

We investigate the validity of the Phragmèn-Lindelöf principle for a class of
elliptic equations with a potential, posed on infinite graphs. Consequently,
we get uniqueness, in the class of solutions satisfying a suitable growth
condition at infinity. We suppose that the outer degree (or outer curvature)
of the graph is bounded from above, and we allow the potential to go to zero
at infinity in a controlled way. Finally, we discuss the optimality of the
conditions on the potential and on the outer degree on special graphs.

###### Key words and phrases:

Graphs, Phragmèn-Lindelöf, sub–supersolutions, comparison principle, Laplace
operator on graphs

###### 2020 Mathematics Subject Classification:

35A01, 35A02, 35B53, 35J05, 35R02

##  1\. Introduction

Let $(G,\omega,\mu)$ be a fixed infinite weighted graph, with edge-weight
$\omega$ and node (or vertex) measure $\mu$. In this paper we study uniqueness
of possibly unbounded solutions to elliptic equations with a potential of the
following form:

$\Delta u-Vu=f\quad\text{ in }\;G,$ (1.1)

where the potential $V$ is a nonnegative function defined in $G$ and $\Delta$
denotes the Laplace operator on $G$.

Recently, the analysis of partial differential equations on infinite graphs
have attracted great attention (see, e.g., [7, 15, 25]). Qualitative
properties of solutions to elliptic equations have been addressed in [1, 2, 4,
9, 10, 17, 19, 20, 22, 29] for elliptic equations, while in [3, 5, 6, 8, 12,
14, 18, 23, 24, 32] for parabolic equations. In particular,
$\ell^{p}$-Liouville theorems for equation

$\Delta u-Vu=0\quad\text{ in }\;G,$ (1.2)

with $V\equiv 0$ have been established in [10] (see also [2, 19, 9], and [11]
for regular Dirichlet forms). Furthermore, Liouville type results for equation
(1.2), with a potential $V$, can also be deduced from Schnol’s theorem for
$p=2$; a discrete version of such a result can be found in [15] (see also
[17]). In [20], Liouville type theorems have been stated for equation (1.2) in
weighted $\ell^{p}$ spaces, supposing that $\inf_{G}V>0.$ Such hypothesis has
been relaxed in [4], where it is assumed that $V$ can decay at infinity with a
certain rate, but only bounded solutions are considered.

The aim of this paper is to obtain a uniqueness result for equation (1.1), for
possible unbounded solutions, allowing the potential $V$ to decay at infinity,
even if in a controlled way. More precisely, we obtain a Phragmèn-Lindelöf
principle for equation (1.2) (see Proposition 3.2, Theorems 3.3, 3.10). Then a
Liouville type theorem for equation (1.2) and uniqueness of solutions for
equation (1.1) immediately follow (see Corollaries 3.4, 3.5, 3.11).

There is a huge literature regarding Phragmèn-Lindelöf theorems for linear
elliptic equations in $\mathbb{R}^{n}$. We limit ourselves to quote [16, 21,
26, 27, 28, 30, 31].

First we state a general Phragmèn-Lindelöf principle (see Proposition 3.2), in
which the existence of a suitable subsolution is assumed. Thus it is implicit
in its character. Then we show that on a class of graphs, such a subsolution
can be constructed. Hence we obtain an explicit Phragmèn-Lindelöf theorem (see
Theorem 3.3), supposing that the potantial $V$ can decay at infinity with a
certain rate. Moreover, we assume that on such graphs, a relevant quantity,
called outer degree (or outer curvature), is bounded from above. Then we see
that the hypothesis on $V$ and on the outer degree are optimal on spherically
symmetric trees. On the other hand, on the integer lattice $\mathbb{Z}^{n}$ we
can allow a faster decay of the potential $V$ (see Theorem 3.10), and we show
that such condition is sharp (see Theorem 3.12).

The paper is organized as follows. In Section 2 the mathematical background is
introduced, then in Section 3 and main results are stated. In Section 4 we
recall or obtain some preliminary results. The general Phragmèn-Lindelöf
principle is proved in Section 5; finally, in Section 6 the results posed on
the integer lattice $\mathbb{Z}^{n}$ are shown.

##  2\. Mathematical framework

###  2.1. The graph setting

Let $G$ be a countably infinite set, and let $\mu:G\to(0,+\infty)$ be a
measure on $G$ satisfying $\mu(\\{x\\})<+\infty$ for every $x\in G$ (so that
$(G,\mu)$ becomes a measure space). Furthermore, let

$\omega:G\times G\to[0,+\infty)$

be a symmetric, with zero diagonal and finite sum function, i.e.

$\displaystyle\text{(i)}\,\,\omega(x,y)=\omega(y,x)\quad$
$\displaystyle\text{for all}\,\,\,(x,y)\in G\times G;$ (2.1)
$\displaystyle\text{(ii)}\,\,\omega(x,x)=0\quad\quad\quad\,\,$
$\displaystyle\text{for all}\,\,\,x\in G;$
$\displaystyle\text{(iii)}\,\,\sum_{y\in G}\omega(x,y)<\infty\quad$
$\displaystyle\text{for all}\,\,\,x\in G\,.$

Thus, we define weighted graph the triplet $(G,\omega,\mu)$, where $\omega$
and $\mu$ are the so called edge weight and node measure, respectively.
Observe that assumption $(ii)$ corresponds to ask that $G$ has no loops.

Let $x,y$ be two points in $G$; we say that

  * •

$x$ is connected to $y$ and we write $x\sim y$, whenever $\omega(x,y)>0$;

  * •

the couple $(x,y)$ is an edge of the graph and the vertices $x,y$ are called
the endpoints of the edge whenever $x\sim y$;

  * •

a collection of vertices $\\{x_{k}\\}_{k=0}^{n}\subset G$ is a path if
$x_{k}\sim x_{k+1}$ for all $k=0,\ldots,n-1.$

We are now ready to list some properties that the weighted graph
$(G,\omega,\mu)$ may satisfy.

######  Definition 2.1.

We say that the weighted graph $(G,\omega,\mu)$ is

  * (i)

locally finite if each vertex $x\in G$ has only finitely many $y\in G$ such
that $x\sim y$;

  * (ii)

connected if, for any two distinct vertices $x,y\in G$ there exists a path
joining $x$ to $y$;

For any $x\in G$, we define

  * •

the degree of $x$ as

$\operatorname{deg}(x):=\sum_{y\in G}\omega(x,y);$

  * •

the weighted degree of $x$ as

$\operatorname{Deg}(x):=\frac{\operatorname{deg}(x)}{\mu(x)}.$

Let $d:G\times G\to[0,+\infty)$ be a distance on $G.$ For any $x_{0}\in G$ and
$r>0$ we define the ball $B_{r}(x_{0})$ with respect to the metric $d$ as

$B_{r}(x_{0}):=\\{x\in G\,:\,d(x,x_{0})<r\\}\,.$

Sometimes we shall use specific metrics.

