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

Abstract

We address some forward and inverse problems involving indefinite eigenvalues for discrete 𝑝-Laplacian operators with potential terms. These indefinite eigenvalues are the discrete analogues of 𝑝-Laplacians on Riemannian manifolds with potential terms. We first define and discuss some fundamental properties of the indefinite eigenvalue problems for discrete 𝑝-Laplacian operators with potential terms with respect to some given weight functions. We then discuss resonance problems, anti-minimum principles, and inverse conductivity problems for the discrete 𝑝-Laplacian operators with potential terms involving the smallest indefinite eigenvalues.

1. Introduction

In this paper, we study a generalized version of spectral theory, resonance problems, antiminimum principles, and inverse problems for discrete 𝑝-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 𝐺=𝐺(𝑆⊍𝜕𝑆,𝐸,𝜔) 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_𝑝,𝜔𝜙(𝑥)+𝑉(𝑥)𝜙(𝑥)^𝑝−2𝜙(𝑥)=𝜆_ℎℎ(𝑥)𝜙(𝑥)^𝑝−2𝜙(𝑥)𝜙(𝑧),𝑥∈𝑆=0,𝑧∈𝜕𝑆, (1) where 𝑉 and ℎ are real valued functions on a network 𝑆 with boundary 𝜕𝑆. Here, Δ_𝑝,𝜔 is the discrete 𝑝-Laplacian on a network 𝑆 with weight 𝜔 defined by Δ_𝑝,𝜔𝑢(𝑥)∶=_𝑦∈𝑆𝑢𝑦−𝑢(𝑥)^𝑝−2𝑢𝑦−𝑢(𝑥)𝜔𝑥,𝑦,𝑥∈𝑆 (2) for 1<𝑝<∞. 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 𝜆_ℎ,0 as follows:(i)the variationally expressed form of 𝜆_ℎ,0,(ii)the positivity of eigenfunctions corresponding to 𝜆_ℎ,0,(iii)the multiplicity of 𝜆_ℎ,0.

Moreover, we also show that 𝜆_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 𝜔 is not guaranteed from 𝜆_ℎ,0. This implies that there can be different conductivities 𝜔₁ and 𝜔₂ on edges such that the smallest indefinite eigenvalues of networks for 𝜔₁ and 𝜔₂ 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 Schrӧdinger or discrete 𝑝-Laplacian operators (see, e.g., [3–8]) which are generalizations of the discrete Laplacian. In [9], in particular, Amghibech introduces the indefinite eigenvalue problem for the case where 𝑉≡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.

2. Preliminaries

In this section, we describe the theoretic graph notations frequently used throughout this paper.

By a graph  𝐺=𝐺(𝒱,𝐸) we refer to a finite set 𝒱 of vertices with a set 𝐸 of two-element subsets of 𝒱 whose elements are called edges.

For notational convenience, we denote by 𝑥∈𝐺 the fact that 𝑥 is a vertex in 𝐺(𝒱,𝐸). A graph 𝐺^′=𝐺^′(𝒱^′,𝐸^′) is said to be a subgraph of 𝐺(𝒱,𝐸) if 𝑉^′⊂𝑉 and 𝐸^′⊂𝐸. If 𝐸^′ consists of all the edges from 𝐸 which connect the vertices of 𝒱^′ in 𝐺, then 𝐺^′ is called an induced subgraph. Throughout this paper, we assume that the graph 𝐺(𝒱,𝐸) is finite, simple, and connected.

A weight on a graph 𝐺(𝒱,𝐸) is a function 𝜔∶𝒱×𝒱→[0,∞) satisfying(i)𝜔(𝑥,𝑥)=0,𝑥∈𝒱, (ii)𝜔(𝑥,𝑦)=𝜔(𝑦,𝑥)>0 if {𝑥,𝑦}∈𝐸,(iii)𝜔(𝑥,𝑦)=0 if and only if {𝑥,𝑦}∉𝐸,

and a graph 𝐺(𝒱,𝐸) with a weight 𝜔 is called a network 𝐺(𝒱,𝐸,𝜔). The integration of a function 𝑢∶𝒱→ℝ is defined by _𝒱𝑢∶=_𝑥∈𝒱𝑢(𝑥). (3)

For an induced subgraph 𝑆 of 𝐺(𝑉,𝐸), by 𝑆∶=𝑆⊍𝜕𝑆 we denote a graph whose vertices and edges are in 𝑆 and vertices in 𝜕𝑆∶={𝑦∈𝑉⧵𝑆∣𝜔(𝑥,𝑦)>0 for some 𝑥∈𝑆}. Here, 𝑆 and 𝜕𝑆 are called interiors and boundaries, respectively.

The 𝑝-gradient  ∇_𝑝,𝜔 of a function 𝑢∶𝑆→ℝ is defined as ∇_𝑝,𝜔𝑢(𝑥)∶=𝐷_{𝑝,𝜔,𝑦}𝑢(𝑥)_𝑦∈𝑆 (4) for 𝑥∈𝑆. In the case of 𝑝=2, we write simply ∇_𝜔 instead of ∇_2,𝜔.

It has been known that for any pair of functions 𝑢∶𝑆→ℝ and 𝑣∶𝑆→ℝ, we have 2_𝑆𝑣−Δ_𝑝,𝜔𝑢=_𝑆∇_𝜔𝑣⋅∇_𝑝,𝜔𝑢, (5) where 𝔸⋅𝔹∶=∑^𝑛_𝑖=1𝑎_𝑖𝑏_𝑖 for 𝔸=(𝑎₁,…,𝑎_𝑛) and 𝔹=(𝑏₁,…,𝑏_𝑛). This fact yields many useful formulas such as the network version of the Green theorem (for details, see [7]).

For the given functions 𝑉 and ℎ∶𝑆→ℝ, if 𝜆_ℎ∈ℝ and 𝜙∶𝑆→ℝ satisfy (1), then 𝜆_ℎ is called the (Dirichlet) indefinite eigenvalue for −ℒ^𝑉_𝑝,𝜔 where ℒ^𝑉_𝑝,𝜔𝑢∶=Δ_𝑝,𝜔𝑢−𝑉|𝑢|^𝑝−2𝑢 and 𝜙 is called an eigenfunction corresponding to 𝜆_ℎ. Moreover, (𝜆_ℎ,𝜙) is called an indefinite eigenpair.

Finally, we recall some known results on discrete 𝑝-Laplacian operators such as the minimum principle and Picone's identity.

Theorem 1 (see [6] minimum principle for −Δ_𝑝,𝜔 on networks). Let 𝑢∶𝑆→ℝ satisfy the differential inequality −Δ_𝑝,𝜔𝑢(𝑥)≥0 for all 𝑥∈𝑆. If 𝑢 attains the minimum at a point in 𝑆, then 𝑢 is constant in 𝑆.

