A Liouville theorem for elliptic equations with a potential on infinite graphs

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is $u\equiv 0$. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.


Introduction
Let (G, ω, µ) be a fixed infinite weighted graph, with edge-weight ω and node (or vertex) measure µ.In this paper we study bounded solutions to elliptic equations with a potential of the following form: ∆u − V u = 0 in G, (1.1) where the potential V is a nonnegative function defined in G and ∆ denotes the Laplace operator on G.
The uniqueness of solutions of equation (1.1) has been investigated by two of us in the recent paper [29]; in this paper, it is proved that u ≡ 0 is the only solution to equation (1.1), whenever that u belongs to a certain ℓ p ϕ (G, µ) space, where ϕ is a weight which tends to 0 at infinity and p ∈ [1, +∞).Also the case u ∈ ℓ ∞ (G, µ) can be considered, provided that the graph satisfies a suitable property.In any case, an essential hypothesis for the arguments used in [29] is the existence of some c 1 > 0 such that V (x) ≥ c 1 for all x ∈ G . (1.2) Hence, the aim of the present paper is to understand when u ≡ 0 is the unique bounded solution of problem (1.1), without supposing hypothesis (1.2).To do this, we have to use completely different methods than those exploited in [29].
We say that the Liouville theorem (or property) holds for equation (1.1), whenever u ≡ 0 is the only bounded solution of the same equation.Thus, in other terms, we are concerned with the validity of the Liouville theorem for equation (1.1).
Before describing our results and methods of proof, let us contextualize our problem within the literature.As it is well-known, many phenomena in various fields of applied sciences can be modeled by means of graphs (see, e.g., [5,24,27]) .For this reason, partial differential equations posed on graphs have recently attracted the attention of many authors.In particular, qualitative properties of solutions to both elliptic and parabolic equations have been addressed, see e.g.[11,12,13,15,18,22,26,34] and [1,4,6,16,17,25,31,38,39], respectively.Moreover, the monographs [10,23,32] are important contributions to this topic.
In [19] and in [21], under suitable assumptions on the graph, it is shown that the parabolic problem has at most one solution fulfilling a suitable growth condition.An analogous result can be found in [4], where the time derivative is replaced by discrete differences.Furthermore, in [23,Theorem 12.15, Corollary 12.16, Theorem 12.17] it is shown that, under suitable assumptions on G, if u is a subsolution of equation (1.1) with V ≡ 0 which satisfies u ∈ ℓ p (G, µ) and u ≥ 0, then u must be constant.Similar uniqueness results have been established also on manifolds (see, e.g., [7,8,9,30,35]), in bounded domains of R n (see [2,3,33,36]), and for nonlocal operators (see [28,37]).Now, let d denote a pseudo metric on G, and let B r (x 0 ) be the ball centered at x 0 ∈ G with radius r (see Section 2 below).Concerning the potential V , we suppose that V (x) ≥ c 0 d −α (x, x 0 ) for all x ∈ G \ B R 0 (x 0 ), for some c 0 > 0, R 0 > 0 and α ∈ [0, 1].Under this assumption on V we prove that, if there exists some constant Λ ∈ (0, 1) such that e −Λd α (x,x 0 ) µ(x) < +∞, (1.3) then u ≡ 0 is the only bounded solution of equation (1.1).
In order to prove this result, we take inspiration from [7].However, since in [7] the problem is posed on a Riemannian manifold, many ideas used in that setting cannot be exploited on graphs, hence important differences arise.More precisely, the line of arguments we follow to show our main result is the following: introducing the function v(x, t) := e t u(x) − 1 for all x ∈ G, t ∈ [0, T ], we first show that the positive part of v, namely v + , is a subsolution of equation (1.1) in an appropriate sense (see Lemma 4.2).Then, we obtain a key a priori estimate for v + (see Proposition 4.1), where test functions ξ = ξ(x, t) and η = η(x) are employed.Then we select a suitable function ξ, which can be regarded as a sort of supersolution of an "adjoint parabolic equation" (see Lemma 5.1).On the other hand, η will be chosen to be a "cut-off" function (see Lemma 5.2).Then the conclusion follows by means of appropriate estimates.