In this paper, we always make the following assumption:

$(G,\omega,\mu)\text{ is a connected, locally finite, weighted graph}.$ (2.2)

###  2.2. Difference and Laplace operators

Let $\mathfrak{F}$ denote the set of all functions $f:G\to\mathbb{R}$ . For
any $f\in\mathfrak{F}$ and for all $x,y\in G$, let us give the following

######  Definition 2.2.

Let $(G,\omega,\mu)$ be a weighted graph. For any $f\in\mathfrak{F}$,

  * •

the difference operator is

$\nabla_{xy}f:=f(y)-f(x)\,;$

  * •

the (weighted) Laplace operator on $(G,\omega,\mu)$ is

$\Delta f(x):=\frac{1}{\mu(x)}\sum_{y\in G}[f(y)-f(x)]\omega(x,y)\quad\text{
for all }\,x\in G\,.$ (2.3)

Clearly,

$\Delta f(x)=\frac{1}{\mu(x)}\sum_{y\in G}(\nabla_{xy}f)\omega(x,y)\quad\text{
for all }x\in G\,.$

###  2.3. Outer and inner degrees

We introduce some basic definitions following [15, Chapter 9]. Let $\rho$ be
the combinatorial graph distance on $G$, that is, for any $x,y\in G,$
$\rho(x,y)$ is the least number of edges in a path between the two vertices
$x$ and $y$.

Let $\Omega\subset G$ be finite subset. Define the distance from any $x\in G$
to the subset $\Omega$

$\rho(x,\Omega):=\min_{y\in\Omega}\rho(x,y)\quad\forall x\in G\,.$

With an abuse of notation we write $\rho(x,y)$ to indicate the distance
between the two points $x,y\in G$, and $\rho(x,\Omega)$ to denote the distance
from the point $x\in G$ to the set $\Omega\subset G$.

For any $r\in\mathbb{N}_{0}$, let

$S_{r}(\Omega):=\\{x\in G\,:\rho(x,\Omega)=r\,\\}.$

For any $f\in\mathfrak{F}$, if

$f(x)=f(y)\quad\text{ whenever }\rho(x,\Omega)=\rho(y,\Omega),$

then we say that $f$ is spherically symmetric w.r.t. $\Omega$; moreover, with
a slight abuse of notation we write

$f(x)=f(r)\quad\forall x\in S_{r}(\Omega)\,.$

For any $x\in G$ with $r\equiv r(x):=\rho(x,\Omega)\geq 1$, let

$\mathfrak{D}_{+}(x):=\frac{1}{\mu(x)}\sum_{y\in
S_{r+1}(\Omega)}\omega(x,y),\quad\mathfrak{D}_{-}(x):=\frac{1}{\mu(x)}\sum_{y\in
S_{r-1}(\Omega)}\omega(x,y)\,.$

The function $\mathfrak{D}_{+}:G\to[0,+\infty)$ is called outer degree w.r.t.
$\Omega$, whereas $\mathfrak{D}_{-}:G\to[0,+\infty)$ is called inner degree
w.r.t. $\Omega.$ Sometimes the outer degree and the inner degree are also
called outer curvature and, respectively, inner curvature (see [1]).

######  Remark 2.3.

We explicitly highlight, for a future reference, the following simple
property: _given any $x\in S_{r}(\Omega)$ (for some $r\geq 1$) and any $y\in
G$, by definition of $\rho$ we have_

$\omega(x,y)>0\,\,\Rightarrow\,\,y\in S_{r-1}(\Omega)\cup S_{r+1}(\Omega).$
(2.4)

The weighted graph $(G,\mu,\omega)$, endowed with the combinatorial distance
$\rho$, is said to be weakly spherically symmetric with respect to a finite
subset $\Omega\subset G$, if the outer and inner degrees $\mathfrak{D}_{\pm}$
are spherically symmetric with respect to $\Omega$. Therefore, on a weakly
symmetric graph,

$\mathfrak{D}_{\pm}(x)=\mathfrak{D}_{\pm}(r)\quad\forall x\in
S_{r}(\Omega)\,.$

##  3\. Main Results

We have already stated in (2.2) the main hypotheses on the weighted graph
$(G,\omega,\mu)$. In order to state our main results, we first fix the
following definition.

######  Definition 3.1.

Let $\mathcal{O}\subseteq G$ be an arbitrary set.

  * 1)

Given any $f:\mathcal{O}\to\mathbb{R}$, we say that a function
$u\in\mathfrak{F}$ is a subsolution _[_ resp. supersolution _]_ of the
equation $\mathcal{L}u=f$ in $\mathcal{O}$ if, _for every $x\in\mathcal{O}$_,
we have

$\frac{1}{\mu(x)}\sum_{y\in
G}\omega(x,y)\left[u(y)-u(x)\right]-V(x)u(x)\geq\,[\text{resp.\,\,$\leq$}]\,\,f(x).$
(3.1)

Moreover, we say that $u$ is a solution of the equation $\mathcal{L}u=f$ in
$\mathcal{O}$ if it is both a subsolution and a supersolution of the same
equation.

  * 2)

Given any $f:\mathcal{O}\to\mathbb{R}$ and
$g:G\setminus\mathcal{O}\to\mathbb{R}$, we say that a function
$u\in\mathfrak{F}$ is a subsolution _[_ resp. supersolution _]_ of the
$\mathcal{L}$ \- Dirichlet problem

$\begin{cases}\mathcal{L}u=f&\text{in $\mathcal{O}$}\\\ u=g&\text{in
$G\setminus\mathcal{O}$}\end{cases}$ (3.2)

if the following conditions hold:

    * i)

$u$ is a subsolution _[_ resp. supersolution _]_ of the equation
$\mathcal{L}u=f$ in $\mathcal{O}$;

    * ii)

$u\leq g$ _[_ resp. $u\geq g$_]_ pointwise in $G\setminus\mathcal{O}$.

Finally, we say that $u$ is a solution of problem (3.2) $u$ is both a
subsolution and a supersolution of this problem.

###  3.1. Phragmèn-Lindelöf principle

The first main result of this paper is a general Phragmèn-Lindelöf type
principle, which reads as follows.