Theorem 2 (see [9] Picone's identity for Δ_𝑝,𝜔 on networks). For an induced subnetwork 𝑆 of a given weighted network 𝐺, let two functions 𝑢₁ and 𝑢₂ be nonnegative and positive on 𝑆, respectively. Then ∇_𝑆,𝜔𝑢₁⋅∇_{𝑆,𝑝,𝜔}𝑢₁−∇_𝑆,𝜔𝑢^𝑝₁𝑢₂^𝑝−1⋅∇_{𝑆,𝑝,𝜔}𝑢₂(𝑥)≥0 (6) for all 𝑥 in 𝑆. Moreover, if the induced subnetwork 𝑆 is connected, then the equality holds if and only if there exists 𝑡>0 such that 𝑢(𝑥)=𝑡𝑣(𝑥) for all 𝑥 in 𝑆.

3. Indefinite Eigenvalue Problems with Positive Weight Functions

In [9], Amghibech introduces the indefinite eigenvalue problems for −Δ_𝑝,𝜔 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 𝑝-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 𝜆_ℎ,0 for −ℒ^𝑉_𝑝,𝜔 when ℎ is positive. We also address some fundamental problems such as the multiplicity of 𝜆_ℎ,0 and its isolation.

It will be shown in the next theorem that 𝜆_ℎ,0 exists and can be variationally expressed as 𝜆_ℎ,0=inf_{𝜙∈𝒜𝜙≢0}(1/2)∫_𝑆∇_𝜔𝜙⋅∇_𝑝,𝜔𝜙+∫_𝑆𝑉𝜙^𝑝∫_𝑆ℎ𝜙^𝑝, (7) where 𝒜∶=𝜙∈𝑆𝜙_𝜕𝑆=0. (8)

Theorem 3. There exists a nonzero function 𝜙₀∈𝒜 such that 𝜆_ℎ,0=(1/2)∫_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+∫_𝑆𝑉𝜙₀^𝑝∫_𝑆ℎ𝜙₀^𝑝. (9) Moreover, 𝜆_ℎ,0 is the smallest eigenvalue for −ℒ^𝑉_𝑝,𝜔 and 𝜙₀ is an eigenfunction corresponding to 𝜆_ℎ,0.

Proof. Note that inf_{𝜙∈𝒜𝜙≢0}(1/2)∫_𝑆∇_𝜔𝜙⋅∇_𝑝,𝜔𝜙+∫_𝑆𝑉ℎ𝜙^𝑝∫_𝑆ℎ𝜙^𝑝=inf_{𝜙∈𝒜𝜙≢0}12_𝑆∇_𝜔𝜙∫_𝑆ℎ𝜙^𝑝⋅∇_𝑝,𝜔𝜙∫_𝑆ℎ𝜙^𝑝+_𝑆𝜙𝑉ℎ∫_𝑆ℎ𝜙^𝑝𝑝=inf_{𝜙∈𝑆1𝑆}12∇_𝜔𝜙⋅∇_𝑝,𝜔𝜙+_𝑆𝑉𝜙^𝑝, (10) where 𝒮₁∶={𝜙∈𝒜|∫_𝑆ℎ|𝜙|^𝑝=1}. Here, we note that 𝑆₁ is closed and bounded (i.e., compact), since it is a subset of vectors in ℝ^𝑛, for 𝑛=|𝑆|, and since ℎ is positive. Therefore, there exists 𝜙₀∈𝒮₁ such that 12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+_𝑆𝑉𝜙₀^𝑝=min_𝜙∈𝒮112_𝑆∇_𝜔𝜙⋅∇_𝑝,𝜔𝜙+_𝑆𝑉𝜙^𝑝. (11) Since it is easily seen from (1) and (5) that 𝜆_ℎ,0≤𝜆_ℎ for each eigenvalues 𝜆_ℎ, it suffices to show that (𝜆_ℎ,0,𝜙₀) is an eigenpair. For any 𝑥∈𝑆, we define a function 𝛿_𝑥∶𝑆→ℝ as 𝛿_𝑥𝑦=1,𝑥=𝑦,0,otherwise. (12) Taking an arbitrary 𝑥₀∈𝑆, we have _𝑆𝜙₁+𝑡𝛿_𝑥0^𝑝≠0 (13) for a sufficiently small 𝑡 and 𝜆_ℎ,0≤12_𝑆∇_𝜔𝜙₀+𝑡𝛿_𝑥0⋅∇_𝑝,𝜔𝜙₀+𝑡𝛿_𝑥0+_𝑆𝑉𝜙₀+𝑡𝛿_𝑥0^𝑝×_𝑆ℎ𝜙₀+𝑡𝛿_𝑥0^𝑝−1. (14) Hence, we have 10≤2_𝑆∇_𝜔𝜙₀+𝑡𝛿_𝑥0⋅∇_𝑝,𝜔𝜙₀+𝑡𝛿_𝑥0+_𝑆𝑉−𝜆_ℎ,0ℎ𝜙₀+𝑡𝛿_𝑥0^𝑝 (15) for a sufficiently small 𝑡. Note that the right-hand side is continuously differentiable with respect to 𝑡 and equals zero at 𝑡=0. Thus, we have 𝑑0=1𝑑𝑡2_𝑆∇_𝜔𝜙₀+𝑡𝛿_𝑥0⋅∇_𝑝,𝜔𝜙₀+𝑡𝛿_𝑥0+_𝑆(𝑉−𝜆_ℎ,0ℎ)𝜙₀+𝑡𝛿_𝑥0^𝑝_𝑡=0=𝑝−Δ_𝑝,𝜔𝜙₁𝑥₀+𝑉𝑥₀−𝜆_ℎ,0ℎ𝜙₀𝑥₀^𝑝−2×𝜙₀𝑥₀𝑑_𝜔𝑥₀. (16) Since 𝑥₀ is chosen arbitrary in 𝑆, we have −Δ_𝑝,𝜔𝜙₀(𝑥)+𝑉(𝑥)𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥)=𝜆_ℎ,0ℎ𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥),𝑥∈𝑆, (17) which completes the proof.

We now prove the simplicity of 𝜆_ℎ,0. To achieve this goal, we first prove a theorem which asserts that there always exists an eigenfunction 𝜙₀ corresponding to 𝜆_ℎ,0 which is positive in 𝑆.

Theorem 4. There exists 𝜙₀∈𝒜 with 𝜙₀>0 in 𝑆 such that (𝜆_ℎ,0,𝜙₀) is an indefinite eigenpair for −ℒ^𝑉_𝑝,𝜔.