A key point in our strategy of proof of uniqueness is to show that if u is a bounded solution of equation (1.1), then there exists a solution v to the same problem such that 0 < v ≤ 1 (see Proposition 3.1).To show this, we first need to establish a weak maximum principle, see Lemma 3.3, and a strong maximum principle, see Lemma 3.4.Let us mention that Proposition 3.1, Lemmas 3.3, 3.4, 4.2 can have an independent interest.We also show that the bound (1.3) with α ∈ [0, 1] is optimal.More precisely, on a special class of graphs, we prove that if V decays like d −α (x, x 0 ) for some α > 1 as d(x, x 0 ) → +∞, then there exist infinitely many bounded solutions to problem (1.1).To be specific, for any γ > 0 there exists a u solution to (1.1) such that u(x) → γ as d(x, x 0 ) → +∞.Its proof is based on the construction of a suitable barrier at infinity, which is related to the class of graphs we consider.To the best of our knowledge, on graphs, such type of result, which consists in prescribing a Dirichlet condition at infinity, and the explicit construction of such kind of barrier are totally new.
The paper is organized as follows.In Section 2 we describe the graph framework and we state our main result.Section 3 is devoted to the auxiliary results for elliptic equations previously described.In Section 4 we obtain the key apriori estimate for v + (x, t).We introduce and study our test functions in Section 5.The main result is proved in Section 6.Finally, in Section 7 we show the optimality of condition (1.3).

Mathematical framework and the main result
2.1.The graph setting.Let G be a countably infinite set and µ : G → (0, +∞) be a given function.Observe that µ can be viewed as a Radon measure on G so that (G, µ) becomes a measure space.Furthermore, let ω : G × G → [0, +∞) be a symmetric, with zero diagonal and finite sum function, i.e.
(2.1) Thus, we define weighted graph the triplet (G, ω, µ), where ω and µ 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 ∼ y, whenever ω(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 ∼ y; for all k = 0, . . ., n − 1.We are now ready to list some properties that the weighted graph (G, ω, µ) may satisfy.Definition 2.1.We say that the weighted graph (G, ω, µ) is (i) locally finite if each vertex x ∈ G has only finitely many y ∈ G such that x ∼ y; (ii) connected if, for any two distinct vertices x, y ∈ G there exists a path joining x to y; (iii) undirected if its edges do not have an orientation.
For any x ∈ G, we define • the degree of x as deg(x) := y∈G ω(x, y); • the weighted degree of x as Deg(x) := deg(x) µ(x) .
A pseudo metric on G is a symmetric, with zero diagonal map, d : G × G → [0, +∞), which also satisfies the triangle inequality In general, d is not a metric, since we can find points x, y ∈ G, x = y such that d(x, y) = 0 .Now, let us consider any path γ ≡ {x k } n k=0 ⊂ G, and a symmetric function σ : G × G → [0, +∞) such that σ(x, y) > 0 if and only if x ∼ y.Then we define the lenght subordinated to σ as Finally, we define the jump size s > 0 of a pseudo metric d as s := sup{d(x, y) : x, y ∈ G, ω(x, y) > 0}. (2.2) For a more detailed understanding of the objects introduced so far, we refer the reader to [14,20,21,29].We conclude the subsection with the following Definition 2.2.A pseudo metric d on (G, ω, µ) is said to be intrinsic if Observe that hypothesis (2.3) can be compared with an analogous condition on Riemannian manifolds.Indeed, given any fixed reference point x 0 ∈ G, let us consider the map Then, we have Therefore, condition (2.3) ensures that Such a property is clearly fulfilled on Riemannian manifolds.
For any x 0 ∈ G and r > 0 we define the ball B r (x 0 ) with respect to any pseudo metric d as In this paper, we always make the following assumptions: (i) (G, ω, µ) is a connected, locally finite, weighted graph; (ii) there exists a pseudo metric d such that the jump size s is finite; (iii) the ball B r (x) with respect to d is a finite set, for any x ∈ G, r > 0; (2.4) here we have used Definitions 2.1 and 2.2.