######  Proposition 3.2.

Let assumption (2.2) be satisfied. Let $V\in\mathfrak{F},V\geq 0,x_{0}\in G$.
Suppose that there exists $Z\in\mathfrak{F}$ bounded from above such that

$\Delta Z\geq-V\quad\text{ in }G,$ (3.3)
$\lim_{d(x,x_{0})\to+\infty}Z(x)=-\infty\,.$ (3.4)

Let $u$ be a subsolution of equation (1.2) fulfilling

$\limsup_{d(x,x_{0})\to+\infty}\frac{u(x)}{|Z(x)|}\leq 0\,.$ (3.5)

Then

$u\leq 0\quad\text{ in }G\,.$

In particular, on graphs with outer degree bounded from above, we have the
next

######  Theorem 3.3.

Let assumption (2.2) be satisfied. Let $\Omega\subset G$ be a finite subset.
Suppose that, for some $k_{0}>0$,

$0\leq\mathfrak{D}_{+}(x)\leq k_{0}\quad\text{ for all }\,x\in G\text{ with
}\rho(x,\Omega)\geq 1\,.$ (3.6)

Suppose that

$V\in\mathfrak{F},\quad V(x)\geq
c_{0}\,[1+\rho(x,\Omega)]^{-\alpha}\quad\textrm{for all}\;\;x\in G,$ (3.7)

for some $\alpha\in[0,1],c_{0}>0.$ Let $u$ be a subsolution of equation (1.2)
fulfilling

$\limsup_{\rho(x,\Omega)\to+\infty}\frac{u(x)}{|\tilde{Z}(x)|}\leq 0\,,$ (3.8)

where

$\tilde{Z}(x):=\begin{cases}-[\rho(x,\Omega)]^{1-\alpha}&\text{ if
}\alpha\in[0,1)\\\ -\log\rho(x,\Omega)&\text{ if }\alpha=1\end{cases},\text{
whenever }\rho(x,\Omega)>1.$ (3.9)

Then

$u(x)\leq 0\quad\forall x\in G.$

We can immediately infer the following uniqueness results.

######  Corollary 3.4.

Let assumption (2.2) be satisfied. Let $V\in\mathfrak{F},V\geq 0,x_{0}\in G$;
let $f\in\mathfrak{F}$. Suppose that there exists $Z\in\mathfrak{F}$ bounded
from above such that (3.3) and (3.4) hold. Then there exists at most one
solution to equation (1.1) such that

$\lim_{d(x,x_{0})\to+\infty}\frac{u(x)}{Z(x)}=0\,.$

######  Corollary 3.5.

Let assumption (2.2) be satisfied; let $f\in\mathfrak{F}$. Assume that (3.6)
and (3.7) are fulfilled. Then there exists at most one solution to equation
(1.1) such that

$\lim_{d(x,\Omega)\to+\infty}\frac{u(x)}{\tilde{Z}(x)}=0\,,$

where $\tilde{Z}$ is given by (3.9).

###  3.2. Optimality on spherically symmetric trees

In this subsection we consider a special kind of weakly symmetric graphs, the
so called spherically symmetric trees, and we show that the results in Theorem
3.3 and Corollary 3.5 are sharp. More precisely, we show that _both_ the
choice $\alpha\in[0,1]$ in assumption (3.7) and the boundedness assumption
(3.6) on $\mathfrak{D}_{+}$ _are optimal_ , that is, infinitely many bounded
solutions exist whenever $\alpha>1$ or whenever $\mathfrak{D}_{+}$ is not
bounded.

Let $(G,\omega,\mu)$ be a weakly symmetric graph w.r.t. $\Omega=\\{o\\}$, for
some fixed point $o\in G$ (which is usually referred to as the _root of $G$_).
Suppose that

  * •

$\omega:G\times G\to\\{0,1\\}$;

  * •

$\omega|_{S_{r}(\Omega)\times S_{r}(\Omega)}=0$;

  * •

$\mu(x)=1\text{ for every }x\in G;$

  * •

there exists $b:\mathbb{N}\to\mathbb{N}$, which is called the branching
function, such that

$\mathfrak{D}_{+}(x)=b(r),\quad\mathfrak{D}_{-}(x)=1\quad\text{for every $x\in
S_{r}(\Omega)$ and $r\in\mathbb{N}$}.$

As regards the optimality of the choice $\alpha\in[0,1]$ in (3.7), from [4,
Corollary 7.4] we immediately have the following result.

######  Corollary 3.6.

Let $(G,\omega,\mu)$ be a spherically symmetric tree as above, with _constant
branching function_ $b(r)=b_{0}\geq 2$. Moreover, let $V\in\mathfrak{F},\,V>0$
on $G$. Assume that,

$V(x)\leq C_{0}\,\rho^{-\alpha}(x,\Omega)\quad\text{ for any }\;x\in
G\setminus B_{R_{0}}\,.$ (3.10)

for some $\alpha>1$ and some $R_{0}\geq 2$.

Then there exist infinitely many bounded solutions $u$ of problem (1.2). In
particular, for any $\gamma\in{\mathbb{R}}$, $\gamma>0$, there exists a
solution $u$ to equation (1.2) such that

$\lim_{x\to\infty}u(x)=\gamma.$

As regards the optimality of the bounded assumption (3.6), instead, by relying
on a general result established in [4] (see, precisely, Proposition 4.3 below)
we will establish in the next Section 5 the following theorem.

######  Theorem 3.7.

Let $(G,\omega,\mu)$ be a spherically symmetric tree, with _branching
function_

$b(r)=(r+1)^{p}$, for some $p\geq 1$.

Moreover, let $0\leq\alpha\leq 1$ and let

$V:G\to{\mathbb{R}},\qquad V(x)=(1+\rho(x,\Omega))^{-\alpha}.$

If $\alpha+p>1$, then there exist infinitely many bounded solutions $u$ of
problem (1.2). In particular, for any $\gamma\in{\mathbb{R}}$, $\gamma>0$,
there exists a solution $u$ to equation (1.2) such that

$\lim_{x\to\infty}u(x)=\gamma.$

######  Remark 3.8.

Let $(G,\omega,\mu)$ be a spherically symmetric tree as above, with branching
function $b(r)$. Moreover, let $\alpha\in[0,1]$ be fixed, and let

$V:G\to{\mathbb{R}},\qquad V(x)=(1+d(x,\Omega))^{-\alpha}.$

On account of Theorem 3.3 we see that, _if $b(r)$ is bounded_, then there
exists at most one bounded solution to equation (1.2); on the other hand, if
$b(r)=(1+r)^{p}$ (for some $p\geq 2$), it follows from Theorem 3.7 that there
exists _infinitely many bounded solutions_ of (1.2).

