Various Half-Eigenvalues of Scalar 𝑝-Laplacian with Indefinite Integrable Weights

Abstract

Consider the half-eigenvalue problem (𝜙_𝑝(𝑥^′))^′+𝜆𝑎(𝑡)𝜙_𝑝(𝑥₊)−𝜆𝑏(𝑡)𝜙_𝑝(𝑥₋)=0 a.e. 𝑡∈[0,1], where 1<𝑝<∞, 𝜙_𝑝(𝑥)=|𝑥|^𝑝−2𝑥, 𝑥_±(⋅)=max{±𝑥(⋅),0} for 𝑥∈𝒞⁰∶=𝐶([0,1],ℝ), and 𝑎(𝑡) and 𝑏(𝑡) are indefinite integrable weights in the Lebesgue space ℒ^𝛾∶=𝐿^𝛾([0,1],ℝ),1≤𝛾≤∞. We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in (𝑎,𝑏)∈(ℒ^𝛾,𝑤_𝛾)², where 𝑤_𝛾 denotes the weak topology in ℒ^𝛾 space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in (𝑎,𝑏)∈(ℒ^𝛾,‖⋅‖_𝛾)², where ‖⋅‖_𝛾 is the 𝐿^𝛾 norm of ℒ^𝛾.

1. Introduction

Fučik spectrum and half-eigenvalues are useful for solving problems with "jumping nonlinearities." Compared with Fučik spectrum, half-eigenvalues have not been paid much research. However, as half-eigenvalues are concerned with only one parameter, we think the theory on half-eigenvalues will lead to better knowledge for Fučik spectrum and jumping nonlinearities.

Given an exponent 𝑝∈(1,∞), let 𝜙_𝑝(𝑥)∶=|𝑥|^𝑝−2𝑥 for 𝑥∈ℝ which is used to define the scalar 𝑝-Laplacian. Denote 𝑥_±=max{±𝑥,0} for 𝑥∈ℝ and 𝑥_±(⋅)=max{±𝑥(⋅),0} for any 𝑥∈𝒞⁰∶=𝐶([0,1],ℝ). For a pair of indefinite (sign-changing) integrable weights (𝑎,𝑏) with 𝑎,𝑏∈ℒ^𝛾∶=𝐿^𝛾([0,1],ℝ), 1≤𝛾≤∞, we are concerned with the half-eigenvalue problem (𝜙_𝑝(𝑥^′))^′+𝜆𝑎(𝑡)𝜙_𝑝(𝑥₊)−𝜆𝑏(𝑡)𝜙_𝑝(𝑥₋)=0,a.e.𝑡∈[0,1], (1.1) associated with the periodic boundary condition 𝑥(0)−𝑥(1)=𝑥^′(0)−𝑥^′(1)=0, (1.2) the antiperiodic boundary condition 𝑥(0)+𝑥(1)=𝑥^′(0)+𝑥^′(1)=0, (1.3) the Dirichlet boundary condition 𝑥(0)=𝑥(1)=0, (1.4) or the Neumann boundary condition 𝑥^′(0)=𝑥^′(1)=0, (1.5) respectively. Values of 𝜆 for which (1.1) has nontrivial solutions 𝑥 satisfying the boundary condition (1.2), (1.3), (1.4), or (1.5) will be called periodic, antiperiodic, Dirichlet, or Neumann half-eigenvalues, respectively, while the corresponding solutions 𝑥 will be called half-eigenfunctions. Subscripts 𝑃, 𝐴, 𝐷, and 𝑁 will be used to indicate periodic, antiperiodic, Dirichlet and Neumann boundary conditions, respectively. The set of the half-eigenvalues under each of the preceding boundary conditions will be denoted by Σ_𝑃, Σ_𝐴, Σ_𝐷, and Σ_𝑁, respectively. Notice that these spectra are dependent on weights (𝑎,𝑏).

This article studies the half-eigenvalue problem (1.1) from two aspects. One is the structure of Σ_𝑃, Σ_𝐴, Σ_𝐷, and Σ_𝑁, and the other is the dependence of all these spectra on weights.

Generally speaking, for the 𝑝-Laplacian eigenvalue problem, if separated boundary conditions (such as Dirichlet and Neumann) are imposed and if the weight is definite, the spectrum structure is similar to that of the classical linear case. However, compared with separated boundary conditions, definite weights, and ℒ^∞ weights (or potentials), more difficulty will always be brought on by nonseparated boundary conditions, indefinite weights, and ℒ^𝛾 (1≤𝛾<∞) weights (or potentials), respectively. It has been proved very recently in [1] that with an indefinite weight 𝑎∈ℒ^𝛾, 1≤𝛾≤∞, the spectra structures of (𝜙_𝑝(𝑥^′))^′+𝜆𝑎(𝑡)𝜙_𝑝(𝑥)=0a.e.𝑡∈[0,1] (1.6) under both boundary conditions (1.4) and (1.5) are the same as when 𝑝=2. Special case of (1.6) with indefinite weight 𝑎∈ℒ^∞ is also studied by Anane et al. [2], Cuesta [3], and Eberhard and Elbert [4]. Various spectra (including half-eigenvalues) of 𝑝-Laplacian with potentials are studied by Binding and Rynne in [5]. It is shown that compared with the linear case 𝑝=2, some new phenomena occur in the periodic problems when 𝑝≠2. The existence of nonvariational periodic and antiperiodic eigenvalues of 𝑝-Laplacian for some potentials in [6] can partially explain these new phenomena.

We will use the shooting method to obtain spectra structures. To this end, (1.1) is studied in 𝑝-polar coordinates (2.5), and the associated argument function plays a crucial role. Careful analysis on the argument function leads to partial characterization of Σ_𝑃 and Σ_𝐴 and complete characterization of Σ_𝐷 and Σ_𝑁, see Theorems 3.9, 4.2, and 5.2. Some quasimonotonicity of the argument function is proved by using the Fréchet derivatives and the boundary conditions, see Lemmas 3.3, 4.1, and 5.1. The uniformly asymptotical result is obtained from the 𝑤^∗ compactness in the Banach-Alaoglu theorem as done in [1], see Lemma 2.3. Besides the properties of the argument functions, the Hamiltonian structure of the problem is essential for us to obtain variational periodic and antiperiodic half-eigenvalues, see Lemma 3.2. After introducing the rotation number function, we can easily obtain the ordering of these variational periodic half-eigenvalues, see (3.38).

For regular self-adjoint linear Sturm-Liouville problems, the continuous dependence of eigenvalues on weights or potentials in the usual 𝐿^𝛾 topology is well understood, and so is the Fréchet differentiable dependence. Many of these results are summarized in [7]. However, since the space of potentials (or weights) is infinite-dimensional, such a continuity result cannot answer many basic questions. For example, if potentials or weights are confined to a bounded set or a noncompact set, are the eigenvalues finite? To answer such kind of questions, a stronger continuity result is obtained in [8] for Sturm-Liouville operators and Hill's operators. That is, the eigenvalues are continuous in potentials in weak topology 𝑤_𝛾. Based on such a stronger continuity and the differentiability, variational method and singular integrals are applied in [9] to obtain the extremal value of smallest eigenvalues of Hill's operators with potentials confined to 𝐿¹ balls. The continuity result in weak topology are generalized to scalar 𝑝-Laplacian for eigenvalues on potentials (see [10]), for (separated) eigenvalues on indefinite weights (see [1]), and for half-eigenvalues on potentials (see [11]). Some elementary applications are also presented in [1, 10].

In this paper, we will prove that the variational periodic or antiperiodic half-eigenvalues 𝜆^𝐿_𝑚 and 𝜆^𝑅_𝑚 (defined by (3.22) and (3.25), resp.), and all the half-eigenvalues in Σ_𝐷 and Σ_𝑁, are continuous in weights (𝑎,𝑏)∈(ℒ^𝛾,𝑤_𝛾)². See Theorems 3.12, 4.3, and 5.3. Moreover, the Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in weights (𝑎,𝑏)∈(ℒ^𝛾,‖⋅‖_𝛾)², see Theorems 4.4 and 5.3. Due to the so-called parametric resonance [12] or the so-called coexistence of periodic and antiperiodic eigenvalues [13], periodic and antiperiodic half-eigenvalues are, in general, not differentiable in weights (𝑎,𝑏).

If 𝜆 is a half-eigenvalue of (1.1) corresponding to weights (𝑎,𝑏) and satisfying the boundary condition (1.2), (1.3), (1.4), or (1.5), then −𝜆 is also a half-eigenvalue of (1.1) corresponding to weights (−𝑎,−𝑏) and satisfying the same boundary condition. So we need only consider nonnegative half-eigenvalues of (1.1). Some preliminary results are given in Section 2. However, Sections 3, 4, and 5 are devoted to Σ_𝐴/𝑃, Σ_𝐷, and Σ_𝑁, respectively.

2. Preliminary Results

Given 𝑝∈(1,∞), denote by (𝐶_𝑝(𝜃),𝑆_𝑝(𝜃)) the unique solution of the initial value problem 𝑑𝑥𝑑𝜃=−𝜙_𝑝∗(𝑦),𝑑𝑦𝑑𝜃=𝜙_𝑝(𝑥),(𝑥(0),𝑦(0))=(1,0). (2.1) The functions 𝐶_𝑝(𝜃) and 𝑆_𝑝(𝜃) are the so-called 𝑝-cosine and 𝑝-sine, because they possess some properties similar to those of cosine and sine functions, such as

(i)both 𝐶_𝑝(𝜃) and 𝑆_𝑝(𝜃) are 2𝜋_𝑝-periodic, where 𝜋_𝑝=2𝜋(𝑝−1)^1/𝑝;𝑝sin(𝜋/𝑝) (2.2)(ii)𝐶_𝑝(𝜃)=0 if and only if 𝜃=𝜋_𝑝/2+𝑚𝜋_𝑝, 𝑚∈ℤ, and 𝑆_𝑝(𝜃)=0 if and only if 𝜃=𝑚𝜋_𝑝, 𝑚∈ℤ;(iii)one has |𝐶_𝑝(𝜃)|^𝑝+(𝑝−1)|𝑆_𝑝(𝜃)|^𝑝∗≡1. (2.3)