2.2.
Difference and Laplace operators.Let F denote the set of all functions f : G → R .For any f ∈ F and for all x, y ∈ G, let us give the following Definition 2.3.Let (G, ω, µ) be a weighted graph.For any f ∈ F, • the difference operator is • the (weighted) Laplace operator on (G, ω, µ) is We also define the gradient squared of f ∈ F (see [4]) It is straightforward to show, for any f, g ∈ F, the validity of • the product rule for all x, y ∈ G ; • the integration by parts formula provided that at least one of the functions f, g ∈ F has finite support.
2.3.The main result.We have already stated in (2.4) the main hypotheses on the weighted graph (G, ω, µ).Concerning the potential V , we suppose that (2.7) for some We can now state the main result of this paper.
Since γ > 0 is arbitrary, in particular nonuniqueness for equation (1.1) follows.See Section 7 for more details.To the best of our knowledge, in the literature such methods used on graphs cannot be found.

Auxiliary Results
In this section we collect several preliminary results of independent interest which shall be used in the proof of our main result, namely Theorem 2.4.Proposition 3.1.Let assumption (2.4)-(i) be fulfilled.Assume that there exists a nontrivial bounded solution of equation (1.1).Then there exists a solution v of equation (1.1) such that Analogously to [2,7], the proof of Proposition 3.1 is crucially based on the unique solvability of the Dirichlet problem for that is, (where Ω ⊆ G is an arbitrary finite set and f, g ∈ F), together with some maximum principles for L. Since we were not able to find a precise reference for these results, and in order to make the paper as self-contained as possible, we present here below the full proofs.
To begin with, we give the following definition Definition 3.2.We say that u ∈ F is a solution of equation and u ≡ g in G \ Ω.Moreover, we say that u is a supersolution (subsolution) to equation We now establish the following Weak Maximum Principle.
Lemma 3.3.Let assumption (2.4)-(i) be fulfilled.Let Ω ⊆ G be a finite set, and let u ∈ F be such that Proof.We proceed essentially as in the proof of [10,Lemma 1.39].We set m := min Ω u; observe that m is well-defined since the set Ω ⊆ G is finite.Suppose, by contradiction, that m < 0. Then the set Indeed, let x ∈ F be fixed, hence u(x) = m < 0. Due to (2.7), (3.4) and recalling that ω(x, y) > 0 if y ∼ x, we have from which we derive that In view of (3.6), since u ≥ m in G, we conclude that u(y) = m for every y ∈ G, y ∼ x, i.e. (3.5).Now, let us consider some x ∈ F and y ∈ G \ Ω, hence u(x) = m < 0 and u(y) ≥ 0. Due to (2.4), there exist a path {x k } n k=0 such that x 0 = x, x n = y.
Since x 0 = x ∈ F , we can apply (3.5) and infer that x 1 ∈ F .By repeating this argument, we get that x i ∈ F for every i = 0, ..., n, hence in particular that x n = y ∈ F and thus u(y) = m < 0 which yields a contradiction.
We now prove the following Strong Maximum Principle for L-harmonic functions.
Lemma 3.4.Let assumption (2.4)-(i) be fulfilled.Let u ∈ F be such that Then either u ≡ 0 in G or u > 0 in G.
Proof.Let u be a solution of (3.7).Then, due to Lemma 3.3, u ≥ 0 in G. Let us now assume that there exists x 0 ∈ G such that u(x 0 ) = 0.Moreover, we consider the set Observe that x 0 ∈ F .By arguing as in the proof of Lemma 3.3, by using that min G u = 0, we get, equivalently to (3.5), that if y ∈ G and y ∼ x 0 , then y ∈ F.
Consequently, since G is connected, we conclude that F = G, and hence u ≡ 0 in G.
Due to Lemmas 3. Proof.We begin by observing that, given any u ∈ F, we can write where Thus, we see that u ∈ F is a solution of (3.2) if and only if u ≡ g in G \ Ω and We now claim that the validity of (3.9), which only involves the values attained by u on Ω, can be rephrased as a linear equation in a suitable finite-dimensional vector space.