###  3.3. Further results on $\mathbb{Z}^{n}$

Consider the $n-$dimensional integer lattice graph. Then $G=\mathbb{Z}^{n}$;
furthermore, $x\sim y$ if and only if there exists $k\in\\{1,\ldots,n\\}$ such
that $x_{k}=y_{k}\pm 1$ and $x_{i}=y_{i}$ for $i\neq k$. The edge weight is

$\omega:\mathbb{Z}^{n}\times\mathbb{Z}^{n}\to[0,+\infty)$
$\omega(x,y)=\begin{cases}1&\text{ if }y\sim x\\\ 0&\text{ if }y\not\sim
x\,,\end{cases}$

whereas the node measure is

$\mu(x)=\sum_{y\in\mathbb{Z}^{n}}\omega(x,y)=2n\,.$

We equip the graph $(\mathbb{Z}^{n},\omega,\mu)$ with the euclidean distance

$|x-y|=\left(\sum_{k=1}^{n}|x_{k}-y_{k}|^{2}\right)^{\frac{1}{2}}\quad(x,y\in\mathbb{Z}^{n})\,.$

######  Remark 3.9.

We remark that $\mathbb{Z}^{n}$ with the euclidean distance is not a weakly
symmetric graph, since in the definition of weakly symmetric graph only the
combinatorial graph distance is considered. Furthermore, it is easily seen
that $\mathbb{Z}^{n}$ endowed with the combinatorial metric is not a weakly
symmetric graph.

In this case the condition on $\alpha$ made in (3.7) is not optimal. In fact,
the critical value is now $\alpha=2$, and not more $\alpha=1$.

More precisely, we can prove the next results.

######  Theorem 3.10.

Let $G=\mathbb{Z}^{n}$. Suppose that, for some $\alpha\in[0,2]$ and $c_{0}>0$,

$V\in\mathfrak{F},\quad V(x)\geq c_{0}\,(1+|x|)^{-\alpha}\quad\textrm{for
all}\;\;x\in G.$ (3.11)

Let $u$ be a subsolution of equation (1.2) fulfilling (3.5), with $x_{0}=0$,
$d(x,y)=|x-y|$ and

$Z(x):=\begin{cases}-|x|^{2-\alpha}&\text{ if }\alpha\in[0,2)\\\
-\log|x|&\text{ if }\alpha=2\end{cases},\text{ whenever }|x|>2.$ (3.12)

Then

$u(x)\leq 0\quad\forall x\in G.$

A direct consequence of Theorem 3.2 is the following uniqueness result.

######  Corollary 3.11.

Let $G=\mathbb{Z}^{n},f\in\mathfrak{F}$. Assume that (3.11) holds. Then there
exists at most one solution to equation (1.1) such that

$\lim_{|x|\to+\infty}\frac{u(x)}{Z(x)}=0\,,$

where $Z$ is given by (3.12).

The value $\alpha=2$ in Corollary 3.11 is optimal. This follows from the next
nonuniqueness result.

######  Theorem 3.12.

Let $G=\mathbb{Z}^{n},n\geq 3$. Assume that

$V\in\mathfrak{F},\quad 0<V(x)\leq c_{0}\,(1+|x|)^{-\alpha}\quad\textrm{for
all}\;\;x\in G,$ (3.13)

for some $\alpha>2$. Then there exist infinitely many bounded solutions to
equation (1.2). More precisely, for any $\gamma\in\mathbb{R}$ there exists a
bounded solution $u_{\gamma}$ such that

$\lim_{|x|\to+\infty}u_{\gamma}(x)=\gamma\,.$

##  4\. Auxiliary Results

In this section we collect some well-known facts which will be exploited in
the proofs of our main results. To begin with, we prove the following simple
lemma.

######  Lemma 4.1.

Let assumption (2.2) be satisfied. Let $\Omega\subset G$ be a finite set and
let $f\in\mathfrak{F}$ be a spherically symmetric function with respect to
$\Omega$. Then

$\Delta
f(x)=\mathfrak{D}_{+}(x)[f(r+1)-f(r)]+\mathfrak{D}_{-}(x)[f(r-1)-f(r)]$ (4.1)

for any $x\in G$ with $r\equiv r(x):=\rho(x,\Omega)\geq 1.$

###### Proof.

Let $x\in G$ with $r\equiv r(x):=\rho(x,\Omega)\geq 1.$ From (2.3) and Remark
2.3 we have

$\displaystyle\Delta f(x)$ $\displaystyle=\Delta
f(r(x))=\frac{1}{\mu(x)}\sum_{y\in G}[f(y)-f(x)]\omega(x,y)$ (4.2)
$\displaystyle=\frac{1}{\mu(x)}\sum_{y\in S_{r+1}(\Omega)\cup
S_{r-1}(\Omega)}[f(y)-f(x)]\omega(x,y)$
$\displaystyle=\frac{1}{\mu(x)}\Big{\\{}\sum_{y\in
S_{r+1}(\Omega)}[f(y)-f(x)]\omega(x,y)$ $\displaystyle\qquad\qquad+\sum_{y\in
S_{r-1}(\Omega)}[f(y)-f(x)]\omega(x,y)\Big{\\}}$
$\displaystyle=\frac{1}{\mu(x)}\sum_{y\in
S_{r+1}(\Omega)}[f(r+1)-f(r)]\omega(x,y)$
$\displaystyle\qquad\qquad+\frac{1}{\mu(x)}\sum_{y\in
S_{r-1}(\Omega)}[f(r-1)-f(r)]\omega(x,y)$
$\displaystyle=[f(r+1)-f(r)]\frac{1}{\mu(x)}\sum_{y\in
S_{r+1}(\Omega)}\omega(x,y)$
$\displaystyle\qquad\qquad+[f(r-1)-f(r)]\frac{1}{\mu(x)}\sum_{y\in
S_{r-1}(\Omega)}\omega(x,y)$
$\displaystyle=\mathfrak{D}_{+}(x)[f(r+1)-f(r)]+\mathfrak{D}_{-}(x)[f(r-1)-f(r)]\,.$

This ends the proof. ∎

We then proceed by reviewing some results proved in [4]. Throughout the rest
of the paper, to simplify the notation we denote by $\mathcal{L}$ the operator

$\mathcal{L}:=\Delta-V(x).$

First of all, we state the following _Weak Maximum Principle_ (see, e.g., [4,
Lemma 3.3]).

######  Lemma 4.2.

Let assumption (2.2) be in force. Let $\Omega\subseteq G$ be a finite set, and
let $u\in\mathfrak{F}$ be a supersolution of the _homogeneous Dirichlet
problem_

$\begin{cases}\mathcal{L}u=0&\text{in $\Omega$}\\\ u=0&\text{in
$G\setminus\Omega$}\end{cases}$ (4.3)

_(_ in the sense of Definition 3.1 _)_. Then

$u\geq 0\quad\text{ in }\Omega.$

Then, we recall a general non-uniqueness criterion (see [4, Proposition 7.1]).