Given 𝑎,𝑏∈ℒ^𝛾, 𝛾∈[1,∞], let 𝑦=−𝜙_𝑝(𝑥^′). (2.4) In the 𝑝-polar coordinates 𝑥=𝑟^2/𝑝𝐶_𝑝(𝜃),𝑦=𝑟^2/𝑝∗𝑆_𝑝(𝜃), (2.5) the scalar equation (𝜙_𝑝(𝑥^′))^′+𝑎(𝑡)𝜙_𝑝(𝑥₊)−𝑏(𝑡)𝜙_𝑝(𝑥₋)=0,a.e.𝑡∈[0,1], (2.6) is transformed into the following equations for 𝑟 and 𝜃𝜃^′=𝐴(𝑡,𝜃;𝑎,𝑏)∶=𝑎(𝑡)|𝐶_𝑝(𝜃)|^𝑝+(𝑝−1)|𝑆_𝑝(𝜃)|^𝑝∗,if𝐶_𝑝(𝜃)≥0,𝑏(𝑡)|𝐶_𝑝(𝜃)|^𝑝+(𝑝−1)|𝑆_𝑝(𝜃)|^𝑝∗,if𝐶_𝑝(𝜃)<0,(log𝑟)^′𝑝=𝐺(𝑡,𝜃;𝑎,𝑏)∶=2(𝑎(𝑡)−1)𝜙_𝑝(𝐶_𝑝(𝜃))𝜙_𝑝∗(𝑆_𝑝(𝜃)),if𝐶_𝑝𝑝(𝜃)≥0,2(𝑏(𝑡)−1)𝜙_𝑝(𝐶_𝑝(𝜃))𝜙_𝑝∗(𝑆_𝑝(𝜃)),if𝐶_𝑝(𝜃)<0. (2.7)(2.8) For any 𝜗₀∈ℝ, denote by (𝜃(𝑡;𝜗₀,𝑎,𝑏),𝑟(𝑡;𝜗₀,𝑎,𝑏)), 𝑡∈[0,1], the unique solution of (2.7) + (2.8) satisfying 𝜃(0;𝜗₀,𝑎,𝑏)=𝜗₀ and 𝑟(0;𝜗₀,𝑎,𝑏)=1. Let Θ(𝜗₀,𝑎,𝑏)∶=𝜃(1;𝜗₀,𝑎,𝑏),𝑅(𝜗₀,𝑎,𝑏)∶=𝑟(1;𝜗₀,𝑎,𝑏). (2.9) As 𝐴(𝑡,𝜃;𝑎,𝑏) is independent of 𝑟 and is 2𝜋_𝑝-periodic in 𝜃, we have 𝜃(𝑡;𝜗₀+2𝑚𝜋_𝑝,𝑎,𝑏)≡𝜃(𝑡;𝜗₀,𝑎,𝑏)+2𝑚𝜋_𝑝 (2.10) for all 𝑡∈[0,1], 𝜗₀∈ℝ, and 𝑚∈ℤ.

An important property of the argument solution 𝜃 is the quasimonotonicity as in the following lemma.

Lemma 2.1 (see [14]). Let 𝜃(𝑡)=𝜃(𝑡;𝜗₀,𝑎,𝑏) be a solution of (2.7). Then 𝜋𝜃(𝑡)≥−^𝑝2+𝑚𝜋_𝑝at𝜋𝑡∈[0,1)⇒𝜃(𝑠)>−^𝑝2+𝑚𝜋_𝑝∀𝑠∈(𝑡,1]. (2.11)

Denote by 𝒞⁰∶=𝐶([0,1],ℝ) the space of continuous functions from [0,1] to ℝ. Some dependence results of solutions 𝑟 and 𝜃 on (𝑎,𝑏) are collected in the following theorem.

Theorem 2.2 (see [11]). (i) The functional ℝ×(ℒ^𝛾,𝑤_𝛾)²→ℝ,(𝜗,𝑎,𝑏)↦Θ(𝜗,𝑎,𝑏) (2.12) is continuous. Here 𝑤_𝛾 denotes the weak topology in ℒ^𝛾.
(ii) The functional ℝ×(ℒ^𝛾,‖⋅‖_𝛾)²→ℝ,(𝜗,𝑎,𝑏)↦Θ(𝜗,𝑎,𝑏) (2.13) is continuously differentiable. The derivatives of Θ(𝜗,𝑎,𝑏) at 𝜗, at 𝑎∈ℒ^𝛾, and at 𝑏∈ℒ^𝛾 (in the Fréchet sense), denoted, respectively, by 𝜕_𝜗Θ, 𝜕_𝑎Θ, and 𝜕_𝑏Θ, are 𝜕_𝜗1Θ(𝜗,𝑎,𝑏)=𝑅²,(𝜗,𝑎,𝑏)𝜕_𝑎Θ(𝜗,𝑎,𝑏)=𝑋^𝑝₊∈𝒞⁰⊂(ℒ^𝛾,‖⋅‖_𝛾)^∗,𝜕_𝑏Θ(𝜗,𝑎,𝑏)=𝑋^𝑝₋∈𝒞⁰⊂(ℒ^𝛾,‖⋅‖_𝛾)^∗, (2.14)(2.15)(2.16) where 𝑋=𝑋(𝑡)=𝑋(𝑡;𝜗,𝑎,𝑏)∶={𝑟(𝑡;𝜗,𝑎,𝑏)}^2/𝑝𝐶_𝑝(𝜃(𝑡;𝜗,𝑎,𝑏)){𝑟(1;𝜗,𝑎,𝑏)}^2/𝑝 (2.17) is a solution of (2.6).

To characterize the spectrum structure of (1.1) via shooting method, we need careful analysis on the associated argument function Θ(𝜗,𝜆𝑎,𝜆𝑏). Given 𝑎₁,𝑎₂,𝑏₁,𝑏₂∈ℒ^𝛾, write 𝑎₁≻𝑏₁ if 𝑎₁≥𝑏₁ and 𝑎₁(𝑡)>𝑏₁(𝑡) holds for 𝑡 in a subset of [0,1] of positive measure. Write (𝑎₁,𝑏₁)≥(𝑎₂,𝑏₂) if 𝑎₁≥𝑎₂ and 𝑏₁≥𝑏₂. Write (𝑎₁,𝑏₁)≻(𝑎₂,𝑏₂) if (𝑎₁,𝑏₁)≥(𝑎₂,𝑏₂) and both 𝑎₁(𝑡)>𝑎₂(𝑡) and 𝑏₁(𝑡)>𝑏₂(𝑡) hold for 𝑡 in a common subset of [0,1] of positive measure. Denote 𝒲^𝛾₊∶={(𝑎,𝑏)∣𝑎,𝑏∈ℒ^𝛾,(𝑎₊,𝑏₊)≻(0,0)}. (2.18) The following asymptotical property of Θ(𝜗,𝜆𝑎,𝜆𝑏) in 𝜆∈ℝ plays a crucial role in the characterization of half-eigenvalues.

Lemma 2.3. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, one has lim_𝜆→∞(Θ(𝜗,𝜆𝑎,𝜆𝑏)−𝜗)=∞ (2.19) uniformly in 𝜗∈ℝ.