Proof. It follows from Theorem 3 that there exists an eigenfunction 𝜙₀ corresponding to 𝜆_ℎ,0 satisfying −Δ_𝑝,𝜔𝜙₀(𝑥)+𝑉(𝑥)𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥)=𝜆_ℎ,0ℎ(𝑥)𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥),𝑥∈𝑆. (18) Let 𝜓(𝑥)∶=|𝜙₀(𝑥)|,𝑥∈𝑆. Then _𝑆𝜓^𝑝=_𝑆𝜙₀^𝑝,12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+_𝑆𝑉𝜙₀^𝑝≥12_𝑆∇_𝜔𝜓⋅∇_𝑝,𝜔𝜓+_𝑆𝑉𝜓^𝑝. (19) Thus, we have 𝜆_ℎ,0=(1/2)∫_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+∫_𝑆𝑉𝜙₀^𝑝∫_𝑆ℎ𝜙₀^𝑝≥(1/2)∫_𝑆∇_𝜔𝜓⋅∇_𝑝,𝜔𝜓+∫_𝑆𝑉𝜓^𝑝∫_𝑆ℎ𝜓^𝑝. (20) Otherwise, by the definition of 𝜆_ℎ,0, 𝜆_ℎ,0≤(1/2)∫_𝑆∇_𝜔𝜓⋅∇_𝑝,𝜔𝜓+∫_𝑆𝑉𝜓^𝑝∫_𝑆ℎ𝜓^𝑝. (21) Thus, 𝜆_ℎ,0=(1/2)∫_𝑆∇_𝜔𝜓⋅∇_𝑝,𝜔𝜓+∫_𝑆𝑉𝜓^𝑝∫_𝑆ℎ𝜓^𝑝. (22) It follows from Theorem 3 that (𝜆_ℎ,0,𝜓) is an indefinite eigenpair. Now it suffices to show that 𝜓>0 in 𝑆. Suppose, to the contrary, that 𝜓(𝑥₀)=0 for some 𝑥₀∈𝑆. It will be shown that 𝜓≡0. Since 𝜓 is an eigenvalue, it follows from (1) that _𝑦∈𝑆𝜓𝑦^𝑝−2𝜓𝑦𝜔𝑥₀,𝑦=0 (23) and thus 𝜓(𝑦)=0 for all 𝑦∼𝑥₀ where 𝑦∼𝑥 means that two vertices 𝑥 and 𝑦 are connected by an edge. By repeating the above process for 𝑦∼𝑥₀, we conclude that 𝜓(𝑧)=0 for each 𝑧∼𝑦. Since the network 𝑆 is assumed to be connected, 𝜓(𝑥)=0 for all 𝑥∈𝑆.

Using the above theorem, we prove the simplicity of 𝜆_ℎ,0 as follows.

Theorem 5. If (𝜆_ℎ,0,𝜙₀) is an indefinite eigenpair for −ℒ^𝑉_𝑝,𝜔, then sgn𝜙₀(𝑥)=sgn𝜙₀𝑦,𝑥,𝑦∈𝑆. (24)

Proof. As shown in the proof for Theorem 4, if (𝜆_ℎ,0,𝜙₀) is indefinite, then (𝜆_ℎ,0,|𝜙₀|) is also an indefinite eigenpair. Let 𝜙₀(𝑥)∶=|𝜙₀(𝑥)| for all 𝑥 in 𝑆. Then we have 12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+_𝑆𝑉𝜙₀^𝑝=12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+_𝑆𝑉𝜙₀^𝑝, (25) which implies that _{𝑥,𝑦∈𝑆}𝜙₀𝑦−𝜙₀(𝑥)^𝑝𝑤𝑥,𝑦=_{𝑥,𝑦∈𝑆}𝜙₀𝑦−𝜙₀(𝑥)^𝑝𝑤𝑥,𝑦. (26) Since 𝜙₀𝑦−𝜙₀(𝑥)≥𝜙₀𝑦−𝜙₀(𝑥) (27) for all 𝑥∼𝑦 in 𝑆, we have 𝜙₀𝑦−𝜙₀(𝑥)=𝜙₀𝑦−𝜙₀(𝑥) (28) for all 𝑥, 𝑦 in 𝑆. Hence, either 𝜙₀(𝑥)=𝜙₀(𝑥) or 𝜙₀(𝑥)=−𝜙₀(𝑥) for all 𝑥 in 𝑆.

The above theorem shows that the dimension of the eigenspace corresponding to 𝜆_ℎ,0 is one. Thus, we have the following.

Corollary 6. The multiplicity of 𝜆_ℎ,0 is one.

For linear operators such as −Δ_𝜔 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^𝑉_𝑝,𝜔, 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 𝜆_ℎ,0 is isolated for a set of indefinite eigenvalues.

Theorem 7. The smallest eigenvalue 𝜆_ℎ,0 is isolated.

Proof. We proceed by contradiction. Suppose that for each 𝜖>0, there exists 𝑢_𝜖 satisfying ∫_𝑆|𝑢_𝜖|^𝑝=1 and −Δ_𝑝,𝜔𝑢_𝜖+𝑉𝑢_𝜖^𝑝−2𝑢_𝜖=𝜆_ℎ+𝜖ℎ𝑢_𝜖^𝑝−2𝑢_𝜖,in𝑆𝑢_𝜖=0,on𝜕𝑆. (29) Since the multiplicity of 𝜆_ℎ,0 is one, there exists an eigenfunction 𝜙₀ corresponding to 𝜆_ℎ,0 with 𝜙₀ in 𝑆 such that 𝑢_𝜖→𝜙₀>0 in 𝑆 as 𝜖→0. Hence, for sufficiently small 𝜖>0 we have 𝑢_𝜖>0 in 𝑆. Since Δ𝑉=^𝑝𝑢𝜖𝑢_𝜖^𝑝−1+𝜆_ℎ,0+𝜖ℎin𝑆, (30) we have −Δ_𝑝𝜙₀+𝜙₀^𝑝−1𝑢_𝜖^𝑝−1Δ_𝑝𝑢_𝜖+𝜆_ℎ,0ℎ𝜙₀^𝑝−1+𝜖ℎ𝜙₀^𝑝−1=𝜆_ℎ,0ℎ𝜙₀^𝑝−1on𝑆. (31) That is, −𝜖𝜙₀^𝑝−1=−Δ_𝑝,𝜔𝜙₀−𝜙₀^𝑝−1𝑢_𝜖^𝑝−1−Δ_𝑝,𝜔𝑢_𝜖on𝑆. (32) Multiplying 𝜙₀ and integrating over 𝑆 on both sides (32) and using Picone's identity, we have a contradiction.

3.2. 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 𝑉∶𝑆→ℝ and a nonnegative source term 𝑔∶𝑆→[0,∞), we consider the following equation: −Δ_𝑝,𝜔𝑢+𝑉|𝑢|^𝑝−2𝑢−𝜆_ℎ,0ℎ|𝑢|^𝑝−2𝑢=𝑔in𝑆,𝑢=0on𝜕𝑆. (33) It is clear that the above equation has a solution (in fact, an eigenvalue) if 𝑔≡0 in 𝑆. The next result shows that if 𝜆_ℎ,0>0, then the converse of the statement also holds. Thus, there is no solution of the above equation if 𝑔 is nonzero in 𝑆.

Theorem 8 (resonance problem). Suppose that a function 𝑉 satisfies the condition that the smallest indefinite eigenvalue 𝜆_ℎ,0 is positive. Then (33) has a solution if and only if 𝑔≡0.

