Positive Solutions of a General Discrete Dirichlet Boundary Value Problem

A steady state equation of the discrete heat diffusion can be obtained by the heat diffusion of particles or the difference method of the elliptic equations. In this paper, the nonexistence, existence, and uniqueness of positive solutions for a general discrete Dirichlet boundary value problem are considered by using the maximum principle, eigenvalue method, suband supersolution technique, and monotone method. All obtained results are new and valid on any n-dimension finite lattice point domain. To the best of our knowledge, they are better than the results of the corresponding partial differential equations. In particular, the methods of proof are different.


Introduction
In this paper, we consider the discrete Dirichlet boundary value problem −Δ () =  (,  ()) ,  ∈ Ω,  () = 0,  ∈ Ω, where Ω ⊂   is a finite domain and  defined on Ω ×  is a continuous function about the second variable.To understand the above problem, we need some terminology.Denote the set of integers by .A lattice point  = ( 1 ,  2 , . . .,   ) ∈   is defined as a point with integer coordinates.Two lattice points are said to be neighbors if their Euclidean distance is one.The lattice points  1 ,  2 , . . .,   are said to form a path with terminals  1 and   if  1 is a neighbor of  2 ,  2 is a neighbor of  3 , and so forth.A set of lattice points is said to be connected if any two of its points are terminals of a path of points contained in the set.A nonempty connected set of lattice points is called a domain.Given a domain Ω, a lattice point is an exterior boundary point of Ω if it does not belong to Ω but has at least one neighbor in Ω.We will denote the set of exterior boundary points of Ω by Ω, and the set Ω ∪ Ω of lattice points will be denoted by Ω.For a finite domain Ω, the number of the lattice points is denoted by |Ω|.
Given a sequence {()}, the partial differences of () are defined by ∇ 1  () =  ( 1 + 1,  2 , . . .,   ) −  ( 1 ,  2 , . . .,   ) , ∇ 2  () =  ( 1 ,  2 + 1, . . .,   ) −  ( 1 ,  2 , . . .,   ) , . . . ( The discrete Laplace operator will be denoted by Δ which is defined by Problem (1) can be regarded as the discrete analogue of the elliptic boundary problem of the form where Φ ⊂   and  satisfy some conditions and Δ is the Laplacian operator defined by There exist many differences between (1) and ( 4).As a simplest example, let us consider the continuous linear equation: It is well known that the operator −Δ has a sequence of eigenvalues: when the weight function  : Φ →  + is assumed to be in   (Φ) with  > /2 and () > 0 on a subset of positive measures of Φ.In particular, from Krein-Rutman theorem, the principal eigenvalue  (1) is simple, in the sense that the eigenspace is one-dimensional, and the corresponding eigenfunction  (1) can be chosen to be positive.At the same time, we also have lim  → ∞  () = ∞.Note that when Ω is a finite domain, the discrete problem is only a finite dimension problem.This will induce some essential differences between ( 6) and (8); see [1,2] and the listed references.First of all, we need to ask  ∈   (Φ) for eigenvalue problem (6); however, corresponding eigenvalue problem (8) does not need to add any conditions; secondly, problem (8) has at most |Ω| eigenvalues; however, the number of eigenvalues of ( 6) is infinite; thirdly, in general case, the existence of solutions for (4) will be considered in Sobolev space  1 0 (Φ) (see [3][4][5][6]), but (1) is not necessary, and so forth.Thus, it is necessary to consider the existence of solutions for problem (1).
In this paper, we will consider the nonexistence, existence, and uniqueness of positive solutions of (1) by using the maximum principle, eigenvalue method, sub-and supersolution technique, and monotone method.To this end, the elementary theory will be introduced in the next section.The sub-and supersolution technique, the nonexistence, existence, and uniqueness will be considered in Section 3. In the final section, we will give some applications.
Our results are suitable for any -dimension space and the shape of domain Ω is not restrained.

Preliminaries
In this section, we hope to obtain some preliminaries results.They are important for establishing our main results.The obtained results seem to be similar to the corresponding continuous elliptic boundary problem; however, their conditions are different.In particular, the methods of proof are also different.
Consider first the eigenvalue problem where  = {()} ∈Ω is real and Ω ⊂   is a finite domain.
It is well known that the maximum principle is important.Thus, we firstly give a maximum principle.

Lemma 1.
Assume that  = {()} ∈Ω is nonnegative and  = {()} ∈Ω satisfies the difference inequality Then  cannot achieve a nonnegative maximum (nonpositive minimum) in the interior of Ω unless it is constant.
In the following, we will consider eigenvalue problem (9).

Lemma 2. Eigenvalue problem (9) has |Ω| real eigenvalues
where  1 () is simple and the corresponding eigenfunction  1 can be chosen to be positive.Moreover, the other eigenfunctions must change sign in Ω.
Clearly, the matrix 2− is positive definite and symmetric.The rest of the proof is the same as Lemma 1 in [26].
A comparison result will be obtained in the following.
has a unique solution.Moreover, if {()} ∈Ω is nonnegative and {()} ∈Ω is nonnegative and not identically zero, the solution is positive.
Consider now the eigenvalue problem where  = {()} ∈Ω is nonnegative and not identically zero in Ω and Ω ⊂   is a finite domain.
For eigenvalue problem (19), we have the following result.
Lemma 5.The first eigenvalue  (1) () of problem ( 19) is simple and the corresponding eigenfunction  (1) can be chosen to be positive.In particular, if  is a positive constant, then Proof.Consider the eigenvalue problem −Δ () −  (1) ()  ()  (z) =  () ,  ∈ Ω, It is easy to see that  (1) () is the first eigenvalue for (19) with corresponding eigenfunction  (1) if and only if 0 is the first eigenvalue of ( 21) with corresponding eigenfunction  (1) .In view of Lemma 2, we complete the proof.Lemma 6.Let  satisfy the condition of (19) and  <  (1) ().Then for any sequence {()} ∈Ω , the discrete Dirichlet boundary value problem has a unique solution.Moreover, if {()} ∈Ω is nonnegative and not identically zero, then the solution is positive.

Nonexistence, Existence, and Uniqueness of Positive Solutions
First of all, we give a nonexistence result.
Theorem 7. Assume that holds.Then problem (1) has no nonzero solution.
Proof.By contradiction, suppose that  is a nonzero solution of (1); then we have or where In view of [27], we know that max In view of (36) and (38), we complete the proof.
Definition 8.A sequence  = {()} ∈Ω is said to be a supersolution (subsolution) of (1) if (42) By using (40) and the maximum principle, we can obtain In the following, we claim that (44) This implies by the maximum principle that  0 () −  1 () ≤ 0 in Ω.A similar argument gives V 1 ≤ V 0 .The monotonicity of  gives the rest.So there exist  and V such that lim The proof is complete.
Proof.Note that  (1) ( ∞ ()) > 1.In view of Lemma 6, the Dirichlet problem has a unique positive solution.It is a supersolution of (1) and any nonnegative solution  of (1) satisfies  ≤ .On the other hand, the eigenvalue problem has the first eigenvalue  (1) ( 0 ()) < 1 and the corresponding positive eigenfunction  (1) ().Choose an enough small positive number  such that  1 is a subsolution of (1).In view of Theorem 9 and condition (H 3 ), problem (1) has a positive solution.
Remark 12.For the partial differential equations, the corresponding result has been obtained in [28] when  ∈   .) .