Proof. The proof is a slight extension of that in [1, Lemma 3.4]. We write it out in detail for the convenience of the readers.
Besides (2.4), we introduce another transformation 𝑥_𝜆=𝑥,𝑦_𝜆=−𝜆^−1/𝑝∗𝜙_𝑝(𝑥^′)=𝜆^−1/𝑝∗𝑦 (2.20) when 𝜆>0. However, (1.1) becomes 𝑥_𝜆=−𝜆^1/𝑝𝜙_𝑝∗(𝑦_𝜆),𝑦^′_𝜆=𝜆^1/𝑝{𝑎(𝑡)𝜙_𝑝((𝑥_𝜆)₊)+𝑏(𝑡)𝜙_𝑝((𝑥_𝜆)₋)}. (2.21) In the 𝑝-polar coordinates of the (𝑥_𝜆,𝑦_𝜆)-plane 𝑥_𝜆∶=𝑟_𝜆^2/𝑝𝐶_𝑝(𝜃_𝜆),𝑦_𝜆∶=𝑟^2/𝑝∗_𝜆𝑆_𝑝(𝜃_𝜆), (2.22) the equation for the new argument 𝜃_𝜆 is 𝜃^′_𝜆=𝜆^1/𝑝{1+𝑎(𝑡)(𝐶_𝑝(𝜃_𝜆))^𝑝₊+𝑏(𝑡)(𝐶_𝑝(𝜃_𝜆))^𝑝₋−|𝐶_𝑝(𝜃_𝜆)|^𝑝}. (2.23) Denote by 𝜃_𝜆(𝑡;𝜗,𝑎,𝑏) the solution of (2.23) satisfying the initial condition 𝜃_𝜆(0)=𝜗. By (2.20), the argument functions of (1.1), 𝜃_𝜆(𝑡)=𝜃_𝜆(𝑡;𝜗,𝑎,𝑏) and 𝜃(𝑡)=𝜃(𝑡;𝜗,𝜆𝑎,𝜆𝑏) are related by an orientation-preserving homeomorphism ℋ∶ℝ→ℝ, which fixes the points {𝑚𝜋_𝑝,𝑚𝜋_𝑝+𝜋_𝑝/2∶𝑚∈ℤ}, in the following relation 𝜃(𝑡;ℋ(𝜗),𝑎,𝑏)=ℋ(𝜃_𝜆(𝑡;𝜗,𝑎,𝑏)). (2.24)
If lemma is not true, then there exist {𝜆_𝑛}_𝑛∈ℕ and {𝜗_𝑛}_𝑛∈ℕ such that 𝜆_𝑛↑∞ and {Θ(𝜗_𝑛,𝜆_𝑛𝑎,𝜆_𝑛𝑏)−𝜗_𝑛}_𝑛∈ℕ is bounded from above. Combining (2.11), there exists some 𝑚₀∈ℕ such that −𝜋_𝑝<Θ(𝜗_𝑛,𝜆_𝑛𝑎,𝜆_𝑛𝑏)−𝜗_𝑛<𝑚₀𝜋_𝑝,∀𝑛∈ℕ. (2.25) By (2.11) again, we have −𝜋_𝑝<𝜃(𝑡;𝜗_𝑛,𝜆_𝑛𝑎,𝜆_𝑛𝑏)−𝜗_𝑛<(𝑚₀+1)𝜋_𝑝,∀𝑡∈[0,1],∀𝑛∈ℕ. (2.26) By (2.10), we may assume 𝜗_𝑛∈[0,2𝜋_𝑝],∀𝑛∈ℕ. (2.27) Hence, −𝜋_𝑝<𝜃(𝑡;𝜗_𝑛,𝜆_𝑛𝑎,𝜆_𝑛𝑏)<(𝑚₀+3)𝜋_𝑝,∀𝑡∈[0,1],∀𝑛∈ℕ. (2.28)
Denote 𝜃_𝑛(𝑡)∶=𝜃_𝜆𝑛(𝑡;𝜗_𝑛,𝑎,𝑏) for simplicity. By the conjugacy (2.24) and the estimate (2.28), we have −𝜋_𝑝<𝜃_𝑛(𝑡)<(𝑚₀+3)𝜋_𝑝,∀𝑡∈[0,1],∀𝑛∈ℕ. (2.29)
Denote 𝐼₀∶={𝑓∈ℒ^∞∶0≤𝑓≤1} and define 𝑓_𝑛(𝑡)∶=(𝐶_𝑝(𝜃_𝑛(𝑡)))^𝑝₊∈𝐼₀,𝑔_𝑛(𝑡)∶=(𝐶_𝑝(𝜃_𝑛(𝑡)))^𝑝₋∈𝐼₀,ℎ_𝑛(𝑡)∶=|𝐶_𝑝(𝜃_𝑛(𝑡))|^𝑝∈𝐼₀, (2.30) for any 𝑛∈ℕ. By the Banach-Alaoglu theorem [15, pages 229-230], the unit ball 𝐵₁∶={𝑓∈ℒ^∞∶‖𝑓‖_∞≤1} is sequentially compact in (ℒ^∞,𝑤_∞) by considering ℒ^∞ as the dual space of the (separable) Banach space (ℒ¹,‖⋅‖₁). The order interval 𝐼₀ is a closed subset of (ℒ^∞,𝑤_∞). Hence, 𝐼₀(⊂𝐵₁) is also sequentially compact in (ℒ^∞,𝑤_∞). Consequently, passing to a subsequence, we may assume 𝑓_𝑛→𝑓₀∈𝐼₀,𝑔_𝑛→𝑔₀∈𝐼₀,ℎ_𝑛→ℎ₀∈𝐼₀,in(ℒ^∞,𝑤_∞). (2.31) From (2.23), 𝜃_𝑛(𝑡) satisfies 𝜃^′_𝑛=𝜆_𝑛^1/𝑝{1+𝑎(𝑡)𝑓_𝑛(𝑡)+𝑏(𝑡)𝑔_𝑛(𝑡)−ℎ_𝑛(𝑡)}a.e.𝑡∈[0,1]. (2.32) For any 𝑡∈[0,1], we have 𝜃_𝑛(𝑡)−𝜗_𝑛𝜆_𝑛^1/𝑝=^𝑡₀{1+𝑎(𝑠)𝑓_𝑛(𝑠)+𝑏(𝑠)𝑔_𝑛(𝑠)−ℎ_𝑛(𝑠)}𝑑𝑠=𝑡+¹₀𝑎(𝑠)𝜒_[0,𝑡](𝑠)⋅𝑓_𝑛(𝑠)𝑑𝑠+¹₀𝑏(𝑠)𝜒_[0,𝑡](𝑠)⋅𝑔_𝑛(𝑠)𝑑𝑠−¹₀𝜒_[0,𝑡](𝑠)⋅ℎ_𝑛(𝑠)𝑑𝑠. (2.33) Let 𝑛→∞, by (2.27) and (2.29), the left-hand side tends to 0. Since 𝑎(⋅)𝜒_[0,𝑡](⋅),𝑏(⋅)𝜒_[0,𝑡](⋅),𝜒_[0,𝑡](⋅)∈ℒ¹, (2.34) we can use (2.31) to deduce from the above equality 0=𝑡+¹₀𝑎(𝑠)𝜒_[0,𝑡](𝑠)⋅𝑓₀(𝑠)𝑑𝑠+¹₀𝑏(𝑠)𝜒_[0,𝑡](𝑠)⋅𝑔₀(𝑠)𝑑𝑠−¹₀𝜒_[0,𝑡](𝑠)⋅ℎ₀=(𝑠)𝑑𝑠_𝑡0{1+𝑎(𝑠)𝑓₀(𝑠)+𝑏(𝑠)𝑔₀(𝑠)−ℎ₀(𝑠)}𝑑𝑠 (2.35) for all 𝑡∈[0,1]. Thus 1+𝑎(𝑡)𝑓₀(𝑡)+𝑏(𝑡)𝑔₀(𝑡)−ℎ₀(𝑡)=0,a.e.𝑡∈[0,1]. (2.36) As ℎ_𝑛=𝑓_𝑛+𝑔_𝑛 and the weak* limits 𝑓₀,𝑔₀,ℎ₀ satisfy 0≤𝑓₀(𝑡),𝑔₀(𝑡),ℎ₀(𝑡)≤1 a.e. 𝑡∈[0,1], this equality implies that ℎ₀(𝑡)=𝑓₀(𝑡)+𝑔₀(𝑡)≠0,a.e.𝑡∈[0,1]. (2.37) Therefore, 𝑎(𝑡)𝑓₀(𝑡)+𝑏(𝑡)𝑔₀(𝑡)=ℎ₀(𝑡)−1≤0,a.e.𝑡∈[0,1], (2.38) which contradicts the assumption (𝑎,𝑏)∈𝒲^𝛾₊.

Remark 2.4. Let 𝑎≻0 and 𝑏(𝑡)=0, a.e. 𝑡∈[0,1] in (1.1). Then the equation for the corresponding argument function is 𝜃^′=𝜆𝑎(𝑡)(𝐶_𝑝(𝜃))^𝑝₊+(𝑝−1)|𝑆_𝑝(𝜃)|^𝑝∗ (2.39) (cf. (2.7). The solution 𝜃(𝑡) takes a constant length of time, say 𝑇₀>0, to start from 𝜋_𝑝/2+2𝑚𝜋_𝑝 and to reach at 3𝜋_𝑝/2+2𝑚𝜋_𝑝 for any 𝑚∈ℤ. Consequently, Θ(−𝜋_𝑝/2,𝜆𝑎,𝜆⋅0) is bounded for 𝜆∈[0,∞) and (2.19) does not hold in this case.

Notice that in the potential case [11], the associated argument function Θ(𝜗,𝜆+𝑎,𝜆+𝑏) is strictly increasing in 𝜆. However, as we are considering sign-changing weights (𝑎,𝑏), the monotonicity of Θ(𝜗,𝜆𝑎,𝜆𝑏) in 𝜆 does not hold any more. We will develop some quasi-monotonicity of Θ(𝜗,𝜆𝑎,𝜆𝑏) in 𝜆 by employing the Fréchet derivatives of Θ and the boundary conditions, see Lemmas 3.3, 4.1, and 5.1.

3. Periodic and Antiperiodic Spectrum

3.1. Structure of Periodic and Antiperiodic Half-Eigenvalues

Given 𝑎,𝑏∈ℒ^𝛾, 1≤𝛾≤∞, by (2.9), (2.10) and Theorem 2.2(i), one sees that Θ(𝑎,𝑏)∶=max_{𝜗0∈[0,2𝜋𝑝]}(Θ(𝜗₀,𝑎,𝑏)−𝜗₀)=max_𝜗0∈ℝ(Θ(𝜗₀,𝑎,𝑏)−𝜗₀),Θ(𝑎,𝑏)∶=min_{𝜗0∈[0,2𝜋𝑝]}(Θ(𝜗₀,𝑎,𝑏)−𝜗₀)=min_𝜗0∈ℝ(Θ(𝜗₀,𝑎,𝑏)−𝜗₀) (3.1)(3.2) are well-defined. Moreover, we have the following lemma.

Lemma 3.1. Given 𝛾∈[1,∞], the functionals (ℒ^𝛾,𝑤_𝛾)²→ℝ,(𝑎,𝑏)↦Θ(𝑎,𝑏),(ℒ^𝛾,𝑤_𝛾)²→ℝ,(𝑎,𝑏)↦Θ(𝑎,𝑏) (3.3) are continuous.

Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, we write Θ(𝜆)∶=Θ(𝜆𝑎,𝜆𝑏),Θ(𝜆)∶=Θ(𝜆𝑎,𝜆𝑏) (3.4) for simplicity if there is no ambiguity. By Lemma 3.1, Θ(𝜆) and Θ(𝜆) are continuous in 𝜆∈ℝ. By Lemma 2.3, one has lim_𝜆→∞Θ(𝜆)=lim_𝜆→∞Θ(𝜆)=∞. (3.5) Moreover, by setting 𝑎=𝑏=0 in (2.7), we know that Θ(0)∈(0,𝜋_𝑝),Θ(0)=0. (3.6) Now we are considering the following two sequences of equations for 𝜆Θ(𝜆)=𝑚𝜋_𝑝,𝑚∈ℕ,Θ(𝜆)=𝑚𝜋_𝑝,𝑚∈ℤ⁺∶=ℕ∪{0}. (3.7)(3.8)

Lemma 3.2. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞.
(i)For any 𝑚 specified, (3.7) and (3.8) always have solutions in ℝ⁺∶={𝜆∈ℝ∣𝜆≥0}.(ii)All solutions of (3.7) and (3.8) are periodic half-eigenvalues or antiperiodic half-eigenvalues of (1.1) if 𝑚 is even or odd, respectively, while the corresponding half-eigenfunctions have precisely 𝑚 zeroes in the interval [0,1).

