# Various Half-Eigenvalues of Scalarp-Laplacian with Indefinite Integrable Weights

Authors: Wei Li; Ping Yan
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

Consider the half-eigenvalue problem(ϕp(x′))′+λa(t)ϕp(x+)−λb(t)ϕp(x−)=0 a.e.
t∈[0,1], where 1<p<∞, ϕp(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 ℒγ:=Lγ([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)∈(ℒγ,wγ)2, where wγ denotes the weak topology in ℒγ space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in (a,b)∈(ℒγ,∥⋅∥γ)2, where ∥⋅∥γ is the Lγ norm of ℒγ.


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## Body

## 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 exponentp∈(1,∞), let ϕp(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∈ℒγ:=Lγ([0,1],ℝ), 1≤γ≤∞, we are concerned with the half-eigenvalue problem
(1.1)(ϕp(x′))′+λa(t)ϕp(x+)-λb(t)ϕp(x-)=0,a.e.t∈[0,1],
associated with the periodic boundary condition
(1.2)x(0)-x(1)=x′(0)-x′(1)=0,
the antiperiodic boundary condition
(1.3)x(0)+x(1)=x′(0)+x′(1)=0,
the Dirichlet boundary condition
(1.4)x(0)=x(1)=0,
or the Neumann boundary condition
(1.5)x′(0)=x′(1)=0,
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 ΣP, ΣA, ΣD, and ΣN, 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 ΣP, ΣA, ΣD, and ΣN, and the other is the dependence of all these spectra on weights.Generally speaking, for thep-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 ℒγ (1≤γ<∞) weights (or potentials), respectively. It has been proved very recently in [1] that with an indefinite weight a∈ℒγ, 1≤γ≤∞, the spectra structures of
(1.6)(ϕp(x′))′+λa(t)ϕp(x)=0a.e.t∈[0,1]
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 ΣP and ΣA and complete characterization of ΣD and ΣN, 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 usualLγ 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 wγ. 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-eigenvaluesλ̲mL and λ¯mR (defined by (3.22) and (3.25), resp.), and all the half-eigenvalues in ΣD and ΣN, are continuous in weights (a,b)∈(ℒγ,wγ)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)∈(ℒγ,∥·∥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).Ifλ 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 ΣA/P, ΣD, and ΣN, respectively.


