Indefinite Eigenvalue Problems for p-Laplacian Operators with Potential Terms on Networks

and Applied Analysis 3 It will be shown in the next theorem that λh,0 exists and can be variationally expressed as λh,0 = inf φ∈A φ ̸ ≡ 0 (1/2) ∫ S ∇ωφ ⋅ ∇p,ωφ + S V 󵄨󵄨󵄨󵄨φ 󵄨󵄨󵄨󵄨 p ∫ S h 󵄨󵄨󵄨󵄨φ 󵄨󵄨󵄨󵄨 p , (7)


Introduction
In this paper, we study a generalized version of spectral theory, resonance problems, antiminimum principles, and inverse problems for discrete -Laplacian operators with potential terms on a network.We define a network as a way of interconnecting any pair of users or nodes by means of some meaningful links.Therefore, we represent a network by a weighted graph  = (;, , ) with a weight function.
The main goal of this paper is to characterize the indefinite eigenvalues and to solve the inverse conductivity problems for the equations − Δ ,  () +  ()      ()     −2  () =  ℎ ℎ ()      ()     −2  () ,  ∈   () = 0,  ∈ , where  and ℎ are real valued functions on a network  with boundary .Here, Δ , is the discrete -Laplacian on a network  with weight  defined by Δ ,  () for 1 <  < ∞.To address these problems, many researchers have especially concentrated on spectral graph theory which has been one of the most significant tools used in studying graphs.This has led to noteworthy progress in the study of these questions (see, e.g., [1,2]).In this paper, we are primarily concerned with indefinite eigenvalue problems.
In particular, we deal with these problems under the assumptions that ℎ is positive and that ℎ has both positive and negative values.For each case, we present properties for the smallest indefinite eigenvalue  ℎ,0 as follows: (i) the variationally expressed form of  ℎ,0 , (ii) the positivity of eigenfunctions corresponding to  ℎ,0 , (iii) the multiplicity of  ℎ,0 .Moreover, we also show that  ℎ,0 is isolated.Using these properties, we then discuss resonance problems, antiminimum principles, and the inverse conductivity problems.Note that the uniqueness of the conductivity  is not guaranteed from  ℎ,0 .This implies that there can be different conductivities  1 and  2 on edges such that the smallest indefinite eigenvalues of networks for  1 and  2 are the same.Therefore, to guarantee the uniqueness of the conductivity, we impose the additional constraint, the monotonicity condition, on conductivity of the edges.The result for the case that ℎ is positive is Theorem 10 and the results for the other case of ℎ are Theorems 18 and 19.
Recently, in order to expand the results on spectral graph theory with respect to the above viewpoint, great efforts have been concentrated on studying the properties of graphs involving eigenvalues of operators such as discrete Schr ö dinger or discrete -Laplacian operators (see, e.g., [3][4][5][6][7][8]) which are generalizations of the discrete Laplacian.In [9], in particular, Amghibech introduces the indefinite eigenvalue problem for the case where  ≡ 0 and ℎ > 0 in (1) and gives some characterizations of the smallest indefinite eigenvalue.The author also addresses a resonance problem, an antiminimum principle, and an inverse problem.
This paper is organized as follows.In Section 2, we recall some basic terminology and properties of networks.In Section 3, for the case that ℎ is positive, we give some characterizations of the smallest positive indefinite eigenvalue, and we study the resonance problems, the antiminimum principles, and the inverse conductivity problems.Finally, in Section 4, we discuss the same problems discussed in Section 3 under the assumption that ℎ has both positive and negative values.

Preliminaries
In this section, we describe the theoretic graph notations frequently used throughout this paper.
By a graph  = (V, ) we refer to a finite set V of vertices with a set  of two-element subsets of V whose elements are called edges.
For notational convenience, we denote by  ∈  the fact that  is a vertex in (V, ).A graph   =   (V  ,   ) is said to be a subgraph of (V, ) if   ⊂  and   ⊂ .If   consists of all the edges from  which connect the vertices of V  in , then   is called an induced subgraph.Throughout this paper, we assume that the graph (V, ) is finite, simple, and connected.
A weight on a graph (V, ) is a function  : and a graph (V, ) with a weight  is called a network (V, , ).The integration of a function  : V → R is defined by For an induced subgraph  of (, ), by  := ; we denote a graph whose vertices and edges are in  and vertices in  := { ∈  \  | (, ) > 0 for some  ∈ }.Here,  and  are called interiors and boundaries, respectively.
The -gradient ∇ , of a function  :  → R is defined as for  ∈ .In the case of  = 2, we write simply ∇  instead of ∇ 2, .It has been known that for any pair of functions  :  → R and V :  → R, we have where A ⋅ B := ∑  =1     for A = ( 1 , . . .,   ) and B = ( 1 , . . .,   ).This fact yields many useful formulas such as the network version of the Green theorem (for details, see [7]).
Finally, we recall some known results on discrete -Laplacian operators such as the minimum principle and Picone's identity.
Theorem 1 (see [6] minimum principle for −Δ , on networks).Let  :  → R satisfy the differential inequality −Δ , () ≥ 0 for all  ∈ .If  attains the minimum at a point in , then  is constant in .
Theorem 2 (see [9] Picone's identity for Δ , on networks).For an induced subnetwork  of a given weighted network , let two functions  1 and  2 be nonnegative and positive on , respectively.Then ) ⋅ ∇ ,,  2 ) () ≥ 0 (6) for all  in .Moreover, if the induced subnetwork  is connected, then the equality holds if and only if there exists  > 0 such that () = V() for all  in .