In fact, if we denote by F Ω the set of all real-valued functions defined on Ω, it is immediate to recognize that F Ω is a real vector space, and (here, χ A stands for the indicator function of the set A ⊆ G).Thus, since Ω is finite, we derive that F Ω has finite dimension n = card(Ω).On this space F Ω , we then define the map Clearly, A is linear; moreover, identity (3.9) can be rewritten as Summing up, we have that u ∈ F is a solution of problem (3.2) if and only if Using this 'abstract' formulation of the Dirichlet problem (3.2), we can easily complete the proof of the proposition.First of all we observe that, owing to the Weak Maximum Principle in Lemma 3.3, the linear operator A is injective: indeed, if u ∈ F Ω is such that Au = 0 (that is, Au(x) = 0 for every x ∈ Ω) and if define û := u1 Ω , from (3.8) we have thus, since û ≡ 0 in G \ Ω, an immediate application of Lemma 3.3 shows that û ≡ 0 in G, and hence u = 0 in F Ω .From this, since F Ω has finite dimension, we derive that A is also surjective, and thus there exists a unique function u ∈ F Ω such that and, for every n ∈ N, we let v n ∈ F be the unique solution of problem where p ∈ G is a point arbitrarily chosen.The existence and uniqueness of v n for each n ∈ N is guaranteed by Proposition 3.5, since the balls B r (x) are finite sets, see (2.4).
We now claim that, for every n ∈ N, the following properties holds: Taking this claim for granted for a moment, we can easily complete the proof of the proposition.In fact, owing to (i)-(ii) we deduce that the sequence {v n } n is increasing and bounded on G; as a consequence, the function is well-defined, and it satisfies 0 ≤ v ≤ 1 on G.Moreover, since and since the sum which defines the Laplacian ∆ is actually a finite sum (recall that the graph G is locally finite), by letting n → +∞ in (3.12) we readily obtain and thus v is a solution of problem (1.1).Finally, reminding that 0 ≤ v ≤ 1 on G and using the Strong Maximum Principle in Lemma 3.4, we conclude that v ∈ F is a solution of problem (1.1) also fulfilling (3.1).Hence, we are left to prove the claimed (i)-(ii).Let us show (i).We first observe that, since Lv n = 0 in B n (p) and since v n = (u − M ) + ≥ 0 in G \ B n (p), from the Weak Maximum Principle in Lemma 3.3 we infer that v n ≥ 0 in G. On the other hand, since the constant function ζ ≡ 1 satisfies  Finally, let η ∈ F, and ξ : S T → R be such that To prove Proposition 4.1 we need an auxiliary result (of independent interest) which shows that, if v is any solution of the parabolic equation then v + := max{v; 0} is a subsolution of the same equation (in a suitable sense).This is the content of the following Lemma 4.2.Let v : S T → R be a solution of equation (4.6) such that the map t → v(x, t) is C 1 ([0, T ]) for any x ∈ G.Then, for any fixed x ∈ G, Proof.We separately consider three cases.

Some distinguished test functions
Let us now prove the existence of suitable test functions ξ and η which are admissible in (4.5) and which satisfy some ad-hoc properties.
Proof.To ease the readability, we split the proof into two steps.
Step I: In this first step we prove the following estimate for every x, y ∈ G with x ∼ y. (5.4) To this end, it is useful to distinguish two cases.
(i) x ∈ B r−s (x 0 ).In this case, by triangle's inequality and (2.(ii) x ∈ G \ B r−s (x 0 ).In this case we first notice that, since the function is Lipschitz-continuous with Lipschitz constant L = 1, by the Mean Value Theorem and again the triangle inequality we can write where σ ≥ 0 is a suitable point between d(x) and d(y).On the other hand, since we assuming that x / ∈ B r−s (x 0 ) (hence, d(x) ≥ r − s), by (2.2) we have Recalling that β ≤ 1, from (5.5)-(5.6)we immediately obtain which is exactly the desired (5.4).