Proof. Suppose that a function 𝑢₀ is a solution to the equation and we define a function 𝑢₀ as 𝑢₀(𝑥)∶=max−𝑢₀(𝑥),0,𝑥∈𝑆. (34) Since it is obvious that if 𝑢₀≡0, then 𝑔≡0; we assume that 𝑢₀≢0. Then we have 0≤_𝑆𝑔𝑢₀=_𝑆−Δ_𝑝𝑢₀+𝑉𝑢₀^𝑝−2𝑢₀−𝜆_ℎ,0ℎ𝑢₀^𝑝−2𝑢₀𝑢₀1≤−2_{𝑥,𝑦∈𝑆}𝑢₀𝑦−𝑢₀(𝑥)^𝑝−_𝑥∈𝑆𝑣(𝑥)𝑢₀(𝑥)^𝑝+𝜆_{ℎ,0𝑥∈𝑆}ℎ(𝑥)𝑢₀(𝑥)^𝑝, (35) which implies that 𝑢₀≡𝑘𝜙₀ for some 𝑘≥0. If 𝑘>0 then 𝑢₀ is an eigenfunction corresponding to 𝜆_ℎ,0 so that 𝑔≡0. Now suppose that 𝑘=0 so 𝑢₀≥0. Since 𝑢₀≢0 and 𝑔≥0, we have −Δ_𝑝,𝜔𝑢₀+𝑣𝑢₀^𝑝−2𝑢₀≥0,but−Δ_𝑝,𝜔𝑢₀+𝑣𝑢₀^𝑝−2𝑢₀≢0in𝑆. (36) Thus, by using a similar method that we used in the proof for Theorem 4, it is easy to we show that the solution 𝑢₀ is positive in 𝑆. Using Picone's identity, we have 10≤2_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀−∇_𝜔𝜙^𝑝₀𝑢₀^𝑝−1⋅∇_𝑝,𝜔𝑢₀=12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀−_𝑆𝜙^𝑝₀𝑢₀^𝑝−1−𝑣𝑢₀^𝑝−1+𝜆₀ℎ𝑢₀^𝑝−1=1+𝑔2_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+_𝑆𝑣𝜙^𝑝₀−𝜆_0𝑆ℎ𝜙^𝑝₀−_𝑆𝜙^𝑝₀𝑢₀^𝑝−1𝑔 (37) which implies that 𝑔≡0.

The next theorem is the antiminimum principle. From it, we see that each (nonconstant) solution for the following equation −Δ_𝑝,𝜔𝑢+𝑉|𝑢|^𝑝−2𝑢−𝜆ℎ|𝑢|^𝑝−2𝑢≥0in𝑆𝑢=0on𝜕𝑆. (38) has its minimum in 𝑆 if 𝜆>𝜆_ℎ,0.

Theorem 9 (antiminimum principle). For a nonnegative source term 𝑔∶𝑆→[0,∞), suppose 𝑢_𝜆 is a solution to the following equation: −Δ_𝑝,𝜔𝑢+𝑉|𝑢|^𝑝−2𝑢−𝜆ℎ|𝑢|^𝑝−2𝑢=𝑔𝑖𝑛𝑆𝑢=0𝑜𝑛𝜕𝑆. (39) If 𝜆>𝜆_ℎ,0, then 𝑢_𝜆(𝑥₀)<0 for some 𝑥₀∈𝑆.

Proof. By virtue of Theorem 8, it suffices to show that if there exist a nonnegative solution 𝑢_𝜆 for (75), then 𝜆<𝜆_ℎ,0. Suppose 𝑢_𝜆 is a solution to (75) with 𝑢_𝜆(𝑥)≥0, 𝑥∈𝑆. Using a similar method that we used in the proof of Theorem 4, we can easily show that if 𝑢_𝜆(𝑥₀)=0 for some 𝑥₀∈𝑆, then 𝑢_𝜆≡0. Thus, we may assume that 𝑢_𝜆 is positive in 𝑆. By Picone’s identity, we have 10≤2_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀−∇_𝜔𝜙^𝑝₀𝑢_𝜆^𝑝−1⋅∇_𝑝,𝜔𝑢_𝜆=12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀−𝜙^𝑝₀𝑢_𝜆^𝑝−1−Δ_𝑝,𝜔𝑢_𝜆=12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀−𝜙^𝑝𝑢_𝜆^𝑝−1−𝑉𝑢_𝜆^𝑝−1+𝜆ℎ𝑢_𝜆^𝑝−1+𝑔, (40) where 𝜙₀ is the positive eigenfunction corresponding to 𝜆_ℎ,0. Thus, we have 0<𝜆_ℎ,0−𝜆_𝑆ℎ𝜙^𝑝₀. (41) Since ∫_𝑆ℎ𝜙^𝑝₀>0, we finally have 𝜆_ℎ,0>𝜆, 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) 𝜔 of the network by the smallest indefinite eigenvalue 𝜆_ℎ,0 for −ℒ_𝑝,𝜔 with respect to ℎ. Note that the uniqueness of the conductivity 𝜔 is not guaranteed by 𝜆_ℎ,0. This implies that there can be different conductivities 𝜔₁ and 𝜔₂ on the edges which induces the same eigenvalue 𝜆_ℎ,0 for the operators −ℒ_{𝑝,𝜔𝑖},𝑖=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 𝜔₁ and 𝜔₂ on the edges satisfying 𝜔₁≤𝜔₂ in 𝑆×𝑆 which induce the same smallest indefinite eigenvalue 𝜆_ℎ,0.

Theorem 10 (inverse conductivity problem). For networks 𝐺(𝑆,𝐸_𝑖,𝜔_𝑖) for 𝑖=1,2, let 𝜆^𝜔𝑖_ℎ,0 be the smallest indefinite eigenvalue for −ℒ^𝑉_{𝑝,𝜔𝑖}. If the weight functions satisfy 𝜔₁≤𝜔₂𝑖𝑛𝑆×𝑆, (42) then one has 𝜆^𝜔1_ℎ,0≤𝜆^𝜔2_ℎ,0. (43) Moreover, 𝜆^𝜔1_ℎ,0=𝜆^𝜔2_ℎ,0 if and only if one has(i)𝜙₁=𝜙₂ on 𝑆,(ii)𝜔₁(𝑥,𝑦)=𝜔₂(𝑥,𝑦) whenever 𝜙₁(𝑥)≠𝜙₁(𝑦) or 𝜙₂(𝑥)≠𝜙₂(𝑦)where 𝜙_𝑖 is the eigenfunction corresponding to 𝜆^𝜔𝑖_ℎ,0, 𝑖=1,2.