Proof. (i) The solvability of (3.7) and (3.8) follows immediately from (3.5), (3.6) and the continuity of Θ(𝜆) and Θ(𝜆) in 𝜆∈ℝ⁺.
(ii) Suppose 𝜆∈ℝ⁺ satisfies (3.7) or (3.8). Then there exists 𝜗₀∈[0,2𝜋_𝑝] such that Θ(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀=𝑚𝜋_𝑝,𝑑𝑑𝜗₀Θ(𝜗₀,𝜆𝑎,𝜆𝑏)|_𝜗0=𝜗₀=1. (3.9)(3.10) By (2.14), the latter equation is 𝑅(𝜗₀,𝜆𝑎,𝜆𝑏)=1. (3.11) Geometrically, (3.9) and (3.11) imply that in 𝑝-polar coordinates (2.5), the solution of (1.1) starting at the point (𝑟,𝜃)=(1,𝜗₀) arrives at (𝑟,𝜃)=(1,𝜗₀+𝑚𝜋_𝑝) after one period. Thus 𝜆 is a periodic or an antiperiodic half-eigenvalue of (1.1) if 𝑚 is even or odd, respectively.
Denote by 𝑥 the half-eigenfunction corresponding to 𝜆. Then 𝑥(𝑡)=0 if and only if 𝜃(𝑡)∶=𝜃(𝑡;𝜗₀𝜋,𝜆𝑎,𝜆𝑏)=−^𝑝2+𝑘𝜋_𝑝,𝑘∈ℤ. (3.12) Notice that 𝜃(0)=𝜗₀, 𝜃(1)=𝜗₀+𝑚𝜋_𝑝 (see (3.9), and 𝜃(𝑡) is quasimonotone in 𝑡 (see (2.11). Consequently, if 𝜗₀=−𝜋_𝑝/2+𝑙₀𝜋_𝑝 for some 𝑙₀∈ℤ, then (3.12) holds only for 𝑘=𝑙₀,𝑙₀+1,𝑙₀+2,…,𝑙₀+𝑚 at (0=)𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑚(=1). If −𝜋_𝑝/2+𝑙₀𝜋_𝑝<𝜗₀<−𝜋_𝑝/2+(𝑙₀+1)𝜋_𝑝, then (3.12) holds only for 𝑘=𝑙₀+1,𝑙₀+2,…,𝑙₀+𝑚 at (0<)𝑡₁<𝑡₂<⋯<𝑡_𝑚(<1). In both cases 𝑥 has precisely 𝑚 zeroes in [0,1).

The following lemma will be used for further study on solutions of (3.7) and (3.8).

Lemma 3.3. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞.
(i)If Θ(𝜇)=𝑚𝜋_𝑝 for some 𝜇>0 and 𝑚∈ℕ, then there exists 𝛿>0 such that Θ(𝜆)>𝑚𝜋_𝑝,∀𝜆∈(𝜇,𝜇+𝛿). (3.13)(ii)If Θ(𝜇)=𝑚𝜋_𝑝 for some 𝜇>0 and 𝑚∈ℤ⁺, then there exists 𝛿∈(0,𝜇) such that Θ(𝜆)<𝑚𝜋_𝑝,∀𝜆∈(𝜇−𝛿,𝜇). (3.14)

Proof. We only prove (i) and the proof of (ii) is analogous.
Assume that Θ(𝜇)=𝑚𝜋_𝑝. By Lemma 3.2, 𝜇 is a periodic or an antiperiodic half-eigenvalue of (1.1). Then there exists some 𝜗₀∈[0,2𝜋_𝑝] such that Θ(𝜗₀,𝜇𝑎,𝜇𝑏)−𝜗₀=𝑚𝜋_𝑝, (3.15) and the corresponding half-eigenfunction 𝑋(𝑡)=𝑋(𝑡;𝜗₀,𝜇𝑎,𝜇𝑏) satisfies (𝜙_𝑝(𝑋^′))^′+𝜇𝑎(𝑡)𝜙_𝑝(𝑋₊)−𝜇𝑏(𝑡)𝜙_𝑝(𝑋₋)=0,a.e.𝑡∈[0,1], (3.16) and the boundary condition (1.2) or (1.3). Applying (2.15) and (2.16), we have 𝑑𝑑𝜆Θ(𝜗₀,𝜆𝑎,𝜆𝑏)|_𝜆=𝜇=¹₀(𝑎𝑋^𝑝₊+𝑏𝑋^𝑝₋)𝑑𝑡, (3.17) Multiplying (3.16) by 𝑋 and integrating over [0,1], one has ¹₀(𝑎𝑋^𝑝₊+𝑏𝑋^𝑝₋1)𝑑𝑡=𝜇(¹₀|𝑋^′|^𝑝𝑑𝑡−𝑋(𝑡)𝜙_𝑝(𝑋^′(𝑡))|¹_𝑡=0)=1𝜇¹₀|𝑋^′|^𝑝𝑑𝑡>0, (3.18) and, therefore, 𝑑𝑑𝜆Θ(𝜗₀,𝜆𝑎,𝜆𝑏)|_𝜆=𝜇>0. (3.19) Then there exists 𝛿>0 such that Θ(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀>𝑚𝜋_𝑝∀𝜆∈(𝜇,𝜇+𝛿). (3.20) Consequently, we have Θ(𝜆)≥Θ(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀>𝑚𝜋_𝑝,∀𝜆∈(𝜇,𝜇+𝛿). (3.21)

Definition 3.4. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, define 𝜆^𝐿_𝑚∶=min{𝜆>0∣Θ(𝜆)=𝑚𝜋_𝑝},𝑚∈ℕ,𝜆^𝑅_𝑚∶=max{𝜆>0∣Θ(𝜆)=𝑚𝜋_𝑝},𝑚∈ℕ,𝜆^𝐿_𝑚∶=min{𝜆≥0∣Θ(𝜆)=𝑚𝜋_𝑝},𝑚∈ℤ⁺,𝜆^𝑅_𝑚∶=max{𝜆≥0∣Θ(𝜆)=𝑚𝜋_𝑝},𝑚∈ℤ⁺. (3.22)(3.23)(3.24)(3.25)

These values 𝜆_𝑚^𝐿/𝑅 and 𝜆_𝑚^𝐿/𝑅 are well defined. By Lemma 3.2, they are periodic or antiperiodic half-eigenvalues. These are what we are interested in. By Lemma 3.3, we have the following results.

Corollary 3.5. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, one has Θ(𝜆)<𝑚𝜋_𝑝,if0≤𝜆<𝜆^𝐿_𝑚∈[𝑚𝜋_𝑝,𝑚𝜋_𝑝+𝜋_𝑝),if𝜆^𝐿_𝑚≤𝜆≤𝜆^𝑅_𝑚>𝑚𝜋_𝑝,if𝜆>𝜆^𝑅_𝑚∀𝑚∈ℕ,Θ(𝜆)<𝑚𝜋_𝑝,if0≤𝜆<𝜆^𝐿_𝑚∈(𝑚𝜋_𝑝−𝜋_𝑝,𝑚𝜋_𝑝],if𝜆^𝐿_𝑚≤𝜆≤𝜆^𝑅_𝑚>𝑚𝜋_𝑝,if𝜆>𝜆^𝑅_𝑚.∀𝑚∈ℤ⁺. (3.26)

Combining (3.5), (3.6), and (3.26), one has the following ordering for these half-eigenvalues: (0<)𝜆^𝐿₁≤𝜆^𝑅₁<𝜆^𝐿₂≤𝜆^𝑅₂<⋯<𝜆^𝐿_𝑚≤𝜆^𝑅_𝑚<⋯(→∞),(0=)𝜆^𝐿₀≤𝜆^𝑅₀<𝜆^𝐿₁≤𝜆^𝑅₁<⋯<𝜆^𝐿_𝑚≤𝜆^𝑅_𝑚<⋯(→∞). (3.27) Moreover, by the definition of 𝜆_𝑚^𝐿/𝑅 and 𝜆_𝑚^𝐿/𝑅, one has 𝜆^𝐿_𝑚≤𝜆^𝑅_𝑚,𝜆^𝑅_𝑚,𝜆^𝐿_𝑚∈[𝜆^𝐿_𝑚,𝜆^𝑅_𝑚],∀𝑚∈ℕ. (3.28) Notice that the ordering between 𝜆^𝑅_𝑚 and 𝜆^𝐿_𝑚 is not determinate.

Till now, we still do not know the ordering between 𝜆^𝑅_𝑚 and 𝜆^𝐿_𝑛 with 𝑛>𝑚. This will be partially settled by associating (1.1) with the rotation number function defined by 𝜌(𝜆)=𝜌(𝜆,𝑎,𝑏)∶=lim_𝑛→∞Θ^𝑛(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀2𝑛𝜋_𝑝, (3.29) where Θ^𝑛 is the 𝑛th iteration of Θ, namely, Θ^𝑛(𝜗₀,𝑎,𝑏)=Θ(Θ^𝑛−1(𝜗₀,𝑎,𝑏),𝑎,𝑏). (3.30) It can be proved that the rotation number function 𝜌(𝜆) in (3.29) is well defined and is independent of 𝜗₀ (cf. [16, Theorem 2.1]).

Lemma 3.6. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, one has
(i)𝜌(𝜆)≥0 and 𝜌(0)=0;(ii)lim_𝜆→∞𝜌(𝜆)=∞;(iii)𝜆∈[0,𝜆^𝐿_𝑚),𝑚∈ℕ⇒𝜌(𝜆)<𝑚/2;(iv)𝜆∈(𝜆^𝑅_𝑚,∞),𝑚∈ℤ⁺,⇒𝜌(𝜆)>𝑚/2;(v)𝜆∈[𝜆^𝐿_2𝑚,𝜆^𝑅_2𝑚],𝑚∈ℕ⇒𝜌(𝜆)=𝑚;(vi)𝜆=𝜆^𝑅₀⇒𝜌(𝜆)=0.