Step II: In this second step we establish (5.3).To begin with, we point out that (e a − 1) 2 ≤ a 2 e 2|a| ∀ a ∈ R; this inequality, together with (5.4), allows us to write and this estimate holds for every x, y ∈ G and every t ∈ (0, T ).Then, using (2.2) and recalling that d is intrinsic (hence, (2.3) holds), for every x ∈ G and t ∈ (0, T ) we obtain . (5.7) To proceed further, we now fix β = α and we exploit assumption (2.7): taking into account the piecewise definition of ρ, see (5.2), it is easy to recognize that (5.8) as a consequence, by combining (5.7)-(5.8)we conclude that This ends the proof.Now that we have proved Lemma 5.1, we turn to prove the existence of a suitable 'cut-off' function η.To this end, taking for fixed all the notation introduced so far, we choose and we define the function Owing to [28, Lemma 5.2] (with the choice δ = 1/2), we obtain the following result.

Proof of Theorem 2.4
Due to the results established in Sections 3, 4 and 5, we are ready to provide the proof of Theorem 2. 4. In what follows, we take for fixed all the notation introduced so far.
Proof of Theorem 2.4.By contradiction, suppose that there exists a non-trivial bounded solution of equation (1.1).Then, due to Proposition 3.1, we know that there exists a solution u to to same equation (1.1) such that Let us now define v(x, t) := e t u(x) − 1, for any (x, t) ∈ S, for S as in (4.1).We want to show that v(x, t) ≤ 0 for every x ∈ supp(V ) and t > 0. ( To do so, let us fix T > 0 (to be chosen conveniently small in a moment), and we arbitrarily choose r > 2s + R 0 , for R 0 > 0 and 0 < s < +∞ as in (2.7) and (2.2), respectively.Let ξ be as in (5.2) with β = α, λ > 1 and with M = T chosen as in Lemma 5.1.Moreover, let us fix r 1 > 0 in such a way that r 1 ≥ 2r + 8s, ( and let η be as in Lemma 5.2.Now, we observe that η and ξ obviously satisfy conditions (4.2) and (4.3); furthermore, also (4.4) is fulfilled, since both η and ξ(•, t) are non-increasing functions of d.Therefore, form (4.5) we obtain (6.4) On the other hand, by Lemma 5.1 there exists T 0 = T 0 (λ) > 0, such that provided that T ≤ T 0 .As a consequence, by combining (6.4) and (6.5), we obtain We then proceed by estimating both sides of (6.6).
-Estimate of the left-hand side.First of all we observe that, owing to the definition of η in (5.9), we have η ≥ 0 pointwise on G and η ≡ 1 on B r (x 0 ) = {x : d(x) < r}.Now we observe that, due to (6.1), where we have used the shorthand notation γ r,Λ = e − r α λ−1 > 0.
(6.10) Furthermore, we recall that r 1 was arbitrarily fixed, hence by taking the limit as r 1 → +∞ in (6.10) and by assumption (2.8), we get Thus, we readily derive that where we recall that the number T 0 depends on λ, which is by now fixed.From this, recalling also that r ≥ 2s + R 0 was arbitrarily fixed, we then obtain Now, let us introduce the 'shifted' function Clearly, it is still a solution of the parabolic problem (4.6).In addition, by (6.11), v 1 (x, 0) ≤ 0 for every x ∈ supp(V ).By applying the very same argument exploited so far, we can infer that By iterating this argument, and by using in a crucial way the fact that T 0 (λ) > 0 is a universal number remaining unchanged at any iteration (as λ > 0 is fixed), we conclude that This yelds (6.2).
We can now easily conclude the proof of the theorem, in fact, due to (6.2) and exploiting the definition of v, we have 0 < u ≤ e −t for every x ∈ supp(V ) and t > 0.
Then, by letting t → +∞, we deduce that u ≤ 0 on supp(V ) = ∅, but this is clearly in contradiction with (6.1).This completes the proof.