Proof. By definition of the smallest eigenvalue, we have 𝜆^𝜔1_0,ℎ≤(1/2)∫_𝑆∇_𝑝,𝑤1𝜙₂⋅∇_𝑤1𝜙₂+∫_𝑆𝑣𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝. (44) It follows from 𝜔₁≤𝜔₂ that (1/2)∫_𝑆∇_𝑝,𝑤1𝜙₂⋅∇_𝑤1𝜙₂+∫_𝑆𝑣𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝≤(1/2)∫_𝑆∇_𝑝,𝑤2𝜙₂⋅∇_𝑤2𝜙₂+∫_𝑆𝑣𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝. (45) Hence we have 𝜆^𝜔1_ℎ,0≤𝜆^𝜔2_ℎ,0. Now, we suppose that 𝜆^𝜔1_ℎ,0=𝜆^𝜔2_ℎ,0. Then 𝜆^𝜔2_ℎ,0=(1/2)∫_𝑆∇_𝑝,𝑤2𝜙₂⋅∇_𝑤2𝜙₂+∫_𝑆𝑉𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝≥(1/2)∑_{𝑥,𝑦∈𝑆}𝜙₂𝑦−𝜙₂(𝑥)^𝑝𝜔₂𝑥,𝑦−𝜔₁𝑥,𝑦∫_𝑆ℎ𝜙₂^𝑝+𝜆^𝜔1_ℎ,0. (46) Since 𝜆^𝜔1_ℎ,0=𝜆^𝜔2_ℎ,0, we have ()∑0≥1/2_{𝑥,𝑦∈𝑆}𝜙₂𝑦−𝜙₂(𝑥)^𝑝𝜔₂𝑥,𝑦−𝜔₁𝑥,𝑦∫_𝑆ℎ𝜙₂^𝑝. (47) Thus 𝜔₁(𝑥,𝑦)=𝜔₂(𝑥,𝑦) whenever 𝜙₂(𝑥)≠𝜙₂(𝑦), which implies that (1/2)∫_𝑆∇_𝑝,𝑤1𝜙₂⋅∇_𝑤1𝜙₂+∫_𝑆𝑉𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝≤𝜆^𝜔1_ℎ,0. (48) Thus 𝜙₁≡𝜙₂. Hence 𝜔₁(𝑥,𝑦)=𝜔₂(𝑥,𝑦) whenever 𝜙_𝑖(𝑥)≠𝜙_𝑖(𝑦), 𝑖=1,2. If 𝜔₁(𝑥,𝑦)=𝜔₂(𝑥,𝑦) whenever 𝜙_𝑖(𝑥)≠𝜙_𝑖(𝑦), 𝑖=1,2, then 𝜆^𝜔1_ℎ,0=(1/2)∑_{𝑥,𝑦∈𝑆}𝜙₁𝑦−𝜙₁(𝑥)^𝑝𝜔₁𝑥,𝑦+∫_𝑆𝑉𝜙₁^𝑝∫_𝑆ℎ𝜙₁^𝑝=(1/2)∑_{𝑥,𝑦∈𝑆}𝜙₁𝑦−𝜙₁(𝑥)^𝑝𝜔₂𝑥,𝑦+∫_𝑆𝑉𝜙₁^𝑝∫_𝑆ℎ𝜙₁^𝑝≥𝜆^𝜔2_ℎ,0. (49) Thus we have 𝜆^𝜔1_ℎ,0=𝜆^𝜔2_ℎ,0.

4. 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∶𝑆→ℝ satisfies ℎ⁺≢0,ℎ⁻≢0, (50) where ℎ⁺(𝑥)∶=max{ℎ(𝑥),0},ℎ⁻(𝑥)∶=−min{ℎ(𝑥),0} (51) for 𝑥∈𝑆.

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 𝜆⁺_ℎ,0 and 𝜆⁻_ℎ,0 defined by 𝜆⁺_ℎ,0∶=inf_{∫𝑆ℎ|𝑢|𝑝>0}(1/2)∫_𝑆∇_𝜔𝑢⋅∇_𝑝,𝜔𝑢+∫_𝑆𝑉|𝑢|^𝑝∫_𝑆ℎ|𝑢|^𝑝,𝜆⁻_ℎ,0∶=sup_{∫𝑆ℎ|𝑢|𝑝<0}(1/2)∫_𝑆∇_𝜔𝑢⋅∇_𝑝,𝜔𝑢+∫_𝑆𝑉|𝑢|^𝑝∫_𝑆ℎ|𝑢|^𝑝. (52)

Theorem 11. If functions 𝑉∶𝑆→ℝ and ℎ∶𝑆→ℝ satisfy either 𝑉≥0 or 𝑉≥ℎ in 𝑆, then there exists 𝜙₀∈𝐴 such that 𝜆⁺_ℎ,0=(1/2)∫_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+∫_𝑆𝑉𝜙₀^𝑝∫_𝑆ℎ𝜙₀^𝑝, (53) where 𝐴∶=𝑢∶𝑆⟶ℝ∣_𝑆ℎ|𝑢|^𝑝>0,𝑢|_𝜕𝑆=0. (54) Moreover 𝜆⁺_ℎ,0 is the smallest positive eigenvalue for −ℒ^𝑉_𝑝,𝜔 and 𝜙₀ is an eigenfunction corresponding to 𝜆⁺_ℎ,0.

Proof. Define 𝐴₁∶=𝑢∶𝑆→ℝ∣_𝑆|𝑢|^𝑝=1. (55) We note that 𝐴∩𝐴₁ is not compact and its boundary 𝜕(𝐴∩𝐴₁) is given by 𝜕𝐴∩𝐴₁=𝑢∶𝑆⟶ℝ∣_𝑆|𝑢|^𝑝=1,_𝑆ℎ|𝑢|^𝑝=0,𝑢|_𝜕𝑆=0, (56) so 𝐴∩𝐴₁ is compact. Now we take {𝑢_𝑛}^∞_𝑛=1⊂𝐴∩𝐴₁ such that {𝑢_𝑛} converges at some point 𝑢₀∈𝜕(𝐴∩𝐴₁). Since {𝑢_𝑛} converges, (1/2)∫_𝑆∇_𝜔𝑢_𝑛⋅∇_𝑝,𝜔𝑢_𝑛+∫_𝑆𝑉|𝑢_𝑛|^𝑝 also converges. From ∫_𝑆ℎ|𝑢_𝑛|→0 and the function 𝑉 which satisfies either 𝑉≥0 or 𝑉≥ℎ, we easily show that (1/2)∫_𝑆∇_𝜔𝑢_𝑛⋅∇_𝑝𝑢_𝑛+∫_𝑆𝑉𝑢_𝑛^𝑝∫_𝑆ℎ𝑢_𝑛^𝑝↗∞. (57) Therefore, there exists 𝜙₀∈𝐴∩𝐴₁ such that (1/2)∫_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+∫_𝑆𝑉𝜙₀^𝑝∫_𝑆ℎ𝜙₀^𝑝=min_{𝜙∈𝐴∩𝐴1}(1/2)∫_𝑆∇_𝜔𝜙⋅∇_𝑝,𝜔𝜙+∫_𝑆𝑉𝜙^𝑝∫_𝑆ℎ𝜙^𝑝. (58) Now we take an arbitrary 𝑥₀∈𝑆. Since ∫_𝑆ℎ|𝜙₀+𝑡𝛿_𝑥0|^𝑝≠0 for sufficiently small 𝑡>0, by definition of 𝜆⁺_ℎ,1, we have 𝜆⁺_ℎ,0≤12_𝑆∇_𝜔𝜙₀+𝑡𝛿_𝑥0⋅∇_𝑝,𝜔𝜙₀+𝑡𝛿_𝑥0+_𝑆𝑉𝜙₀+𝑡𝛿_𝑥0^𝑝×_𝑆ℎ𝜙₀+𝑡𝛿_𝑥0^𝑝−1. (59) Thus, 0≤_𝑆12∇_𝜔𝜙₀+𝑡𝛿_𝑥0⋅∇_𝑝𝜙₀+𝑡𝛿_𝑥0+_𝑆𝑉𝜙₀+𝑡𝛿_𝑥0^𝑝−𝜆⁺_ℎ,0ℎ𝜙₀+𝑡𝛿_𝑥0^𝑝 (60) for a sufficiently small 𝑡>0. The right-hand side is continuously differentiable with respect to 𝑡 and is equal to zero at 𝑡=0. Thus, 𝑑0=𝑑𝑡_𝑆12∇_𝜔𝜙₀+𝑡𝛿_𝑥0⋅∇_𝑝,𝜔𝜙₀+𝑡𝛿_𝑥0+_𝑆𝑉𝜙₀+𝑡𝛿_𝑥0^𝑝−𝜆⁺_ℎ,0ℎ𝜙₀+𝑡𝛿_𝑥0^𝑝_𝑡=0=−𝑝_{𝑥,𝑦∈𝑆}𝜙₀𝑦−𝜙₀(𝑥)^𝑝−2𝜙₀𝑦−𝜙₀(𝑥)𝛿_𝑥0(𝑥)𝜔𝑥,𝑦+𝑝_𝑥∈𝑆𝑉(𝑥)𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥)𝛿_𝑥0(𝑥)+𝑝_𝑥∈𝑆−𝜆⁺_ℎ,0ℎ𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥)𝛿_𝑥0(𝑥)=𝑝−Δ_𝑝𝜙₀𝑥₀+𝑉(𝑥)𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥)−𝜆⁺_ℎ,0ℎ𝜙₀𝑥₀^𝑝−2𝜙₀𝑥₀𝑥₀. (61) Since the above equations hold for an arbitrary 𝑥₀∈𝑆, we have −Δ_𝑝,𝜔𝜙⁺₀(𝑥)+𝑉(𝑥)𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥)=𝜆⁺_ℎ,0ℎ𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥),𝑥∈𝑆. (62)