Proof. (i) By (2.11), one has Θ^𝑛(0,𝜆𝑎,𝜆𝑏)>−𝜋_𝑝/2 for any 𝑛∈ℕ. Thus 𝜌(𝜆)≥0 since 𝜌(𝜆) defined by (3.29) is independent of 𝜗₀. If 𝜆=0, the equation for the argument function associated with (1.1) becomes 𝜃^′=(𝑝−1)|𝑆_𝑝(𝜃)|^𝑝∗, which has equilibria 𝜃=𝑘𝜋_𝑝, 𝑘∈ℤ. Thus Θ^𝑛(−𝜋_𝑝/2,0⋅𝑎,0⋅𝑏)∈(−𝜋_𝑝/2,0) for all 𝑛∈ℕ and consequently 𝜌(0)=0.
(ii) Given 𝑀>0 arbitrarily large, it follows from (3.5) that there exists Λ>0 such that Θ(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀>2𝑀𝜋_𝑝,∀𝜆>Λ,∀𝜗₀∈ℝ. (3.31) Thus Θ^𝑛(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀=[Θ(Θ^𝑛−1(𝜗₀,𝜆𝑎,𝜆𝑏),𝜆𝑎,𝜆𝑏)−Θ(Θ^𝑛−2(𝜗₀,𝜆𝑎,𝜆𝑏),𝜆𝑎,𝜆𝑏)]+[Θ(Θ^𝑛−2(𝜗₀,𝜆𝑎,𝜆𝑏),𝜆𝑎,𝜆𝑏)−Θ(Θ^𝑛−3(𝜗₀,𝜆𝑎,𝜆𝑏),𝜆𝑎,𝜆𝑏)]+⋯+[Θ(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀]≥2𝑛𝑀𝜋_𝑝 (3.32) for any 𝜆>Λ and any 𝜗₀∈ℝ. Consequently, 𝜌(𝜆)≥𝑀 for any 𝜆>Λ.
(iii) Given 𝜆∈[0,𝜆^𝐿_𝑚),𝑚∈ℕ, it follows from (3.26) that there exists 𝛿=𝛿(𝜆)>0 such that Θ(𝜆)<(𝑚−𝛿)𝜋_𝑝. (3.33) By the definition of Θ(𝜆)=Θ(𝜆𝑎,𝜆𝑏) (see (3.1), this implies that Θ(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀<(𝑚−𝛿)𝜋_𝑝,∀𝜗₀∈ℝ. (3.34) Now similar arguments as in (ii) show that 𝜌(𝜆)≤(𝑚−𝛿)/2<𝑚/2.
(iv) The proof is analogous to that of (iii).
(v) Given 𝜆∈[𝜆^𝐿_2𝑚,𝜆^𝑅_2𝑚],𝑚∈ℕ, it follows from (3.26) that Θ(𝜆)≤2𝑚𝜋_𝑝≤Θ(𝜆). (3.35) Then there exists 𝜗₀ such that Θ(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀=2𝑚𝜋_𝑝, and therefore 𝜌(𝜆)=𝑚.
(vi) By (3.25), if 𝜆=𝜆^𝑅₀, then there exists 𝜗₀ such that Θ(𝜗₀,𝜆𝑎,𝜆𝑏)−𝜗₀=0. Hence 𝜌(𝜆)=0.

Corollary 3.7. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, one has 𝜆^𝐿_2𝑚=min{𝜆>0∣𝜌(𝜆)=𝑚},∀𝑚∈ℕ,𝜆^𝑅_2𝑚=max{𝜆≥0∣𝜌(𝜆)=𝑚},∀𝑚∈ℤ⁺, (3.36)(3.37) and the ordering (0≤)𝜆^𝑅₀<𝜆^𝐿₂≤𝜆^𝑅₂<⋯<𝜆^𝐿_2𝑚≤𝜆^𝑅_2𝑚<𝜆^𝐿_2𝑚+2≤𝜆^𝑅_2𝑚+2⋯(→∞). (3.38)

Proof. The characterization (3.36) follows from Lemma 3.6(iii) and (v), while (3.37) follows from Lemma 3.6(iv), (v), and (vi).
To prove (3.38), by (3.27) and (3.28), we need only prove 𝜆^𝑅_2𝑚<𝜆^𝐿_2𝑚+2 for any 𝑚∈ℕ. Assume on the contrary that 𝜆^𝑅_2𝑚≥𝜆^𝐿_2𝑚+2 for some 𝑚. Then 𝜆^𝐿_2𝑚<𝜆^𝐿_2𝑚+2≤𝜆^𝑅_2𝑚. (3.39) Now Lemma 3.6(v) shows that 𝑚=𝜌(𝜆^𝐿_2𝑚+2)=𝑚+1, a contradiction.

Corollary 3.8. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, one has 𝜆∈(𝜆^𝑅_2𝑚,𝜆^𝐿_2𝑚+2),𝑚∈ℕ⇒𝜌(𝜆)∈(𝑚,𝑚+1). (3.40)

In summary, we can now partially characterize the set of periodic half-eigenvalues Σ_𝑃 and the set of antiperiodic half-eigenvalues Σ_𝐴 of (1.1). Denote Σ⁺_𝑃∶=Σ_𝑃∩ℝ⁺ and Σ⁺_𝐴∶=Σ_𝐴∩ℝ⁺. For any 𝑚∈ℤ⁺, denote by Σ⁺_𝑚=Σ⁺_𝑚(𝑎,𝑏) the set of nonnegative periodic or antiperiodic half-eigenvalues for which the corresponding half-eigenfunctions have precisely 𝑚 zeroes in the interval [0,1). Then Σ⁺_𝑃=^∞_𝑚=0Σ⁺_2𝑚,Σ⁺_𝐴=^∞_𝑚=0Σ⁺_2𝑚+1. (3.41)

Theorem 3.9. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞.
(i)For periodic half-eigenvalues, one has {0,𝜆^𝑅₀}⊂Σ⁺₀⊂[0,𝜆^𝑅₀],{𝜆^𝐿_2𝑚,𝜆^𝑅_2𝑚,𝜆^𝐿_2𝑚,𝜆^𝑅_2𝑚}⊂Σ⁺_2𝑚⊂[𝜆^𝐿_2𝑚,𝜆^𝑅_2𝑚],∀𝑚∈ℕ. (3.42) Moreover, 𝜆^𝐿/𝑅_2𝑚 and 𝜆^𝐿/𝑅_2𝑚 satisfy (3.27), (3.28), and (3.38).(ii)For antiperiodic half-eigenvalues, one has {𝜆^𝐿_2𝑚+1,𝜆^𝑅_2𝑚+1,𝜆^𝐿_2𝑚+1,𝜆^𝑅_2𝑚+1}⊂Σ⁺_2𝑚+1,∀𝑚∈ℤ⁺. (3.43) Moreover, 𝜆^𝐿/𝑅_2𝑚+1 and 𝜆^𝐿/𝑅_2𝑚+1 satisfy (3.27) and (3.28).

Proof. By Lemma 3.2 and the definitions of 𝜆_𝑚^𝐿/𝑅 and 𝜆_𝑚^𝐿/𝑅 (see (3.22)–(3.25), we need only prove Σ⁺₀⊂[0,𝜆^𝑅₀],Σ⁺_2𝑚⊂[𝜆^𝐿_2𝑚,𝜆^𝑅_2𝑚],∀𝑚∈ℕ. (3.44) If 𝜆∈Σ⁺_2𝑚, 𝑚∈ℤ⁺, then 𝜆 is a periodic half-eigenvalue of (1.1) and the corresponding half-eigenfunction has precisely 2𝑚 zeroes [0,1). Consequently, there exists 𝜗₀∈ℝ such that 𝜃(𝜗₀,𝜆𝑎,𝜆𝑏)=𝜗₀+2𝑚𝜋_𝑝. (3.45) Hence 𝜌(𝜆)=𝑚. Combining (3.36), and the fact 𝜌(0)=0 we obtain (3.44).

The following theorem gives the necessary and sufficient condition for 𝜆^𝑅₀>0.

Theorem 3.10. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, then 𝜆^𝑅₀>0⟺¹₀𝑎(𝑡)𝑑𝑡<0or¹₀𝑏(𝑡)𝑑𝑡<0. (3.46)

Proof. Assume that 𝜆^𝑅₀>0. By Lemma 3.2, the half-eigenfunction 𝑋=𝑋(𝑡) corresponding to 𝜆^𝑅₀ is nowhere vanishing and satisfies the periodic boundary condition (1.2). If 𝑋(𝑡)>0 for any 𝑡∈[0,1], then (𝜙_𝑝(𝑋^′))^′+𝜆^𝑅₀𝑎(𝑡)𝜙_𝑝(𝑋)=0,𝑋≢const. (3.47) Therefore, ¹₀1𝑎(𝑡)𝑑𝑡=−𝜆^𝑅₀¹⁰(𝜙_𝑝(𝑋^′))^′𝜙_𝑝1(𝑋)𝑑𝑡=−𝜆^𝑅₀¹⁰𝜙_𝑝(𝑋^′1)𝑑(𝜙_𝑝𝜙(𝑋))+^𝑝(𝑋^′(𝑡))𝜆^𝑅₀𝜙_𝑝|(𝑋(𝑡))¹_𝑡=0=−(𝑝−1)𝜆^𝑅₀¹⁰|𝑋^′|^𝑝𝑋^𝑝𝑑𝑡<0. (3.48) Similarly, if 𝑋(𝑡)<0 for any 𝑡∈[0,1], then ∫¹₀𝑏(𝑡)𝑑𝑡<0.
On the other hand, assume that ∫¹₀𝑎(𝑡)𝑑𝑡<0. Since 𝑥(𝑡)≡1 is the periodic half-eigenfunction corresponding to 𝜆=0, by (2.4), we have 𝑦(𝑡)≡0. Hence in the 𝑝-polar coordinates (2.5) 𝜗₀=0 and the argument solution 𝜃(𝑡;𝜗₀,0⋅𝑎,0⋅𝑏)≡0. Then it follows from (2.15) and (2.16) that 𝑑𝑑𝜆Θ(0,𝜆𝑎,𝜆𝑏)|_𝜆=0=¹₀𝑎(𝑡)𝑑𝑡<0. (3.49) Thus there exists 𝛿>0 such that Θ(0,𝜆𝑎,𝜆𝑏)<0,∀𝜆∈(0,𝛿). (3.50) By the definition of Θ(𝜆)=Θ(𝜆𝑎,𝜆𝑏) (see (3.2), we get Θ(𝜆)<0,∀𝜆∈(0,𝛿). (3.51) By (3.5), (3.6) and the continuity of Θ(𝜆) in 𝜆∈ℝ⁺ (see Lemma 3.1), one has 𝜆^𝑅₀>0. Similarly one can obtain 𝜆^𝑅₀>0 if ∫¹₀𝑏(𝑡)𝑑𝑡<0.

Notice that for the antiperiodic half-eigenvalues, it does not hold Σ_2𝑚+1⊂[𝜆^𝐿_2𝑚+1,𝜆^𝑅_2𝑚+1],∀𝑚∈ℤ⁺, (3.52) for general 𝑎,𝑏∈ℒ^𝛾. This is because 𝐴(𝑡,𝜃;𝑎,𝑏) in (2.7) is 2𝜋_𝑝-periodic in 𝜃 but not 𝜋_𝑝-periodic in 𝜃. Hence (2.10) holds but one can not obtain 𝜃(𝑡;𝜗₀+𝑚𝜋_𝑝,𝑎,𝑏)≡𝜃(𝑡;𝜗₀,𝑎,𝑏)+𝑚𝜋_𝑝,∀𝑡∈[0,1],𝜗₀∈ℝ,𝑚∈ℤ. (3.53) However, if 𝑎=𝑏, then 𝐴(𝑡,𝜃;𝑎,𝑎) is 𝜋_𝑝-periodic in 𝜃 and 𝜃(𝑡;𝜗₀+𝑚𝜋_𝑝,𝑎,𝑎)≡𝜃(𝑡;𝜗₀,𝑎,𝑎)+𝑚𝜋_𝑝,∀𝑡∈[0,1],𝜗₀∈ℝ,𝑚∈ℤ. (3.54) The following theorem characterizes periodic and antiperiodic eigenvalues of (1.6). Now Σ⁺_𝑚=Σ⁺_𝑚(𝑎,𝑎), 𝑚∈ℤ⁺, is the set of all those nonnegative periodic or antiperiodic eigenvalues of (1.6) for which the corresponding eigenfunctions have precisely 𝑚 zeroes in [0,1).