######  Proposition 4.3.

Let assumption (2.2) be in force. Moreover, let $V\in\mathfrak{F}$,
$V>0,x_{0}\in G$ and $\hat{R}>0$. Assume that there exists a supersolution of
the equation

$\Delta h=-V\quad\text{in}\,\,\,\mathcal{O}=G\setminus B_{\hat{R}}(x_{0}),$

further satisfying the following properties

  * i)

$h>0$ in $G\setminus B_{\hat{R}}(x_{0})$;

  * ii)

$\lim_{x\to\infty}h(x)=0.$

Then there exist infinitely many bounded solutions $u$ of problem (1.2). In
particular, for any $\gamma\in{\mathbb{R}}$, $\gamma>0$, there exists a
solution $u$ to problem (1.2) such that

$\lim_{d(x,x_{0})\to\infty}u(x)=\gamma.$

##  5\. Proof of Proposition 3.2 and Theorem 3.3

######  Lemma 5.1.

Let $Z$ be a subsolution of equation

$\Delta Z=-V\quad\text{ in }G\,,$ (5.1)

such that, for some $H\in\mathbb{R}$,

$Z\leq H\quad\text{ in }G\,.$ (5.2)

Then

$\bar{Z}:=Z-H-1$ (5.3)

fulfills

$\bar{Z}\leq-1\quad\text{ in }G,$ (5.4) $\Delta\bar{Z}-V\bar{Z}\geq
0\quad\text{ in }G\,.$ (5.5)

###### Proof of Lemma 5.1.

Obviously, (5.3) is an immediate consequence of (5.2). Furthermore, from (5.1)
and (5.4) we get

$\displaystyle\Delta\bar{Z}-V\bar{Z}$ $\displaystyle=\Delta Z-V\bar{Z}$
$\displaystyle\geq-V-V\bar{Z}=V(-1-\bar{Z})$ $\displaystyle\geq
V(-1+1)=0\quad\text{ in }G\,,$

hence (5.5) follows. ∎

###### Proof of Proposition 3.2.

In view of the hypotheses, we can apply Lemma 5.1 to get the existence of
$\bar{Z}$ fulfilling (5.4) and (5.5). From (3.4), (3.5) and (5.4) we can infer
that

$\liminf_{d(x,x_{0})\to+\infty}\frac{u(x)}{\bar{Z}(x)}\geq 0\,.$

Thus there exists a sequence $\\{R_{n}\\}\subset(0,+\infty)$ with
$\displaystyle R_{n}\mathop{\longrightarrow}_{n\to+\infty}+\infty$ such that

$\lim_{n\to+\infty}\inf_{G\setminus
B_{R_{n}}(x_{0})}\frac{u(x)}{\bar{Z}(x)}\geq 0\,.$

This means that for any $\alpha>0$ there exists
$\bar{n}=\bar{n}(\alpha)\in\mathbb{N}$ such that, for all $n>\bar{n}$,

$\frac{u}{\bar{Z}}>-\alpha\quad\text{ in }G\setminus B_{R_{n}}(x_{0}),$

thus

$u<-\alpha\bar{Z}\quad\text{ in }G\setminus B_{R_{n}}(x_{0})\,.$ (5.6)

For any $\alpha>0$ define

$\mathcal{Z}_{\alpha}:=-\alpha\bar{Z}\,.$

By virtue of (5.4),

$\mathcal{Z}_{\alpha}\geq\alpha\quad\text{ in }G\,.$ (5.7)

From (5.5) and (5.7) it follows that for any $\alpha>0,n\in\mathbb{N}$,
$\mathcal{Z}_{\alpha}$ is a supersolution of problem

$\begin{cases}\mathcal{L}u=0&\text{ in }B_{R_{n}}(x_{0})\\\
u=\mathcal{Z}_{\alpha}&\text{ in }G\setminus B_{R_{n}}(x_{0})\,.\end{cases}$
(5.8)

On the other hand, thanks to (5.6), for any $\alpha>0$ and for any $n>\bar{n}$
(with $\bar{n}\in\mathbb{N}$ as above), $u$ is a subsolution of problem (5.8).
By Lemma 4.2,

$u\leq\mathcal{Z}_{\alpha}\quad\text{ in }B_{R_{n}}(x_{0})\,.$

Letting $\alpha\to 0^{+}$, we deduce that

$u\leq 0\quad\text{ in }G\,.$

∎

###### Proof of Theorem 3.3.

Let

$r\equiv r(x):=\rho(x,\Omega)\quad\forall\,x\in G\,.$

To begin with, we consider the case where (3.7) holds with $\alpha\in[0,1)\,.$

Define

$Z(x)\equiv Z(r):=-Mr^{\beta}-1\quad\forall x\in G\,,$

where $M>0$ will be properly determined in the sequel, whereas
$\beta:=1-\alpha$.

In view of (4.1), for all $x\in G,r(x)\geq 1$,

$\displaystyle\Delta Z(x)$
$\displaystyle=M\left\\{-\mathfrak{D}_{+}(x)[(r+1)^{\beta}-r^{\beta}]+\mathfrak{D}_{-}(x)[r^{\beta}-(r-1)^{\beta}]\right\\}$
$\displaystyle\geq
M\left\\{-\mathfrak{D}_{+}(x)[(r+1)^{\beta}-r^{\beta}]\right\\}=M\left\\{-\mathfrak{D}_{+}(x)\beta\xi^{\beta-1}\right\\},$

for some $\xi\in(r,r+1)$. Therefore, since $\beta<1$,

$\Delta Z(x)\geq-\frac{M\beta k_{0}}{r^{1-\beta}}=-\frac{M\beta
k_{0}}{r^{\alpha}}\quad\forall x\in G,r(x)\geq 1\,.$ (5.9)

Combining together (3.7) and (5.9) we can infer that

$\frac{1}{V(x)}\Delta
Z(x)\geq-\frac{1}{c_{0}}\frac{(1+r)^{\alpha}}{r^{\alpha}}M\beta
k_{0}\geq-1\quad\forall x\in G,r(x)\geq 1\,,$ (5.10)

provided that $M>0$ is small enough. On the other hand, since $\Omega$ is a
finite subset of $G$ (and since $V>0$), by shrinking $M>0$ if needed, we also
have that

$\Delta Z(x)=-M\Delta(r^{\beta})\geq-V(x)\quad\forall\,\,x\in\Omega.$ (5.11)

By virtue of (5.10) and (5.11), $Z$ fulfills (3.3).

Now, let $d_{0}$ be the diameter of the finite set $\Omega$, let
$x_{0}\in\Omega$. Select any $x\in G$ with $\rho(x,x_{0})\geq 2d_{0}$. For all
$y\in\Omega$, by triangular inequality,

$\rho(x,y)\geq\rho(x,x_{0})-\rho(y,x_{0})\geq\rho(x,x_{0})-d_{0}.$

Hence

$\rho(x,\Omega)=\min_{y\in\Omega}\rho(x,y)\geq\rho(x,x_{0})-d_{0},$

thus

$\rho(x,x_{0})-d_{0}\leq\rho(x,\Omega)\leq\rho(x,x_{0})\,.$ (5.12)

In view of (5.12), $Z$ satisfies also (3.4). By (5.12) again, since by
assumption $u$ satisfies (3.8), we can infer that also (3.5) holds. Therefore
we are in position to apply Theorem 3.2, with $d=\rho$, to get the thesis,
when $\alpha\in[0,1)$.