Optimality on model trees
We start by showing a general non-uniqueness criterium which holds for any graph (G, ω, µ) such that (2.4) is fulfilled.We write x → ∞ whenever d(x, x 0 ) → +∞, for some reference point x 0 ∈ G.If there exists a supersolution to problem then there exist infinitely many bounded solutions u of problem (1.1).In particular, for any γ ∈ R, γ > 0, there exists a solution u to problem (1.1) such that Proof.Let γ ∈ R, γ > 0. For any j ∈ N, let us consider the following problem Due to assumption (2.4), existence and uniqueness of a solution u j to problem (7.2), in the sense of Definition 3.2, for any j ∈ N is granted by Proposition 3.5.We now claim that 0 ≤ u j (x) ≤ γ for any x ∈ G and for any j ∈ N.
In fact, since ∆u j − V (x)u j = 0 in B j , and since u j = γ ≥ 0 in G \ B j , from Lemma 3.3, we can infer that u j ≥ 0 in G. On the other hand, let v(x) := γ for any x ∈ G.Then, since V (x) > 0 for any x ∈ G, and since γ > 0 For any j ∈ N, let w := v − u j .Due to (7.2) and (7.4), ∆w − V (x)w ≤ 0 in G.
Moreover, w ≥ 0 in G \ B j .Hence, by Lemma 3.3, w ≥ 0 in G and, in particular, Therefore, (7.3) follows.Furthermore, for any j ∈ N, let u j+1 be the solution to problem (7.2) in B j+1 .Thus, in particular, observe that and, by (7.3), u j+1 (x) ≤ γ for any Therefore, by Lemma 3.3, v ≥ 0 in G and, in particular, for any j ∈ N, Hence, from (7.3) and (7.5), we deduce that the sequence {u j } j∈N is decreasing and bounded on G. Therefore, there exists u ∈ F such that Moreover, u j solves problem (7.2) for any j ∈ N and since the sum which defines the Laplacian ∆ is finite, by letting j → +∞ we obtain that u is a solution to problem (1.1).Let h be a supersolution to problem (7.1).Then we define, for any x ∈ G \ B R, with R > 0 as in the assumptions, w(x) := −C h(x) + γ.
For any j ∈ N such that B j ⊃ B R, we show that w is a subsolution to problem (7.2) in the sense of Definition 3.2.Due to (7.1) and since V > 0 in G, we have that ∆w Therefore, by Lemma 3.3, we get By combining together (7.3) and (7.7) we get By letting j → ∞ we get thus, in particular, due to (7.1), lim x→∞ u(x) = γ.
7.2.Model trees and a special supersolution.In this subsection we consider a special kind of graphs, the so called model trees, and we show that the uniqueness result in Theorem 2.4 is sharp for this choice of graph.More precisely, we show that the choice α ∈ [0, 1] in assumption (2.7) is optimal, indeed infinitely many bounded solutions exist whenever α > 1.
Let us first define a model tree.Let m(x) denote the number of edges which have x as endpoint.
Definition 7.2.A graph T will be called a model tree if it contains a vertex x 0 , known as the root of the model, such that m(x) is constant on spheres S r (x 0 ) = S r of radius r about x 0 .Thus we have: Here and hereafter, for any x, x 0 ∈ G, d(x, x 0 ) denotes the standard metric (on a graph), i.e. the number of vertices separating x from x 0 along a path which connects x 0 to x.
Furthermore, we define the branching, b(r), at the distance r from the root as the number of edges connecting each vertex in S r to a vertex in S r+1 ; we also set b(0) = m(x 0 ).Thus, for any r > 0, we have that m(r) = b(r) + 1.We denote a model tree with branching b(r) by T b(r) .We say that a model tree is homogeneous if the branching is constant, i.e. b(r) = b for every r ≥ 0, for some b ∈ N.
Let ω : G × G → R be the edge weight of a graph G as defined in (2.1).Whenever ω(x, y) ∈ {0, 1} for all x, y ∈ G, we say that the graph has standard edge weight and we denote it by ω 0 (x, y).In particular, for each x ∈ G \ {x 0 }, Moreover, we define the weighted counting measure as the measure µ for which there exists c > 0 such that µ(x) = c for all x ∈ G, and we denote it by µ c .Moreover, let us denote the ball B r (x 0 ) of radius r centered at the root x 0 simply by B r .