Theorem 12. For a function 𝑉∶𝑆→ℝ and ℎ∶𝑆→ℝ satisfying either 𝑉≥0 or 𝑉≤ℎ in 𝑆, there exists 𝜙₀∈𝐵 such that 𝜆⁻_ℎ,0=(1/2)∫_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+∫_𝑆𝑉𝜙₀^𝑝∫_𝑆ℎ𝜙₀^𝑝. (63) where 𝐵∶=𝑢∶𝑆→ℝ∣_𝑆ℎ|𝑢|^𝑝<0,𝑢|_𝜕𝑆=0. (64) Moreover 𝜆⁻_ℎ,0 is the largest negative eigenvalue for −ℒ^𝑉_𝑝,𝜔 and 𝜙₀ is an eigenfunction corresponding to 𝜆⁻_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 𝑉≥0 or 𝑉≡ℎ, then there exist 𝜆⁺_ℎ,0 and 𝜆⁻_ℎ,0 at the same time. The specific case of 𝑉≡0 was dealt with in [9].

In the following results, we give some properties of 𝜆⁺_ℎ,0 and its eigenfunction. One also can get similar results for 𝜆⁻_ℎ,0 and its eigenfunction, assuming that the function 𝑉 satisfies either 𝑉≥0 or 𝑉≤ℎ.

Theorem 13. For a function 𝑉∶𝑆→ℝ and a weight function ℎ satisfying either 𝑉≥0 or 𝑉≥ℎ in 𝑆, there exists a positive eigenfunction 𝜙₀ corresponding to the indefinite eigenvalue 𝜆⁺_ℎ,0 for −ℒ^𝑉_𝑝,𝜔.

Proof. It follows from Theorem 11 that there exists an indefinite eigenfunction 𝜙₀ satisfying −Δ_𝑝,𝜔𝜙₀(𝑥)+𝑉(𝑥)𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥)=𝜆⁺_ℎ,0ℎ𝜙₀(𝑥)^𝑝−2𝜙₀(𝑥) (65) for 𝑥∈𝑆. Let 𝜓(𝑥)∶=|𝜙₀(𝑥)| for all 𝑥 in 𝑆. Then ∫_𝑆|𝜓|^𝑝=∫_𝑆|𝜙₀|^𝑝 and 12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+_𝑆𝑉𝜙₀^𝑝≥12_𝑆∇_𝜔𝜓⋅∇_𝑝,𝜔𝜓+_𝑆𝑉𝜓^𝑝. (66) Thus, we have 𝜆⁺_ℎ,0=(1/2)∫_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+∫_𝑆𝑉𝜙₀^𝑝∫_𝑆ℎ𝜙₀^𝑝≥(1/2)∫_𝑆∇_𝜔𝜓⋅∇_𝑝,𝜔𝜓+∫_𝑆𝑉𝜓^𝑝∫_𝑆ℎ𝜓^𝑝. (67) Otherwise, by definition of 𝜆⁺_ℎ,1, 𝜆⁺_ℎ,0≤(1/2)∫_𝑆∇_𝜔𝜓⋅∇_𝑝,𝜔𝜓+∫_𝑆𝑉𝜓^𝑝∫_𝑆ℎ𝜓^𝑝. (68) Thus, 𝜆⁺_ℎ,0=(1/2)∫_𝑆∇_𝜔𝜓⋅∇_𝑝,𝜔𝜓+∫_𝑆𝑉𝜓^𝑝∫_𝑆ℎ𝜓^𝑝. (69) It follows from Theorem 11 that (𝜆⁺_ℎ,0,𝜓) is an indefinite Dirichlet eigenpair. Next we show that 𝜓(𝑥)>0 for all 𝑥 in 𝑆. It is sufficient to prove that if there exists 𝑥₀ in 𝑆 such that 𝜓(𝑥₀)=0, then 𝜓≡0. Since (𝜆⁺_ℎ,0,𝜓) is an indefinite Dirichlet eigenpair, it satisfies (1). This implies that _𝑦∈𝑆𝜓𝑦^𝑝−2𝜓𝑦𝜔𝑥₀,𝑦=0. (70) Hence 𝜓(𝑦)=0 for all 𝑦∼𝑥₀. By repeating the above process for 𝑦∼𝑥₀, we conclude that 𝜓(𝑧)=0 for each 𝑧∼𝑦. Since 𝐺 is a connected network, 𝜓(𝑥)=0 for all 𝑥∈𝐺. However, this contradicts the fact that 𝜙≢0. Thus, 𝜓(𝑥)>0 for all 𝑥 in 𝐺.

Corollary 14. For a function 𝑉∶𝑆→ℝ with 𝑉≤0 in 𝑆 or 𝑉≤ℎ in 𝑆, if 𝜙₀ is an eigenfunction corresponding to the eigenvalue 𝜆⁺_ℎ,0 for −ℒ^𝑉_𝑝,𝜔 with respect to ℎ, then sgn𝜙₀(𝑥)=sgn𝜙₀(𝑦) for all 𝑥, 𝑦 in 𝑆.