It remains to take into account the case where $\alpha=1$. Define

$Z(x)\equiv Z(r):=-M\log(2+r)\quad\forall x\in G\,,$

where $M>0$ will be properly determined in the sequel.

In view of (4.1), for all $x\in G,r(x)\geq 1$,

$\displaystyle\Delta Z(x)$
$\displaystyle=M\left\\{-\mathfrak{D}_{+}(x)[-\log(2+r)+\log(3+r)]+\mathfrak{D}_{-}(x)[\log(2+r)-\log(1+r)]\right\\}$
$\displaystyle\geq
M\left\\{-\mathfrak{D}_{+}(x)[-\log(2+r)+\log(3+r)]\right\\}=-\frac{M}{\xi}\mathfrak{D}_{+}(x),$

for some $\xi\in(2+r,3+r)$. Therefore,

$\Delta Z(x)\geq-\frac{Mk_{0}}{2+r}\quad\forall r\in\mathbb{N}_{0}\,.$ (5.13)

Combining together (3.7) and (5.13) we can infer that

$\frac{1}{V(x)}\Delta
Z(x)\geq-\frac{1}{c_{0}}\frac{1+r}{2+r}Mk_{0}\geq-1\quad\forall x\in
G,r(x)\geq 1,$ (5.14)

provided that $M>0$ is small enough. On the other hand, by proceeding exactly
as in the previous case (and possibly shrinking the constant $M$), we also
have that

$\Delta Z(x)=-M\Delta(\log(2+r))\geq-V(x)\quad\forall\,\,x\in\Omega.$ (5.15)

By virtue of (5.14) and (5.15), $Z$ fulfills (3.3). Similarly to the previous
case, we get the thesis by means of Theorem 3.2, with $d=\rho$. This completes
the proof. ∎

We conclude the section by proving Theorem 3.7.

###### Proof of Theorem 3.7.

Let $\beta>0$ be arbitrarily fixed, and let

$h:G\to{\mathbb{R}},\qquad h(x)=(1+d(x,\Omega))^{-\beta}.$

We claim that it is possible to choose $\beta$ in such a way that

$\frac{1}{V(x)}\Delta h(x)\leq-1\quad\text{for all $x\in G$ with
$r=d(x,\Omega)\geq R_{0}$,}$ (5.16)

provided that $R_{0}>0$ is sufficiently large (recall that $\Omega=\\{o\\}$,
where $o$ is the _root of $G$_). Once this claim has been established, the
thesis of the theorem (i.e., the existence of infinitely many solutions to
problem (1.2)) is direct consequence of Proposition 4.3 (note that $h$
satisfies properties i) - ii) in the statement of this proposition). Hence, we
turn to prove (5.16).

We first observe that, since we are assuming
$\mathfrak{D}_{+}(x)=b(r)=(r+1)^{p},\,\mathfrak{D}_{-}(x)\equiv 1$ (for every
fixed $x\in S_{r}(\Omega)$ and every $r\geq 1$), a direct application of Lemma
6.1 gives

$\displaystyle\Delta h(x)$
$\displaystyle=(1+r)^{p}[(2+r)^{-\beta}-(1+r)^{-\beta}]+[r^{-\beta}-(1+r)^{-\beta}]$
$\displaystyle(\text{by the Mean Value theorem, for some $r+1<\xi<r+2$ and
$r<\eta<r+1$})$
$\displaystyle=-\beta(1+r)^{p}\xi^{-\beta-1}+\beta\eta^{-\beta-1}$
$\displaystyle(\text{since $\xi<r+2\leq 2r$, as $r\geq 2$})$
$\displaystyle\leq-2^{-\beta-1}\beta r^{p-\beta-1}+\beta r^{-\beta-1}.$

From this, we obtain

$\begin{split}\frac{1}{V(x)}\Delta
h(x)&=-2^{-\beta-1}\beta(1+r)^{\alpha}r^{p-\beta-1}+\beta(1+r)^{\alpha}r^{-\beta-1}\\\
&\leq-2^{-\beta-1}\beta r^{p+\alpha-\beta-1}+2^{\alpha}\beta
r^{\alpha-\beta-1},\end{split}$ (5.17)

and this estimate holds for every $x\in G$ with $r=d(x,\Omega)\geq 2$. In view
of (5.17), if we choose $\beta>0$ in such a way that $0<\beta<\alpha+p-1$
(recall that $\alpha+p>1$), we conclude that

$\frac{1}{V(x)}\Delta h(x)\leq-2^{-\beta-1}\beta
r^{p+\alpha-\beta-1}+2^{\alpha}\beta r^{\alpha-\beta-1}\leq-1,$

provided that $r=d(x,\Omega)$ is sufficiently large. This ends the proof. ∎

##  6\. Further results on $\mathbb{Z}^{n}$: proofs

###  6.1. Construction of sub/supersolutions on $\mathbb{Z}^{n}$

######  Lemma 6.1.

Let the assumptions of Theorem 3.10 hold with $0\leq\alpha<2$. Then, for some
$\beta>0,K>0$,

$Z(x):=-K|x|^{2\beta}-1\quad(x\in\mathbb{Z}^{n})$

fulfills

$\frac{1}{V(x)}\Delta Z(x)\geq-1\quad\forall x\in\mathbb{Z}^{n}\,.$ (6.1)

###### Proof of Lemma 6.1.

Let

$\varphi(t):=-Kt^{\beta}-1\quad\forall t\geq 0\,,$

where $K>0,\beta>0.$ Obviously,

$\varphi^{\prime}(t)=-\beta Kt^{\beta-1},$
$\varphi^{\prime\prime}(t)=\beta(1-\beta)Kt^{\beta-2},$
$\varphi^{\prime\prime\prime}(t)=\beta(1-\beta)(\beta-2)Kt^{\beta-3}\,,$
$\varphi(t)=\varphi(r)+\varphi^{\prime}(r)(t-r)+\frac{\varphi^{\prime\prime}(r)}{2}(t-r)^{2}+\frac{\varphi^{\prime\prime\prime}(\eta)}{6}(t-r)^{3}$
(6.2)

for any $t,r\in(0,+\infty),$ for some $\eta$ between $t$ and $r$.