In what follows, we will deal with a model homogeneous tree with branching b, standard edge weight ω 0 and weighted counting measure µ c , that is the triplet (T b , ω 0 , µ c ).Now, on (T b , ω 0 , µ c ), we explicitly construct a function h satisfying all the properties required in Proposition 7.1.Then, there exist infinitely many bounded solutions u of problem (1.1).In particular, for any γ ∈ R, γ > 0, there exists a solution u to problem (1.1) such that lim x→∞ u(x) = γ.7.3.A counterexample.On account of Corollary 7.4, we can easily show that the requirement α ∈ [0, 1] in assumption (2.7) cannot be dropped, that is, this assumption is optimal if one restricts to a particular class of graphs.
To illustrate this fact, let (T b , ω 0 , µ 1 ) be a homogeneous model tree with branching b ≥ 2 (and weighted counting measure µ c ≡ 1), and let x 0 be the root of the model.Given any number α > 1, we then consider the function V ∈ F defined as follows: V (x) := (1 + d(x, x 0 )) −α (where d is the usual distance on trees).Clearly, assumptions (2.3) and (2.4) are satisfied in this context; moreover, V : T b → R is a strictly positive potential on T b , but the last condition is assumption (2.7) is obviously violated (since α > 1).
We now observe that, since α > 1, the series (2.8) is convergent for every choice of Λ ∈ (0, 1): in fact, recalling that µ 1 ≡ 1, we have the following computation On the other hand, since b ≥ 2 and since V is a strictly positive potential on T b satisfying condition (7.10) (with R 0 = 1 and C 0 = 1), we are entitled to apply Corollary 7.4, ensuring that there exist infinitely many non-trivial bounded solutions to equation (1.1).Hence, the condition α ∈ [0, 1] is optimal in this context.

3 and 3 . 4 ,
we can now prove the following Proposition 3.5.Let assumption (2.4)-(i) be fulfilled.Let Ω ⊆ G be a finite set.Let f : Ω → R and g : G \ Ω → R be arbitrary functions.Then there exists a unique solution u ∈ F to problem (3.2) in the sense of Definition 3.2.
by applying the Weak Maximum Principle in Lemma 3.3 to the function ζ −v n we immediately conclude that v n ≤ ζ ≡ 1 pointwise on G. Now, let us prove (ii).First of all we observe that, since ζ = u − M satisfies

Proposition 4 . 1 .
by applying the Weak Maximum Principle in Lemma 3.3 to the function v n+1 − ζ we derive that v n+1 ≥ ζ = u − M in G; thus, since we already know that v n+1 ≥ 0, we obtain v n+1 ≥ (u − M ) + pointwise on G. (3.13) Owing to (3.13), and applying once again the Weak Maximum Principle in Lemma 3.3 to the function v n+1 − v n with Ω = B n (p), we then conclude that v n ≤ v n+1 in G.This ends the proof.4. A useful apriori estimate Now we have established Proposition 3.1, we turn to prove an apriori estimate for nonnegative and bounded solutions to (1.1) which will play a key role in the proof of Theorem 2.4.Throughout what follows, we set S := G × [0, +∞); (4.1) and, for any given T > 0, S T := G × [0, T ] .Let assumption (2.4) be in force.Let u be a solution of equation (1.1) such that 0 ≤ u ≤ 1.Moreover, let T > 0 and define v(x, t) := e t u(x) − 1 for all (x, t) ∈ S T .
.11)Extending this unique function u by setting u(x) = g(x) for every x ∈ G \ Ω, from (3.11) we conclude that (3.10) is satisfied, and thus u is the unique solution of (3.2).
Proof of Proposition 3.1.Let u ∈ F be a non-trivial bounded solution of problem (1.1), and let s := sup G |u| ∈ (0, +∞).Without lost of generality, we may assume s = 1, indeed, it would be sufficient to replace u with u s .Moreover, if u has constant sign on G, then the function v := sgn(u)u, is a solution of (1.1) satisfying (3.1).Indeed, it is immediate to recognize that v is a non-trivial solution of (1.1), and 0 ≤ v ≤ 1 on G.Then, by the Strong Maximum Principle in Lemma 3.4, we conclude that