Proof. Let 𝜙₀ be an eigenfunction corresponding to 𝜆⁺_ℎ,0. Then by Theorem 13, |𝜙₀| is also an eigenfunction corresponding to 𝜆⁺_ℎ,0. Therefore, we have 12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+_𝑆𝑉𝜙₀^𝑝=12_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀+_𝑆𝑉𝜙₀^𝑝. (71) This implies that |𝜙₀(𝑦)−𝜙₀(𝑥)|=||𝜙₀(𝑦)|−|𝜙₀(𝑥)|| for all 𝑥,𝑦∈𝑆. Thus, we have 𝜙₀(𝑥)=|𝜙₀(𝑥)|, 𝑥∈𝑆.

Corollary 15. The multiplicity of 𝜆⁺_ℎ,0 is one.

4.2. Resonance Problems, Antiminimum Principle, and Inverse Problems

As previously mentioned, throughout this section the weight function ℎ is assumed to have both positive and negative values in 𝑆. The next theorem shows that even in this case, we can solve a resonance problem similar to that in Theorem 8.

Theorem 16 (resonance problems). Suppose that a function 𝑉 satisfies 𝜆⁺_ℎ,0>0. For 𝑔∶𝑆→[0,∞), the equation −Δ_𝑝,𝜔𝑢+𝑉|𝑢|^𝑝−2𝑢−𝜆⁺_ℎ,0ℎ|𝑢|^𝑝−2𝑢=𝑔𝑖𝑛𝑆,𝑢=0𝑜𝑛𝜕𝑆 (72) has a solution if and only if 𝑔≡0. Moreover, the solutions are eigenfunctions corresponding to 𝜆⁺_ℎ,0.

Proof. Suppose that a function 𝑢₀ is a solution to (72). If 𝑢₀≡0, then we have 𝑔≡0. Suppose 𝑢₀≢0 and set a function 𝑢₀ as 𝑢₀(𝑥)∶=max{−𝑢₀(𝑥),0} for all 𝑥∈𝑆. Since 𝑢(𝑥)=0 for all 𝑥∈𝜕𝑆, 𝑢₀(𝑥)=0 for all 𝑥∈𝜕𝑆. Since 𝑢₀ is a solution of (72), we have 0≤_𝑆𝑔𝑢₀=_𝑆−Δ_𝑝𝑢₀+𝑉𝑢₀^𝑝−2𝑢₀−𝜆₀ℎ𝑢₀^𝑝−2𝑢₀𝑢₀1≤−2_{𝑥,𝑦∈𝑆}−𝑢₀𝑦−−𝑢₀(𝑥)^𝑝−_𝑥∈𝑆𝑉(𝑥)−𝑢₀(𝑥)^𝑝+𝜆⁺_{ℎ,0𝑥∈𝑆}ℎ(𝑥)−𝑢₀(𝑥)^𝑝 (73) which implies that 𝑢₀≡𝑘𝜙₀ for some 𝑘≥0. Assume 𝑘>0; then 𝑢₀ is an eigenfunction corresponding to 𝜆⁺_ℎ,0 so that 𝑔≡0. Now, assume 𝑘=0. Then 𝑢₀≥0. Suppose 𝑢₀(𝑥₀)=0 for some 𝑥₀∈𝑆. Then we have −Δ_𝑝,𝜔𝑢₀(𝑥₀)=𝑔(𝑥₀). Since 𝑔 is a nonnegative function, we have 𝑢₀(𝑦)=0 for 𝑦∼𝑥₀, so 𝑢₀≡0. This presents a contradiction. Thus, 𝑢₀(𝑥)>0, 𝑥∈𝑆. Let 𝜙₀ be a positive eigenfunction corresponding to 𝜆⁺_ℎ,0. By Picone’s identity, 10≤2_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀−∇_𝜔𝜙^𝑝₀𝑢₀^𝑝−1⋅∇_𝑝,𝜔𝑢₀=_𝑆−Δ_𝑝𝜙₀𝜙₀−−Δ_𝑝𝜙₀𝜙^𝑝₀𝑢₀^𝑝−1=_𝑆−Δ_𝑝𝜙₀𝜙₀−𝜆⁺_ℎ,0−𝑉𝑢₀^𝑝−1𝜙+𝑔^𝑝0𝑢₀^𝑝−1=_𝑆−Δ_𝑝𝜙₀+𝑉𝜙^𝑝₀−_𝑆𝜆⁺_ℎ,0ℎ𝜙^𝑝₀−_𝑆𝑔𝜙^𝑝₀𝑢₀^𝑝−1 (74) which implies that 𝑔≡0.

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 𝑆.

Theorem 17 (antiminimum principle). Let a function 𝑉∶𝑆→ℝ and a weight function ℎ with 𝑉≥0 in 𝑆 or 𝑉≥ℎ in 𝑆 be given. For a nonnegative source term 𝑔∶𝑆→[0,∞), suppose 𝑢_𝜆 is a solution to the following equation: −Δ_𝑝,𝜔𝑢+𝑉|𝑢|^𝑝−2𝑢−𝜆ℎ|𝑢|^𝑝−2𝑢=𝑔𝑖𝑛𝑆𝑢=0𝑜𝑛𝜕𝑆. (75) If 𝜆>𝜆⁺_ℎ,0, then 𝑢_𝜆(𝑥₀)<0 for some 𝑥₀∈𝑆.

Proof. By virtue of Theorem 16, it suffices to show that if there exists a nonnegative solution 𝑢_𝜆 of (75) then 𝜆<𝜆⁺_ℎ,0. Suppose 𝑢_𝜆(𝑥₀)=0 for some 𝑥₀∈𝑆. Then we have −Δ_𝑝,𝜔𝑢₀(𝑥₀)=𝑔(𝑥₀). Since 𝑔 is a nonnegative function, we have 𝑢₀(𝑦)=0 for 𝑦∼𝑥₀, so 𝑢₀≡0. This is a contradiction to the assumption. Thus, we have 𝑢_𝜆>0 in 𝑆. Let 𝜙₀ be a positive eigenfunction corresponding to 𝜆⁺_ℎ,0. By Picone's identity, 10≤2_𝑆∇_𝜔𝜙₀⋅∇_𝑝,𝜔𝜙₀−∇_𝜔𝜙^𝑝₀𝑢_𝜆^𝑝−1⋅∇_𝑝,𝜔𝑢_𝜆=_𝑆−Δ_𝑝𝜙₀𝜙₀−−Δ_𝑝𝜙₀𝜙^𝑝₀𝑢_𝜆^𝑝−1=_𝑆−Δ_𝑝𝜙₀𝜙₀−_𝑆−𝑉𝑢_𝜆^𝑝−1+𝜆ℎ𝑢_𝜆^𝑝−1𝜙+𝑔^𝑝0𝑢_𝜆^𝑝−1=_𝑆𝜆⁺_ℎ,0ℎ𝜙^𝑝₀−_𝑆𝜆ℎ𝜙^𝑝₀−_𝑆𝑔𝜙^𝑝₀𝑢₀^𝑝−1. (76) Since ∫_𝑆𝑔(𝜙^𝑝₀/𝑢₀^𝑝−1)>0, we have 0<𝜆⁺_ℎ,0−𝜆_𝑆ℎ𝜙^𝑝₀. (77)

Finally, we deal with inverse conductivity problems for 𝜆⁺_ℎ,0 and 𝜆⁻_ℎ,0.