By virtue of (6.2) with $t=|y|^{2},\,r=|x|^{2}$ ($x,y\in\mathbb{Z}^{n},|x|\geq
1$),

$\displaystyle\Delta Z(x)$
$\displaystyle=\frac{1}{\mu(x)}\sum_{x\in\mathbb{Z}^{n}}[Z(y)-Z(x)]\omega(x,y)$
(6.3)
$\displaystyle=\frac{K}{2n}\sum_{x\in\mathbb{Z}^{n}}\Big{\\{}\varphi^{\prime}(|x|^{2})[|y|^{2}-|x|^{2}]+\frac{\varphi^{\prime\prime}(|x|^{2})}{2}[|y|^{2}-|x|^{2}]^{2}$
$\displaystyle\qquad\qquad+\frac{\varphi^{\prime\prime\prime}(\eta)}{6}[|y|^{2}-|x|^{2}]^{3}\Big{\\}}\omega(x,y)$
$\displaystyle=\frac{K}{2n}\sum_{y\in\mathbb{Z}^{n}}\Big{\\{}-\beta|x|^{2\beta-2}(|y|^{2}-|x|^{2})+\beta(1-\beta)|x|^{2\beta-4}(|y|^{2}-|x|^{2})^{2}$
$\displaystyle\qquad\qquad+\frac{\beta}{6}(1-\beta)(\beta-2)\eta^{\beta-3}(|y|^{2}-|x|^{2})^{3}\Big{\\}}\omega(x,y),$

for some $\eta$ fulfilling

$\min\\{|x|^{2},|y|^{2}\\}\leq\eta\leq\max\\{|x|^{2},|y|^{2}\\}\,.$ (6.4)

We then turn to estimate from below the right-hand side of (6.3). In what
follows, to ease the readability we denote by $c$ any positive constant,
possibly different from line to line but independent of the fixed $x,y\in G$
with $\omega(x,y)>0$.

Let $x\in\mathbb{Z}^{n}$ be fixed. Define

$\mathcal{O}_{x}:=\\{y\in\mathbb{Z}^{n}\,:\,\omega(x,y)>0\\}.$

It is easy to see that $\mathcal{O}_{x}$ contains $n$ points
$y\in\mathbb{Z}^{n}$ of the form $y=(x_{1},\ldots,x_{k}-1,\ldots,x_{n})$ for
any $k\in\\{1,\ldots,n\\}$, and $n$ points $y\in\mathbb{Z}^{n}$ of the form
$y=(x_{1},\ldots,x_{k}-1,\ldots,x_{n})$ for any $k\in\\{1,\ldots,n\\}$.
Therefore, there are $n$ points $y\in\mathbb{Z}^{n}$ for which

$|y|^{2}-|x|^{2}=|x|^{2}+2x_{k}+1-|x|^{2}=2x_{k}+1\,,$

and $n$ points $y\in\mathbb{Z}^{n}$ for which

$|y|^{2}-|x|^{2}=|x|^{2}-2x_{k}+1-|x|^{2}=-2x_{k}+1\,,$

Consequently,

$\sum_{y\in\mathbb{Z}^{n}}(|y|^{2}-|x|^{2})\omega(x,y)=2n\,.$ (6.5)

Furthermore, we have

$\left|[|y|^{2}-|x|^{2}]^{3}\right|\leq c(|x|^{3}+1)\quad\forall
x,y\in\mathbb{Z}\text{ with }\omega(x,y)>0\,.$ (6.6)

Let now $x,y\in\mathbb{Z}^{n}$ for which $\omega(x,y)>0$, and let $\eta$ be as
in (6.4). If $|x|\geq 1$, then $|y|\geq|x|-1$; as a consequence, we can write

$\eta\geq(|x|-1)^{2}\geq c(|x|^{2}-1).$ (6.7)

Let

$\mathfrak{R}(x):=\frac{\beta}{6}(1-\beta)(\beta-2)\eta^{\beta-3}(|y|^{2}-|x|^{2})^{3}.$

Due to (6.6) and (6.7), for every $x,y\in\mathbb{Z}^{n}$ for which
$\omega(x,y)>0$ we have

$\displaystyle\mathfrak{R}(x)$
$\displaystyle\geq-\frac{c\beta(1-\beta)(\beta-2)}{6}(|x|^{2}-1)^{\beta-3}(1+|x|^{3})$
(6.8) $\displaystyle\geq-c(1+|x|)^{2\beta-3}\,.$

Hence, by virtue of (6.8) and (6.5), choosing $0<\beta<1$, (6.3) yields

$\displaystyle\Delta Z(x)$
$\displaystyle\geq\frac{K}{2n}\sum_{y\in\mathbb{Z}^{n}}\left\\{-\beta|x|^{2\beta-2}(|y|^{2}-|x|^{2})+\mathfrak{R}(x)\right\\}\omega(x,y)$
(6.9)
$\displaystyle\geq-K\left\\{\beta|x|^{2\beta-2}+c(1+|x|)^{2\beta-3}\right\\}\omega(x,y)$
$\displaystyle\geq-cK|x|^{2\beta-2}\quad\forall x\in\mathbb{Z}^{n},|x|\geq
1\,.$

Observe that in view of (3.11), for some $\alpha\in[0,2)$,

$\frac{1}{V(x)}\leq\frac{1}{c_{0}}(1+|x|)^{\alpha}\quad\forall
x\in\mathbb{Z}^{n}\,.$

Therefore, from (6.9) we can infer that

$\frac{1}{V(x)}\Delta Z(x)\geq-cK|x|^{2\beta-2+\alpha}\geq-1\quad\forall
x\in\mathbb{Z}^{n},|x|\geq 1,$ (6.10)

provided that $K\leq\frac{1}{c}$ and $0<\beta<\frac{2-\alpha}{2}.$ On the
other hand, we also have

$\frac{1}{V(0)}\Delta Z(0)=-K\frac{\Delta(|x|^{2\beta})(0)}{V(0)}\geq-1,$
(6.11)

by possibly shrinking $K$. Gathering (6.10) and (6.11), we get that $Z$
satisfies (6.1). ∎

######  Lemma 6.2.

Let the assumptions of Theorem 3.10 hold with $0\leq\alpha\leq 2$. Then, for
some $K>0$, the function

$Z(x):=-K\log\big{(}|x|^{2}+2\big{)}\quad(x\in\mathbb{Z}^{n})$

fulfills (6.1).

###### Proof of Lemma 6.2.

Let

$\psi(t):=-K\log(t+2)\quad\forall t\geq 0\,,$

where $K>0.$ Obviously,

$\psi^{\prime}(t)=-\frac{K}{t+2},$
$\psi^{\prime\prime}(t)=\frac{K}{(t+2)^{2}},$
$\psi^{\prime\prime\prime}(t)=-\frac{2K}{(t+2)^{3}}\,,$
$\psi(t)=\varphi(r)+\psi^{\prime}(r)(t-r)+\frac{\psi^{\prime\prime}(r)}{2}(t-r)^{2}+\frac{\psi^{\prime\prime\prime}(\eta)}{6}(t-r)^{3}$
(6.12)

for any $t,r\in(0,+\infty),$ for some $\eta$ between $t$ and $r$.