Theorem 3.11. Given 𝛾∈[1,∞] and 𝑎=𝑏∈ℒ^𝛾 with 𝑎₊≻0.
(i)All solutions of (3.7) and (3.8) are periodic or antiperiodic eigenvalues of (1.6) if 𝑚 is even or odd, respectively.(ii)One has {0,𝜆^𝑅₀}⊂Σ⁺₀⊂[0,𝜆^𝑅₀],{𝜆^𝐿_𝑚,𝜆^𝑅_𝑚,𝜆^𝐿_𝑚,𝜆^𝑅_𝑚}⊂Σ⁺_𝑚⊂[𝜆^𝐿_𝑚,𝜆^𝑅_𝑚],∀𝑚∈ℕ. (3.55)(iii)One has the ordering (0≤)𝜆^𝑅₀<𝜆^𝐿₁≤𝜆^𝑅₁<𝜆^𝐿₂≤𝜆^𝑅₂<⋯<𝜆^𝐿_𝑚≤𝜆^𝑅_𝑚<⋯(→∞). (3.56)(iv)Finally 𝜆^𝑅₀>0⇔∫¹₀𝑎(𝑡)𝑑𝑡<0.

Proof. By Theorems 3.9 and 3.10, we need only prove (3.52) and 𝜆^𝑅_2𝑚<𝜆^𝐿_2𝑚+1,𝜆^𝑅_2𝑚+1<𝜆^𝐿_2𝑚+2,∀𝑚∈ℤ⁺. (3.57)
If 𝜆∈Σ⁺_𝑚, 𝑚∈ℕ, then there exists 𝜗₀∈ℝ such that Θ(𝜗₀,𝜆𝑎,𝜆𝑎)−𝜗₀=𝑚𝜋_𝑝. By (3.54) and the definition of 𝜌(⋅) in (3.29), one has 𝜆∈Σ⁺_𝑚𝑚⇒𝜌(𝜆)=2, (3.58) in particular 𝜌(𝜆^𝐿_2𝑚+1)=𝜌(𝜆^𝑅_2𝑚+1)=2𝑚+12,∀𝑚∈ℤ⁺. (3.59) Then it follows from Lemma 3.6(iii) and (iv) that 𝜆^𝐿_2𝑚+1=min{𝜆≥0∣𝜌(𝜆)=𝑚+12},𝑚∈ℤ⁺,𝜆^𝑅_2𝑚+1=max{𝜆≥0∣𝜌(𝜆)=𝑚+12},𝑚∈ℤ⁺. (3.60) Hence (3.52) follows immediately from (3.58) and (3.60).
Suppose 𝜆^𝑅_2𝑚≥𝜆^𝐿_2𝑚+1 for some 𝑚∈ℤ⁺. By Lemma 3.6(iii) and (v) we obtain 𝜌(𝜆^𝐿_2𝑚+1)≤𝑚, which contradicts (3.59). Thus 𝜆^𝑅_2𝑚>𝜆^𝐿_2𝑚+1 for any 𝑚∈ℤ⁺. Similarly one can prove 𝜆^𝑅_2𝑚+1<𝜆^𝐿_2𝑚+2, ∀𝑚∈ℤ⁺, proving the desired results in (3.57).

3.2. Continuity of Periodic and Antiperiodic Half-Eigenvalues in Weak Topology

Theorem 3.12. Given 𝛾∈[1,∞], for any admissible 𝑚, the following functionals (𝒲^𝛾₊,𝑤_𝛾×𝑤_𝛾)→ℝ,(𝑎,𝑏)↦𝜆^𝐿_𝑚(𝑎,𝑏)(𝒲^𝛾₊,𝑤_𝛾×𝑤_𝛾)→ℝ,(𝑎,𝑏)↦𝜆^𝑅_𝑚(𝑎,𝑏) (3.61) are continuous. Here 𝑤_𝛾 denotes the weak topology in ℒ^𝛾.

Proof. We only prove the result for 𝜆^𝐿_𝑚(𝑎,𝑏). The result for 𝜆^𝑅_𝑚(𝑎,𝑏) can be attained analogously. Suppose (𝑎_𝑛,𝑏_𝑛)→(𝑎,𝑏) in (𝒲^𝛾₊,𝑤_𝛾×𝑤_𝛾) as 𝑛→∞. Then 𝑎_𝑛→𝑎 and 𝑏_𝑛→𝑏 in (ℒ^𝛾,𝑤_𝛾). Denote 𝜈_𝑛∶=𝜆^𝐿_𝑚(𝑎_𝑛,𝑏_𝑛). We aim to prove 𝜈_𝑛→𝜆^𝐿_𝑚(𝑎,𝑏).
Firstly, we show that the sequence {𝜈_𝑛}_𝑛∈ℕ is bounded. If this is false, without loss of generality, we may assume that 𝜈_𝑛→∞ as 𝑛→∞. Then given any Λ>0, there exists 𝑁∈ℕ such that 𝜈_𝑛>Λ,∀𝑛>𝑁. (3.62) It follows from (3.26) that Θ(Λ𝑎_𝑛,Λ𝑏_𝑛)<𝑚𝜋_𝑝,∀𝑛>𝑁. (3.63) Let 𝑛→∞, we obtain by Lemma 3.1Θ(Λ𝑎,Λ𝑏)≤𝑚𝜋_𝑝, (3.64) which contradicts (3.5) since Λ>0 can be chosen arbitrarily large. Thus {𝜈_𝑛}_𝑛∈ℕ is bounded.
Since {𝜈_𝑛}_𝑛∈ℕ is bounded, passing to a subsequence, we assume that 𝜈_𝑛→𝜈₀ as 𝑛→∞. By the definition of 𝜈_𝑛, we have Θ(𝜈_𝑛𝑎_𝑛,𝜈_𝑛𝑏_𝑛)=𝑚𝜋_𝑝. (3.65) Let 𝑛→∞, by Lemma 3.1 again, we get Θ(𝜈₀𝑎,𝜈₀𝑏)=𝑚𝜋_𝑝. (3.66) Now it suffices to prove 𝜈₀=𝜆^𝐿_𝑚(𝑎,𝑏).
If this is not true, by the definition in (3.22) there holds 𝜈₀>𝜆^𝐿_𝑚(𝑎,𝑏). Then it follows from Lemma 3.3(i) that there exists 𝜆∈(𝜆^𝐿_𝑚(𝑎,𝑏),𝜈₀) (3.67) such that Θ(𝜆𝑎,𝜆𝑏)>𝑚𝜋_𝑝. (3.68) Since Θ(𝜆𝑎_𝑛,𝜆𝑏_𝑛)→Θ(𝜆𝑎,𝜆𝑏) as 𝑛→∞, we may assume, without loss of generality, that Θ(𝜆𝑎_𝑛,𝜆𝑏_𝑛)>𝑚𝜋_𝑝,∀𝑛∈ℕ. (3.69) Therefore, by (3.26), we have 𝜆>𝜆^𝐿_𝑚(𝑎_𝑛,𝑏_𝑛)=𝜈_𝑛,∀𝑛∈ℕ. (3.70) Let 𝑛→∞, one has 𝜆≥𝜈₀, which contradicts the choice of 𝜆 in (3.67).

4. The Dirichlet Spectrum

4.1. Structure of Dirichlet Half-Eigenvalues

Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, in this section we use the following notations for simplicity if there is no ambiguity 𝜋Θ(𝜆)=Θ(𝜆𝑎,𝜆𝑏)=Θ(−^𝑝2𝜋,𝜆𝑎,𝜆𝑏),𝑋(𝑡)=𝑋(𝑡;𝑎,𝑏)=𝑋(𝑡;−^𝑝2,𝑎,𝑏). (4.1)

For 𝑎=𝑏=0, (2.7) is 𝜃^′=(𝑝−1)|𝑆_𝑝(𝜃)|^𝑝∗, which has equilibria 𝜃=𝑘𝜋_𝑝, 𝑘∈ℤ. Hence, we have 𝜋Θ(0)∈(−^𝑝2,0). (4.2) Therefore, 𝜆=0 is not a Dirichlet half-eigenvalue of (1.1), namely, 0∉Σ_𝐷. Due to the jumping nonlinearity of (1.1), 𝜆∈Σ⁺_𝐷∶=Σ_𝐷∩ℝ⁺ if and only if 𝜆>0 is determined by one of the following two equations 𝜋Θ(−^𝑝2𝜋,𝜆𝑎,𝜆𝑏)=−^𝑝2+𝑚𝜋_𝑝,𝜋Θ(^𝑝2𝜋,𝜆𝑎,𝜆𝑏)=^𝑝2+𝑚𝜋_𝑝, (4.3)(4.4) for some 𝑚∈ℤ. The range of 𝑚 will be proved to be ℕ.

The following lemma shows that Θ(𝜆) is quasimonotone in 𝜆∈ℝ⁺.

Lemma 4.1. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, if there exist 𝜇>0 and 𝑚∈ℤ such that 𝜋Θ(𝜇)=−^𝑝2+𝑚𝜋_𝑝, (4.5) then 𝑑𝑑𝜆Θ(𝜆)|_𝜆=𝜇>0,𝜋Θ(𝜆)>(<)−^𝑝2+𝑚𝜋_𝑝∀𝜆>(<)𝜇. (4.6)(4.7)

Proof. If 𝜇>0 satisfies (4.5), then 𝜇 is a Dirichlet half-eigenvalue of (1.1) and the corresponding half-function 𝑋(𝑡)=𝑋(𝑡;𝜆𝑎,𝜆𝑏) satisfies the boundary condition (1.4). By (2.15) and (2.16), 𝑑𝑑𝜆Θ(𝜆)|_𝜆=𝑢=¹₀(𝑎𝑋^𝑝₊+𝑏𝑋^𝑝₋=1)𝑑𝑡𝜇(¹₀|𝑋^′(𝑡)|^𝑝𝑑𝑡−𝑋(𝑡)𝜙_𝑝(𝑋^′(𝑡))|¹_𝑡=0)=1𝜇¹₀|𝑋^′(𝑡)|^𝑝𝑑𝑡>0, (4.8) proving (4.6), and, therefore, (4.7) holds.