Indefinite Eigenvalue Problems with Positive Weight Functions
In [9], Amghibech introduces the indefinite eigenvalue problems for −Δ , on networks with standard weights.In this paper, we study the indefinite eigenvalue problems under more complicated situations than those of Amghibech.More specifically, we look at the -Laplacian operator combined with potential terms and moreover, we do not impose any restrictions on the weight of the networks, further differentiating this paper from [9].We now start this section under the assumption that ℎ is positive.
3.1.The Smallest Indefinite Eigenvalue.In this subsection, we prove the existence of the smallest indefinite eigenvalue  ℎ,0 for −L  , when ℎ is positive.We also address some fundamental problems such as the multiplicity of  ℎ,0 and its isolation.
It will be shown in the next theorem that  ℎ,0 exists and can be variationally expressed as where where S 1 := { ∈ A| ∫  ℎ||  = 1}.Here, we note that  1 is closed and bounded (i.e., compact), since it is a subset of vectors in R  , for  = ||, and since ℎ is positive.Therefore, there exists  0 ∈ S 1 such that Since it is easily seen from ( 1) and ( 5) that  ℎ,0 ≤  ℎ for each eigenvalues  ℎ , it suffices to show that ( ℎ,0 ,  0 ) is an eigenpair.For any  ∈ , we define a function   :  → R as Taking an arbitrary  0 ∈ , we have for a sufficiently small  and Hence, we have for a sufficiently small .Note that the right-hand side is continuously differentiable with respect to  and equals zero at  = 0. Thus, we have Since  0 is chosen arbitrary in , we have which completes the proof.
We now prove the simplicity of  ℎ,0 .To achieve this goal, we first prove a theorem which asserts that there always exists an eigenfunction  0 corresponding to  ℎ,0 which is positive in .

Theorem 4. There exists 𝜙
Proof.It follows from Theorem 3 that there exists an eigenfunction  0 corresponding to  ℎ,0 satisfying Thus, we have Otherwise, by the definition of  ℎ,0 , Thus, It follows from Theorem 3 that ( ℎ,0 , ) is an indefinite eigenpair.Now it suffices to show that  > 0 in .Suppose, to the contrary, that ( 0 ) = 0 for some  0 ∈ .It will be shown that  ≡ 0. Since  is an eigenvalue, it follows from (1) that and thus () = 0 for all  ∼  0 where  ∼  means that two vertices  and  are connected by an edge.By repeating the above process for  ∼  0 , we conclude that () = 0 for each  ∼ .Since the network  is assumed to be connected, () = 0 for all  ∈ .
Using the above theorem, we prove the simplicity of  ℎ,0 as follows.
The above theorem shows that the dimension of the eigenspace corresponding to  ℎ,0 is one.Thus, we have the following.Corollary 6.The multiplicity of  ℎ,0 is one.
For linear operators such as −Δ  on finite networks, it is clear that the number of eigenvalues (including multiplicity) is the same as the number of vertices.However, when we consider nonlinear operators such as −L  , , it becomes significantly more complicated to count the number of eigenvalues.It is not sufficient to simply prove whether the number of eigenvalues is finite of infinite.However, by applying Picone's identity, it is possible to show that the smallest indefinite eigenvalue  ℎ,0 is isolated for a set of indefinite eigenvalues.
Proof.We proceed by contradiction.Suppose that for each  > 0, there exists Since the multiplicity of  ℎ,0 is one, there exists an eigenfunction  0 corresponding to  ℎ,0 with  0 in  such that   →  0 > 0 in  as  → 0. Hence, for sufficiently small  > 0 we have   > 0 in .Since we have That is, Multiplying  0 and integrating over  on both sides (32) and using Picone's identity, we have a contradiction.