By virtue of (6.12) with
$t=|y|^{2},r=|x|^{2}\quad(x,y\in\mathbb{Z}^{n},|x|\geq 1)$,

$\displaystyle\Delta Z(x)$
$\displaystyle=\frac{1}{\mu(x)}\sum_{x\in\mathbb{Z}^{n}}[Z(y)-Z(x)]\omega(x,y)$
(6.13)
$\displaystyle=\frac{K}{2n}\sum_{x\in\mathbb{Z}^{n}}\Big{\\{}\psi^{\prime}(|x|^{2})[|y|^{2}-|x|^{2}]+\frac{\psi^{\prime\prime}(|x|^{2})}{2}[|y|^{2}-|x|^{2}]^{2}$
$\displaystyle\qquad\qquad+\frac{\psi^{\prime\prime\prime}(\eta)}{6}[|y|^{2}-|x|^{2}]^{3}\Big{\\}}\omega(x,y)$
$\displaystyle=\frac{K}{2n}\sum_{y\in\mathbb{Z}^{n}}\Big{\\{}-\frac{1}{|x|^{2}+2}(|y|^{2}-|x|^{2})+\frac{1}{2(|x|^{2}+2)^{2}}(|y|^{2}-|x|^{2})^{2}$
$\displaystyle\qquad\qquad-\frac{1}{3(\eta+2)^{3}}(|y|^{2}-|x|^{2})^{3}\Big{\\}}\omega(x,y),$

for some $\eta$ fulfilling (6.4). As in the proof of Lemma 6.1 we then proceed
by estimating from below the right-hand side of (6.13). In what follows, we
simply denote by $c$ any positive constant, possibly different from line to
line but independent of $x,y$.

Let

$\mathfrak{T}(x):=-\frac{1}{3(\eta+2)^{3}}(|y|^{2}-|x|^{2})^{3}.$

Due to (6.6) and (6.7), for every $x,y\in\mathbb{Z}^{n}$ for which
$\omega(x,y)>0$, we have

$\displaystyle\mathfrak{T}(x)$
$\displaystyle\geq-\frac{c}{3}(|x|^{2}+1)^{-3}(1+|x|^{3})$ (6.14)
$\displaystyle\geq-c(1+|x|)^{-3}\,.$

Hence, by virtue of (6.14) and (6.5), (6.13) yields

$\displaystyle\Delta Z(x)$
$\displaystyle\geq\frac{K}{2n}\sum_{y\in\mathbb{Z}^{n}}\left\\{-\frac{1}{|x|^{2}+2}(|y|^{2}-|x|^{2})+\mathfrak{T}(x)\right\\}\omega(x,y)$
(6.15)
$\displaystyle\geq-K\left\\{\frac{1}{|x|^{2}}+\frac{c}{(1+|x|)^{3}}\right\\}\geq-
cK|x|^{-2}\quad\forall\,\,x\in\mathbb{Z}^{n},|x|\geq 1\,.$

Observe that in view of (3.11), for some $\alpha\in[0,2]$,

$\frac{1}{V(x)}\leq\frac{1}{c_{0}}(1+|x|)^{\alpha}\quad\forall\,\,x\in\mathbb{Z}^{n}\,.$

Therefore, from (6.15) we can infer that

$\frac{1}{V(x)}\Delta Z(x)\geq-
cK|x|^{-2+\alpha}\geq-1\quad\forall\,\,x\in\mathbb{Z}^{n},|x|\geq 1,$ (6.16)

provided that $K\leq\frac{1}{c}$. On the other hand, we also have

$\frac{1}{V(0)}\Delta Z(0)=-K\frac{\Delta(\log(2+r))(0)}{V(0)}\geq-1,$ (6.17)

by possibly shrinking $K$. Gathering (6.16) and (6.17) we get that $Z$
satisfies (6.1). ∎

######  Lemma 6.3.

Let the assumptions of Theorem 3.12 hold. Then, for some
$\sigma>0,K>0,\gamma>0$,

$h(x):=\frac{\sigma}{(K+|x|^{2})^{\gamma}}\quad(x\in\mathbb{Z}^{n})$

fulfills

$\frac{1}{V(x)}\Delta h(x)\leq-1\quad\text{for every $x\in\mathbb{Z}^{n}$ with
$|x|\geq 1$}.$

###### Proof.

By arguing as in the proof of [22, Theorem 6.1] we have that

$\displaystyle\Delta h(x)$
$\displaystyle\leq\frac{\sigma}{(K+|x|^{2})^{\gamma+2}}\Big{\\{}-\left(\gamma-\frac{2\gamma(\gamma+1)}{n}\right)|x|^{2}$
(6.18) $\displaystyle\qquad\qquad-\left(\gamma
K-\frac{\gamma(\gamma+1)}{2}\right)+c(|x|+1)\Big{\\}}\quad\forall\,\,x\in\mathbb{Z}^{n},$

for some ‘absolute’ constant $c>0$. Now we take
$0<\gamma<\frac{n-2}{2},K>\frac{\gamma+1}{2}$, so

$\delta_{0}:=\gamma-\frac{2\gamma(\gamma+1)}{n}>0.$

Hence, from (6.18) and (3.13), we can deduce that

$\displaystyle\frac{1}{V(x)}\Delta h(x)$ $\displaystyle\leq-\frac{\sigma
c}{(K+|x|^{2})^{\gamma+2}}|x|^{2}(1+|x|)^{\alpha}$ (6.19)
$\displaystyle\leq-\sigma c|x|^{\alpha-2-2\gamma}\leq-1\quad\text{for every
$x\in\mathbb{Z}^{n}$ with $|x|\geq 1$},$

provided that $\gamma\leq\frac{\alpha-2}{2}$ and $\sigma>{1}/{c}\,.$ ∎

###  6.2. Proof of Theorems 3.10, 3.12

###### Proof of Theorem 3.10.

The conclusion follows by Theorem 3.2, with $d$ being the Euclidean distance,
combined with Lemma 6.1 when $\alpha\in[0,2)$, and with Lemma 6.2 when
$\alpha\in[0,2].$ ∎

###### Proof of Theorem 3.12.

The conclusion follows by Proposition 4.3, with $d$ being the Euclidean
distance, combined with Lemma 6.3. ∎

Acknowledgement. All authors are member of the “Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni” (GNAMPA) of the “Istituto
Nazionale di Alta Matematica” (INdAM, Italy), and are partially supported by
GNAMPA. The first author is partially supported by the PRIN 2022 project
2022R537CS _$NO^{3}$ \- Nodal Optimization, NOnlinear elliptic equations,
NOnlocal geometric problems, with a focus on regularity_, founded by the
European Union - Next Generation EU. The second author acknowledges that this
work is part of the PRIN project 2022 Geometric-analytic methods for PDEs and
applications, ref. 2022SLTHCE, financially supported by the EU, in the
framework of the ”Next Generation EU initiative”.

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