It follows from (4.2), Lemmas 2.3 and 4.1 that (4.3) has a positive solution if and only if 𝑚∈ℕ. For each 𝑚∈ℕ, (4.3) has the unique positive solution denoted by 𝜆⁺_𝑚(𝑎,𝑏). Moreover, one has (0<)𝜆⁺₁(𝑎,𝑏)<𝜆⁺₂(𝑎,𝑏)<⋯<𝜆⁺_𝑚(𝑎,𝑏)⋯(→∞). (4.9) One can prove similarly that (4.4) has a positive solution if and only if 𝑚∈ℕ, and for each 𝑚∈ℕ, (4.4) has the unique positive solution denoted by 𝜆⁻_𝑚(𝑎,𝑏). Moreover, one has (0<)𝜆⁻₁(𝑎,𝑏)<𝜆⁻₂(𝑎,𝑏)<⋯<𝜆⁻_𝑚(𝑎,𝑏)⋯(→∞). (4.10) Let 𝑧=−𝑥 in (1.1). Then 𝑧₊=𝑥₋, 𝑧₋=𝑥₊, and (𝜙_𝑝(𝑧^′))^′+𝜆𝑏(𝑡)𝜙_𝑝(𝑧₊)−𝜆𝑎(𝑡)𝜙_𝑝(𝑧₋)=0. (4.11) Consequently, we have 𝜆⁻_𝑚(𝑎,𝑏)=𝜆⁺_𝑚(𝑏,𝑎)∀𝑚∈ℕ. (4.12)

In summary, we can completely characterize the structure of Σ⁺_𝐷.

Theorem 4.2. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, then Σ⁺_𝐷 consists of two sequences (4.9) and (4.10), which are related to each other by (4.12). Moreover, the half-eigenfunctions corresponding to 𝜆^±_𝑚(𝑎,𝑏) have precisely 𝑚−1 zeroes in the interval (0,1).

Proof. We need only prove the nodal property, while this can be obtained by (2.11) as done in Lemma 3.2.

4.2. Dependence of Dirichlet Half-Eigenvalues on Weights

By (4.12), we only discuss the dependence of 𝜆⁺_𝑚(𝑎,𝑏) on weights (𝑎,𝑏). Analogous results hold for 𝜆⁻_𝑚(𝑎,𝑏).

Theorem 4.3. Given 𝛾∈[1,∞] and 𝑚∈ℕ, the functional (𝒲^𝛾₊,𝑤_𝛾×𝑤_𝛾)→ℝ,(𝑎,𝑏)↦𝜆⁺_𝑚(𝑎,𝑏), (4.13) is continuous. Here 𝑤_𝛾 denotes the weak topology in ℒ^𝛾.

Proof. Suppose (𝑎_𝑛,𝑏_𝑛)→(𝑎,𝑏) in (𝒲^𝛾₊,𝑤_𝛾×𝑤_𝛾). Denote 𝜈_𝑛∶=𝜆⁺_𝑚(𝑎_𝑛,𝑏_𝑛). We aim to prove 𝜈_𝑛→𝜆⁺_𝑚(𝑎,𝑏) as 𝑛→∞.
We claim that the sequence {𝜈_𝑛}_𝑛∈ℕ is bounded. If this is false, we may assume that 𝜈_𝑛→∞. Then for any Λ>0, there exists 𝑁∈ℕ such that 𝜈_𝑛>Λ for any 𝑛>𝑁. By the definition of 𝜈_𝑛 and Lemma 4.1, we have Θ(Λ𝑎_𝑛,Λ𝑏_𝑛𝜋)<−^𝑝2+𝑚𝜋_𝑝,∀𝑛∈ℕ. (4.14) Let 𝑛→∞, it follows from Theorem 2.2(i) that 𝜋Θ(Λ𝑎,Λ𝑏)≤−^𝑝2+𝑚𝜋_𝑝, (4.15) which contradicts with Lemma 2.3, because Λ can be chosen arbitrarily large. Thus, the claim is true.
Since {𝜈_𝑛}_𝑛∈ℕ is bounded and 𝜈_𝑛>0, without loss of generality, we may assume that 𝜈_𝑛→𝜈₀≥0. By the definition of 𝜈_𝑛, one has Θ(𝜈_𝑛𝑎_𝑛,𝜈_𝑛𝑏_𝑛𝜋)=−^𝑝2+𝑚𝜋_𝑝,∀𝑛∈ℕ. (4.16) Let 𝑛→∞, by Theorem 2.2(i) again, one has Θ(𝜈₀𝑎,𝜈₀𝜋𝑏)=−^𝑝2+𝑚𝜋_𝑝. (4.17) Therefore, 𝜈₀=𝜆⁺_𝑚(𝑎,𝑏)(>0), proving the desired result.

Given (𝑎,𝑏)∈𝒲^𝛾, 1≤𝛾≤∞, and 𝑚∈ℕ, denote 𝐸_𝑚(𝑡)=𝐸_𝑚(𝑡;𝑎,𝑏)∶=𝑋(𝑡){∫¹₀(𝑎𝑋^𝑝₊+𝑏𝑋^𝑝₋)𝑑𝑡}^1/𝑝, (4.18) where 𝑋(𝑡)=𝑋(𝑡;𝜆⁺_𝑚𝑎,𝜆⁺_𝑚𝑏) (see (2.17) is the half-eigenfunction of (1.1) associated with the half-eigenvalue 𝜆⁺_𝑚=𝜆⁺_𝑚(𝑎,𝑏). Then 𝐸_𝑚 is the half-eigenfunction associated with 𝜆⁺_𝑚 and satisfying the normalized condition ¹₀(𝑎(𝐸_𝑚)^𝑝₊+𝑏(𝐸_𝑚)^𝑝₋)𝑑𝑡=1. (4.19)

Theorem 4.4. Given 𝛾∈[1,∞] and 𝑚∈ℕ, the functional (𝒲^𝛾₊,‖⋅‖_𝛾×‖⋅‖_𝛾)→ℝ,(𝑎,𝑏)↦𝜆⁺_𝑚=𝜆⁺_𝑚(𝑎,𝑏) (4.20) is continuously differentiable. Moreover, the Fréchet derivatives of 𝜆⁺_𝑚 at 𝑎 and at 𝑏, denoted by 𝜕_𝑎𝜆⁺_𝑚 and 𝜕_𝑏𝜆⁺_𝑚, respectively, are given by the following functionals: 𝜕_𝑎𝜆⁺_𝑚=−𝜆⁺_𝑚(𝐸_𝑚)^𝑝₊∈𝒞⁰⊂(ℒ^𝛾,‖⋅‖_𝛾)^∗,𝜕_𝑏𝜆⁺_𝑚=−𝜆⁺_𝑚(𝐸_𝑚)^𝑝₋∈𝒞⁰⊂(ℒ^𝛾,‖⋅‖_𝛾)^∗, (4.21)(4.22) where 𝐸_𝑚(⋅)=𝐸_𝑚(⋅;𝑎,𝑏) is the normalized half-eigenfunction associated with 𝜆⁺_𝑚(𝑎,𝑏) as defined in (4.18).

Proof. Consider (4.3) and let 𝐹(𝜆,𝑎,𝑏)∶=Θ(−𝜋_𝑝/2,𝜆𝑎,𝜆𝑏). By Theorem 2.2(ii), 𝐹 is continuously differentiable in (𝜆,𝑎,𝑏)∈ℝ×𝒲^𝛾₊. We have proved in Lemma 4.1 that 𝜕𝐹𝜕𝜆(𝜆⁺_𝑚(𝑎,𝑏),𝑎,𝑏)>0. (4.23) Thus the Implicit Function Theorem implies that the functional 𝜆=𝜆⁺_𝑚=𝜆⁺_𝑚(𝑎,𝑏)(>0) determined by (4.3) is continuously differentiable in (𝑎,𝑏)∈𝒲^𝛾₊. Moreover, for any ℎ∈ℒ^𝛾, one has 𝜕_𝑎Θ∘(𝜆+𝑎𝜕_𝑎𝜆)∘ℎ+𝜕_𝑏Θ∘(𝑏𝜕_𝑎𝜆)∘ℎ=0. (4.24) By (2.15) and (2.16), this reads ¹₀{𝑋^𝑝₊[𝜆ℎ+(𝜕_𝑎𝜆)ℎ𝑎]+𝑋^𝑝₋[(𝜕_𝑎𝜆)ℎ𝑏]}𝑑𝑡=0, (4.25) where 𝑋(⋅)=𝑋(⋅;𝜆⁺_𝑚𝑎,𝜆⁺_𝑚𝑏) is the half-eigenfunction of (1.1) associated with the half-eigenvalue 𝜆⁺_𝑚. Since 𝜕_𝑎𝜆∶ℒ^𝛾→ℝ and (𝜕_𝑎𝜆)ℎ∈ℝ, one has (𝜕_𝑎𝜆)ℎ¹₀(𝑎𝑋^𝑝₊+𝑏𝑋^𝑝₋)𝑑𝑡=−𝜆¹₀𝑋^𝑝₊ℎ𝑑𝑡,(𝜕_𝑎𝜆)ℎ=−𝜆∫¹₀𝑋^𝑝₊ℎ𝑑𝑡∫¹₀(𝑎𝑋^𝑝₊+𝑏𝑋^𝑝₋)𝑑𝑡=−𝜆¹₀(𝐸_𝑚)^𝑝₊ℎ𝑑𝑡, (4.26) which proves (4.21). One can obtain (4.22) analogously.