Resonance Problems, Antiminimum Principle, and Inverse
Problems.In this subsection, we deal with some interesting problems such as the resonance problems, the antiminimum principles, and the inverse conductivity problems with regard to indefinite eigenvalues.We remind the reader that during this section, we assume the weight function ℎ is a positive valued function.
For a given function  :  → R and a nonnegative source term  :  → [0, ∞), we consider the following equation: It is clear that the above equation has a solution (in fact, an eigenvalue) if  ≡ 0 in .The next result shows that if  ℎ,0 > 0, then the converse of the statement also holds.Thus, there is no solution of the above equation if  is nonzero in .
Theorem 8 (resonance problem).Suppose that a function  satisfies the condition that the smallest indefinite eigenvalue  ℎ,0 is positive.Then (33) has a solution if and only if  ≡ 0.
Proof.Suppose that a function  0 is a solution to the equation and we define a function  0 as Since it is obvious that if  0 ≡ 0, then  ≡ 0; we assume that  0 ̸ ≡ 0. Then we have which implies that  0 ≡  0 for some  ≥ 0. If  > 0 then  0 is an eigenfunction corresponding to  ℎ,0 so that  ≡ 0. Now suppose that  = 0 so  0 ≥ 0. Since  0 ̸ ≡ 0 and  ≥ 0, we have Thus, by using a similar method that we used in the proof for Theorem 4, it is easy to we show that the solution  0 is positive in .Using Picone's identity, we have which implies that  ≡ 0.
The next theorem is the antiminimum principle.From it, we see that each (nonconstant) solution for the following equation has its minimum in  if  >  ℎ,0 .
Proof.By virtue of Theorem 8, it suffices to show that if there exist a nonnegative solution   for (75), then  <  ℎ,0 .Suppose   is a solution to (75) with   () ≥ 0,  ∈ .Using a similar method that we used in the proof of Theorem 4, we can easily show that if   ( 0 ) = 0 for some  0 ∈ , then   ≡ 0. Thus, we may assume that   is positive in .By Picone's identity, we have where  0 is the positive eigenfunction corresponding to  ℎ,0 .Thus, we have Since ∫  ℎ  0 > 0, we finally have  ℎ,0 > , which completes the proof.
We now discuss an inverse conductivity problem on networks.The main concern is related to the problem of recovering the conductivity (weight)  of the network by the smallest indefinite eigenvalue  ℎ,0 for −L , with respect to ℎ.Note that the uniqueness of the conductivity  is not guaranteed by  ℎ,0 .This implies that there can be different conductivities  1 and  2 on the edges which induces the same eigenvalue  ℎ,0 for the operators −L ,  ,  = 1, 2. To guarantee the uniqueness of the conductivity, we need to impose some more assumption on the structure of network or on the conductivity.We impose here the additional constraint, called the monotonicity condition, on the conductivity of the edges.The main result of this section shows that there are no different conductivities  1 and  2 on the edges satisfying  1 ≤  2 in ×  which induce the same smallest indefinite eigenvalue  ℎ,0 .
Theorem 10 (inverse conductivity problem).For networks (,   ,   ) for  = 1,2, let    ℎ,0 be the smallest indefinite eigenvalue for −L  ,  .If the weight functions satisfy then one has Moreover, where   is the eigenfunction corresponding to Proof.By definition of the smallest eigenvalue, we have

Indefinite Eigenvalue Problems with Weight Functions Which Have Both Positive and Negative Values
In this section, we address problems for the other case that ℎ has both positive and negative values.Namely, we now assume that the function ℎ :  → R satisfies for  ∈ .
4.1.Indefinite Eigenvalue Problems.We now discuss the indefinite eigenvalue problems with the assumption that ℎ has both positive and negative values and two real values  + ℎ,0 and  − ℎ,0 defined by Theorem 11.If functions  :  → R and ℎ :  → R satisfy either  ≥ 0 or  ≥ ℎ in , then there exists  0 ∈  such that where Moreover  + ℎ,0 is the smallest positive eigenvalue for −L  , and  0 is an eigenfunction corresponding to  + ℎ,0 . Proof.Define We note that  ∩  1 is not compact and its boundary ( ∩  1 ) is given by From ∫  ℎ|  | → 0 and the function  which satisfies either  ≥ 0 or  ≥ ℎ, we easily show that Therefore, there exists  0 ∈  ∩  1 such that Now we take an arbitrary  0 ∈ .Since ∫  ℎ| 0 +   0 |  ̸ = 0 for sufficiently small  > 0, by definition of  + ℎ,1 , we have Thus, for a sufficiently small  > 0. The right-hand side is continuously differentiable with respect to  and is equal to zero at  = 0. Thus, Since the above equations hold for an arbitrary  0 ∈ , we have Theorem 12.For a function  :  → R and ℎ :  → R satisfying either  ≥ 0 or  ≤ ℎ in , there exists  0 ∈  such that where Moreover  − ℎ,0 is the largest negative eigenvalue for −L  , and  0 is an eigenfunction corresponding to  − ℎ,0 .
Proof.Since the proof is similar to that of the previous theorem, we omit it.
We note that it follows from the two above results that if either  ≥ 0 or  ≡ ℎ, then there exist  + ℎ,0 and  − ℎ,0 at the same time.The specific case of  ≡ 0 was dealt with in [9].
In the following results, we give some properties of  + ℎ,0 and its eigenfunction.One also can get similar results for  − ℎ,0 and its eigenfunction, assuming that the function  satisfies either  ≥ 0 or  ≤ ℎ.
The next result that we will discuss is the parallel version of the antiminimum principle discussed in Theorem 9 where the weight function ℎ is assumed to have both positive and negative values in .