Theorem 18 (inverse conductivity problem). For networks 𝐺(𝑆,𝐸_𝑖,𝜔_𝑖), 𝑖=1,2, let 𝜆⁺_0,𝜔𝑖 be the smallest positive indefinite eigenvalue for −ℒ^𝑉_{𝑝,𝜔𝑖}. One can suppose that the given functions 𝑉 and ℎ satisfy either 𝑉≥0 or 𝑉≥ℎ. If the weight functions satisfy 𝜔₁≤𝜔₂𝑖𝑛𝑆×𝑆, (78) then one has 𝜆⁺_0,𝜔1≤𝜆⁺_0,𝜔2. (79) Moreover, 𝜆⁺_0,𝜔1=𝜆⁺_0,𝜔2 if and only if one has(i)𝜙₁=𝜙₂ on 𝑆,(ii)𝜔₁(𝑥,𝑦)=𝜔₂(𝑥,𝑦) whenever 𝜙₁(𝑥)≠𝜙₁(𝑦) or 𝜙₂(𝑥)≠𝜙₂(𝑦),where 𝜙_𝑖 is the eigenfunction corresponding to 𝜆⁺_0,𝜔𝑖, 𝑖=1,2.

Proof. Let 𝜙⁺_𝑖 be an eigenfunction corresponding to 𝜆⁺_0,𝜔𝑖 for 𝑖=1,2. By the definition of the smallest eigenvalue, we have 𝜆⁺_0,𝜔1≤(1/2)∫_𝑆∇_𝑝,𝑤1𝜙₂⋅∇_𝑤1𝜙₂+∫_𝑆𝑉𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝. (80) It follows from 𝜔₁≤𝜔₂ that (1/2)∫_𝑆∇_𝑝,𝑤1𝜙₂⋅∇_𝑤1𝜙₂+∫_𝑆𝑉𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝≤(1/2)∫_𝑆∇_𝑝,𝑤2𝜙₂⋅∇_𝑤2𝜙₂+∫_𝑆𝑉𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝=𝜆⁺_0,𝜔2. (81) Thus, we obtain 𝜆⁺_0,𝜔1≤𝜆⁺_0,𝜔2. Now, we suppose that 𝜆⁺_0,𝜔1=𝜆⁺_0,𝜔2. Then 𝜆⁺_0,𝜔2=(1/2)∫_𝑆∇_𝑝,𝑤2𝜙₂⋅∇_𝑤2𝜙₂+∫_𝑆𝑉𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝≥(1/2)∑_{𝑥,𝑦∈𝑆}𝜙₂𝑦−𝜙₂(𝑥)^𝑝𝜔₂𝑥,𝑦−𝜔₁𝑥,𝑦∫_𝑆ℎ𝜙₂^𝑝+𝜆⁺_0,𝜔1. (82) Since ∑_𝑥∈𝑆ℎ|𝜙₂|^𝑝>0, we have |𝜙₂(𝑦)−𝜙₂(𝑥)|^𝑝(𝜔₂(𝑥,𝑦)−𝜔₁(𝑥,𝑦))=0, 𝑥,𝑦∈𝑆. Then 𝜔₁(𝑥,𝑦)=𝜔₂(𝑥,𝑦) whenever 𝜙₂(𝑥)≠𝜙₂(𝑦). This implies that 𝜆⁺_0,𝜔1=𝜆⁺_0,𝜔2=(1/2)∫_𝑆∇_𝑝,𝑤2𝜙₂⋅∇_𝑤2𝜙₂+∫_𝑆𝑉𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝=(1/2)∫_𝑆∇_𝑝,𝑤2𝜙₂⋅∇_𝑤1𝜙₂+∫_𝑆𝑉𝜙₂^𝑝∫_𝑆ℎ𝜙₂^𝑝. (83) Hence, 𝜙₂ is an eigenfunction of 𝜆⁺_0,𝜔1. Since 𝜆⁺_0,𝜔1 is simple, we have 𝜙₂≡𝜙₁. Therefore, 𝜔₁(𝑥,𝑦)=𝜔₂(𝑥,𝑦) whenever 𝜙_𝑖(𝑥)≠𝜙_𝑖(𝑦), 𝑖=1,2.
If 𝜔₁(𝑥,𝑦)=𝜔₂(𝑥,𝑦) whenever 𝜙_𝑖(𝑥)≠𝜙_𝑖(𝑦), 𝑖=1,2, then 𝜆⁺_0,𝜔1=(1/2)∑_{𝑥,𝑦∈𝑆}𝜙₁𝑦−𝜙₁(𝑥)^𝑝𝜔₁𝑥,𝑦+∫_𝑆𝑉𝜙₁^𝑝∫_𝑆ℎ𝜙₁^𝑝=(1/2)∑_{𝑥,𝑦∈𝑆}𝜙₁𝑦−𝜙₁(𝑥)^𝑝𝜔₂𝑥,𝑦+∫_𝑆𝑉𝜙₁^𝑝∫_𝑆ℎ𝜙₁^𝑝≥𝜆⁺_0,𝜔2. (84) Thus, we have 𝜆⁺_0,𝜔1=𝜆⁺_0,𝜔2.

Theorem 19 (inverse conductivity problem). For networks 𝐺(𝑆,𝐸_𝑖,𝜔_𝑖), 𝑖=1,2, let 𝜆⁻_0,𝜔𝑖 be the largest negative indefinite eigenvalue for −ℒ^𝑉_{𝑝,𝜔𝑖}. One can suppose that the given functions 𝑉 and ℎ satisfy either 𝑉≥0 or 𝑉≤ℎ. If the weight functions satisfy 𝜔₁≤𝜔₂𝑖𝑛𝑆×𝑆, (85) then one has 𝜆⁻_0,𝜔1≥𝜆⁻_0,𝜔2. (86) Moreover, 𝜆⁻_0,𝜔1=𝜆⁻_0,𝜔2 if and only if one has(i)𝜙₁=𝜙₂ on 𝑆,(ii)𝜔₁(𝑥,𝑦)=𝜔₂(𝑥,𝑦) whenever 𝜙₁(𝑥)≠𝜙₁(𝑦) or 𝜙₂(𝑥)≠𝜙₂(𝑦),where 𝜙_𝑖 is the eigenfunction corresponding to 𝜆⁻_0,𝜔𝑖, 𝑖=1,2.

Proof. The proof is similar to that in Theorem 18 and we thus omit it.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MOE) (no. 2012R1A1A2004689).