Corollary 4.5. Given 𝛾∈[1,∞] and (𝑎_𝑖,𝑏_𝑖)∈𝒲^𝛾₊, 𝑖=0,1, one has (𝑎₀,𝑏₀)≻(≥)(𝑎₁,𝑏₁)⇒𝜆⁺_𝑚(𝑎₀,𝑏₀)<(≤)𝜆⁺_𝑚(𝑎₁,𝑏₁),∀𝑚∈ℕ. (4.27)

Proof. Let 𝑎_𝜏=𝜏𝑎₀+(1−𝜏)𝑎₁ and 𝑏_𝜏=𝜏𝑏₀+(1−𝜏)𝑏₁ for any 𝜏∈[0,1]. Although 𝒲^𝛾₊ is not convex, if (𝑎₀,𝑏₀)≥(𝑎₁,𝑏₁), we still have (𝑎_𝜏,𝑏_𝜏)∈𝒲^𝛾₊ for any 𝜏∈[0,1]. Therefore, for any given 𝑚∈ℕ, the function 𝑀∶[0,1]→(0,∞),𝜏↦𝑀(𝜏)=𝜆⁺_𝑚(𝑎_𝜏,𝑏_𝜏), (4.28) is well-defined (by Theorem 4.2) and is continuously differentiable (by Theorem 4.4). Moreover, by (4.21) and (4.22), one has 𝑀^′(𝜏)=−𝑀(𝜏)¹₀{(𝐸_𝑚)^𝑝₊(𝑎₀−𝑎₁)+(𝐸_𝑚)^𝑝₋(𝑏₀−𝑏₁)}𝑑𝑡,∀𝜏∈[0,1], (4.29) where 𝐸_𝑚(⋅)=𝐸_𝑚(⋅;𝑎_𝜏,𝑏_𝜏) is the normalized half-eigenfunction of (1.1) associated with 𝜆⁺_𝑚(𝑎_𝜏,𝑏_𝜏) (cf. (4.18). Consequently, one has 𝑀(𝜏)<(≤)0 if only (𝑎₀,𝑏₀)≻(≥)(𝑎₁,𝑏₁), proving the desired results.

5. The Neumann Spectrum and Positive Principal Eigenvalues

5.1. Structure of Neumann Half-Eigenvalues

Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, in this section we use the following notations for simplicity if there is no ambiguity Θ(𝜆)=Θ(𝜆𝑎,𝜆𝑏)∶=Θ(0,𝜆𝑎,𝜆𝑏),𝑋(𝑡)=𝑋(𝑡;𝑎,𝑏)∶=𝑋(𝑡;0,𝑎,𝑏). (5.1) Notice that 0∈Σ_𝑁 with the constant half-eigenfunctions. Due to the jumping nonlinearity of (1.1), 𝜆∈Σ⁺_𝑁∶=Σ_𝑁∩ℝ⁺ if and only if 𝜆≥0 is determined by one of the following two equations: Θ(0,𝜆𝑎,𝜆𝑏)=𝑚𝜋_𝑝,Θ(𝜋_𝑝,𝜆𝑎,𝜆𝑏)=𝜋_𝑝+𝑚𝜋_𝑝, (5.2)(5.3) for some 𝑚∈ℤ. We still need to specify the range of 𝑚.

The following lemma shows that Θ(𝜆) is quasimonotone in 𝜆∈ℝ⁺. We omit the proof because it is similar to that of Lemma 4.1.

Lemma 5.1. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, if there exist 𝜇>0 and 𝑚∈ℤ such that Θ(𝜇)=𝑚𝜋_𝑝, (5.4) then 𝑑𝑑𝜆Θ(𝜆)|_𝜆=𝜇𝜋>0,Θ(𝜆)>(<)−^𝑝2+𝑚𝜋_𝑝∀𝜆>(<)𝜇. (5.5)

Notice that Θ(0)=0. As a consequence of Lemmas 2.3 and 5.1, (5.2) has a solution if and only if 𝑚∈ℤ⁺. More precisely, for each 𝑚∈ℕ, (5.2) has the unique solution 𝜆=𝜆⁺_𝑚(𝑎,𝑏)>0, and for 𝑚=0, (5.2) has the zero solution 𝜆=0 and has at most one additional positive solution (called positive principal half-eigenvalue) 𝜆=𝜆⁺₀(𝑎,𝑏). Moreover, one has 0<𝜆⁺₀(𝑎,𝑏)<𝜆⁺₁(𝑎,𝑏)<𝜆⁺₂(𝑎,𝑏)<⋯<𝜆⁺_𝑚(𝑎,𝑏)⋯(→∞). (5.6) Similar arguments as in the proof of Theorem 3.10 show that 𝜆⁺₀(𝑎,𝑏)(>0)exists⟺¹₀𝑎(𝑡)𝑑𝑡<0. (5.7) By Lemma 5.1, if the principal half-eigenvalue 𝜆⁺₀ exists, then Θ(𝜆)<0,∀𝜆∈(0,𝜆⁺₀). (5.8)

Similar results hold for (5.3). Denote the positive solution of (5.3) by 𝜆⁻_𝑚(𝑎,𝑏) for any 𝑚∈ℤ⁺. One has 0<𝜆⁻₀(𝑎,𝑏)<𝜆⁻₁(𝑎,𝑏)<𝜆⁻₂(𝑎,𝑏)<⋯<𝜆⁻_𝑚(𝑎,𝑏)⋯(→∞), (5.9) where 𝜆⁻₀(𝑎,𝑏) is also called positive principal half-eigenvalue and 𝜆⁻₀(𝑎,𝑏)(>0)exists⟺¹₀𝑏(𝑡)𝑑𝑡<0. (5.10) It can also be proved that 𝜆⁻_𝑚(𝑎,𝑏)=𝜆⁺_𝑚(𝑏,𝑎),∀𝑚∈ℤ⁺ (5.11) if only both 𝜆⁺_𝑚(𝑏,𝑎) and 𝜆⁻_𝑚(𝑎,𝑏) exist.

In summary, we can completely characterize the structure of Σ⁺_𝑁.

Theorem 5.2. Given (𝑎,𝑏)∈𝒲^𝛾₊, 1≤𝛾≤∞, then Σ⁺_𝑁 consists of two sequences (5.6) and (5.9), which satisfy (5.7), (5.10), and (5.11). The half-eigenfunctions corresponding to 𝜆^±_𝑚(𝑎,𝑏) have precisely 𝑚−1 zeroes in the interval (0,1).

5.2. Dependence of Neumann Half-Eigenvalues on Weights

By (5.11), we only discuss the dependence of 𝜆⁺_𝑚(𝑎,𝑏) on weights (𝑎,𝑏). Analogous results hold for 𝜆⁻_𝑚(𝑎,𝑏). Given (𝑎,𝑏)∈𝒲^𝛾₊ and 𝑚∈ℤ⁺, denote 𝐸_𝑚(𝑡)=𝐸_𝑚(𝑡;𝑎,𝑏)∶=𝑋(𝑡){∫¹₀(𝑎𝑋^𝑝₊+𝑏𝑋^𝑝₋)𝑑𝑡}^1/𝑝, (5.12) where 𝑋(𝑡)=𝑋(𝑡;𝜆⁺_𝑚𝑎,𝜆⁺_𝑚𝑏) is the half-eigenfunction of (1.1) associated with the half-eigenvalue 𝜆⁺_𝑚=𝜆⁺_𝑚(𝑎,𝑏). Then 𝐸_𝑚 is also a half-eigenfunction, and it satisfies the normalized condition (𝑎(𝐸¹_0𝑚)^𝑝₊+𝑏(𝐸_𝑚)^𝑝₋)𝑑𝑡=1. (5.13) Denote 𝒲^𝛾₊₋∶={(𝑎,𝑏)∈𝒲^𝛾₊∣¹₀𝑎(𝑡)𝑑𝑡<0}. (5.14)

Theorem 5.3. Given 𝛾∈[1,∞], the functionals 𝒲^𝛾₊→ℝ,(𝑎,𝑏)↦𝜆⁺_𝑚=𝜆⁺_𝑚𝒲(𝑎,𝑏)(𝑚∈ℕ),_𝛾+−→ℝ,(𝑎,𝑏)↦𝜆⁺₀=𝜆⁺₀(𝑎,𝑏), (5.15) are continuous with respect to the weak topology 𝑤_𝛾×𝑤_𝛾, and are continuously Fréchet differentiable with respect to ‖⋅‖_𝛾×‖⋅‖_𝛾. Moreover, for any 𝑚∈ℤ⁺, the derivatives of 𝜆⁺_𝑚 at 𝑎 and at 𝑏, denoted by 𝜕_𝑎𝜆⁺_𝑚 and 𝜕_𝑏𝜆⁺_𝑚, respectively, are given by 𝜕_𝑎𝜆⁺_𝑚=−𝜆⁺_𝑚(𝐸_𝑚)^𝑝₊∈𝒞⁰⊂(ℒ^𝛾,‖⋅‖_𝛾)^∗,𝜕_𝑏𝜆⁺_𝑚=−𝜆⁺_𝑚(𝐸_𝑚)^𝑝₋∈𝒞⁰⊂(ℒ^𝛾,‖⋅‖_𝛾)^∗, (5.16) where 𝐸_𝑚=𝐸_𝑚(𝑡;𝑎,𝑏) is the normalized half-eigenfunction associated with 𝜆⁺_𝑚(𝑎,𝑏) as defined in (5.12).

Proof. By checking the proof of Theorems 4.3 and 4.4, the only matter to be proved is the continuity of 𝜆⁺₀∶(𝒲^𝛾₊₋,𝑤_𝛾)→(0,∞), (5.17) while this can be attained by using (5.8) and similar arguments as in the proof of Theorem 3.12.

Corollary 5.4. (i) If (𝑎_𝑖,𝑏_𝑖)∈𝒲^𝛾₊,𝑖=0,1, and (𝑎₀,𝑏₀)≻(≥)(𝑎₁,𝑏₁), then 𝜆⁺_𝑚(𝑎₀,𝑏₀)<(≤)𝜆⁺_𝑚(𝑎₁,𝑏₁),∀𝑚∈ℕ. (5.18)
(ii) If (𝑎_𝑖,𝑏_𝑖)∈𝒲^𝛾₊₋,𝑖=0,1, and (𝑎₀,𝑏₀)≻(≥)(𝑎₁,𝑏₁), then 𝜆⁺_𝑚(𝑎₀,𝑏₀)<(≤)𝜆⁺_𝑚(𝑎₁,𝑏₁),∀𝑚∈ℤ⁺. (5.19)

Acknowledgments

This article is supported by the National Basic Research Program of China (Grant no. 2006CB805903), the National Natural Science Foundation of China (Grant no. 10531010), and the 111 Project of China (2007).