## 2. Preliminary Results

Givenp∈(1,∞), denote by (Cp(θ),Sp(θ)) the unique solution of the initial value problem
(2.1)dxdθ=-ϕp*(y),dydθ=ϕp(x),(x(0),y(0))=(1,0).
The functions Cp(θ) and Sp(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)
bothCp(θ) and Sp(θ) are 2πp-periodic, where
(2.2)πp=2π(p-1)1/ppsin(π/p);(ii)
Cp(θ)=0 if and only if θ=πp/2+mπp, m∈ℤ, and Sp(θ)=0 if and only if θ=mπp, m∈ℤ;(iii)
one has(2.3)|Cp(θ)|p+(p-1)|Sp(θ)|p*≡1.Givena,b∈ℒγ, γ∈[1,∞], let
(2.4)y=-ϕp(x′).
In the p-polar coordinates
(2.5)x=r2/pCp(θ),y=r2/p*Sp(θ),
the scalar equation
(2.6)(ϕp(x′))′+a(t)ϕp(x+)-b(t)ϕp(x-)=0,a.e.t∈[0,1],
is transformed into the following equations for r and θ(2.7)θ′=A(t,θ;a,b)≔{a(t)|Cp(θ)|p+(p-1)|Sp(θ)|p*,ifCp(θ)≥0,b(t)|Cp(θ)|p+(p-1)|Sp(θ)|p*,ifCp(θ)<0,(2.8)(logr)′=G(t,θ;a,b)≔{p2(a(t)-1)ϕp(Cp(θ))ϕp*(Sp(θ)),ifCp(θ)≥0,p2(b(t)-1)ϕp(Cp(θ))ϕp*(Sp(θ)),ifCp(θ)<0.
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
(2.9)Θ(ϑ0,a,b)≔θ(1;ϑ0,a,b),R(ϑ0,a,b)≔r(1;ϑ0,a,b).
As A(t,θ;a,b) is independent of r and is 2πp-periodic in θ, we have
(2.10)θ(t;ϑ0+2mπp,a,b)≡θ(t;ϑ0,a,b)+2mπp
for all t∈[0,1], ϑ0∈ℝ, and m∈ℤ.An important property of the argument solutionθ is the quasimonotonicity as in the following lemma.Lemma 2.1 (see [14]).
Letθ(t)=θ(t;ϑ0,a,b) be a solution of (2.7). Then
(2.11)θ(t)≥-πp2+mπpatt∈[0,1)⇒θ(s)>-πp2+mπp∀s∈(t,1].Denote by𝒞0:=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(2.12)ℝ×(ℒγ,wγ)2→ℝ,(ϑ,a,b)↦Θ(ϑ,a,b)
is continuous. Here wγ denotes the weak topology in ℒγ.
(ii)  The functional(2.13)ℝ×(ℒγ,∥·∥γ)2→ℝ,(ϑ,a,b)↦Θ(ϑ,a,b)
is continuously differentiable. The derivatives of Θ(ϑ,a,b) at ϑ, at a∈ℒγ, and at b∈ℒγ (in the Fréchet sense), denoted, respectively, by ∂ϑΘ, ∂aΘ, and ∂bΘ, are
(2.14)∂ϑΘ(ϑ,a,b)=1R2(ϑ,a,b),(2.15)∂aΘ(ϑ,a,b)=X+p∈𝒞0⊂(ℒγ,∥·∥γ)*,(2.16)∂bΘ(ϑ,a,b)=X-p∈𝒞0⊂(ℒγ,∥·∥γ)*,
where
(2.17)X=X(t)=X(t;ϑ,a,b):={r(t;ϑ,a,b)}2/pCp(θ(t;ϑ,a,b)){r(1;ϑ,a,b)}2/p
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∈ℒγ, 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
(2.18)𝒲+γ:={(a,b)∣a,b∈ℒγ,(a+,b+)≻(0,0)}.
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
(2.19)limλ→∞(Θ(ϑ,λa,λb)-ϑ)=∞
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
(2.20)xλ=x,yλ=-λ-1/p*ϕp(x′)=λ-1/p*y
when λ>0. However, (1.1) becomes
(2.21)xλ=-λ1/pϕp*(yλ),yλ′=λ1/p{a(t)ϕp((xλ)+)+b(t)ϕp((xλ)-)}.
In the p-polar coordinates of the (xλ,yλ)-plane
(2.22)xλ:=rλ2/pCp(θλ),yλ:=rλ2/p*Sp(θλ),
the equation for the new argument θλ is
(2.23)θλ′=λ1/p{1+a(t)(Cp(θλ))+p+b(t)(Cp(θλ))-p-|Cp(θλ)|p}.
Denote by θλ(t;ϑ,a,b) the solution of (2.23) satisfying the initial condition θλ(0)=ϑ. By (2.20), the argument functions of (1.1), θλ(t)=θλ(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 {mπp,mπp+πp/2:m∈ℤ}, in the following relation
(2.24)θ(t;ℋ(ϑ),a,b)=ℋ(θλ(t;ϑ,a,b)).
If lemma is not true, then there exist{λn}n∈ℕ and {ϑn}n∈ℕ such that λn↑∞ and {Θ(ϑn,λna,λnb)-ϑn}n∈ℕ is bounded from above. Combining (2.11), there exists some m0∈ℕ such that
(2.25)-πp<Θ(ϑn,λna,λnb)-ϑn<m0πp,∀n∈ℕ.
By (2.11) again, we have
(2.26)-πp<θ(t;ϑn,λna,λnb)-ϑn<(m0+1)πp,∀t∈[0,1],∀n∈ℕ.
By (2.10), we may assume
(2.27)ϑn∈[0,2πp],∀n∈ℕ.
Hence,
(2.28)-πp<θ(t;ϑn,λna,λnb)<(m0+3)πp,∀t∈[0,1],∀n∈ℕ.
Denoteθn(t):=θλn(t;ϑn,a,b) for simplicity. By the conjugacy (2.24) and the estimate (2.28), we have
(2.29)-πp<θn(t)<(m0+3)πp,∀t∈[0,1],∀n∈ℕ.
DenoteI0:={f∈ℒ∞:0≤f≤1} and define
(2.30)fn(t)≔(Cp(θn(t)))+p∈I0,gn(t)≔(Cp(θn(t)))-p∈I0,hn(t)≔|Cp(θn(t))|p∈I0,
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
(2.31)fn→f0∈I0,gn→g0∈I0,hn→h0∈I0,in(ℒ∞,w∞).
From (2.23), θn(t) satisfies
(2.32)θn′=λn1/p{1+a(t)fn(t)+b(t)gn(t)-hn(t)}a.e.t∈[0,1].
For any t∈[0,1], we have
(2.33)θn(t)-ϑnλn1/p=∫0t{1+a(s)fn(s)+b(s)gn(s)-hn(s)}ds=t+∫01a(s)χ[0,t](s)·fn(s)ds+∫01b(s)χ[0,t](s)·gn(s)ds-∫01χ[0,t](s)·hn(s)ds.
Let n→∞, by (2.27) and (2.29), the left-hand side tends to 0. Since
(2.34)a(·)χ[0,t](·),b(·)χ[0,t](·),χ[0,t](·)∈ℒ1,
we can use (2.31) to deduce from the above equality
(2.35)0=t+∫01a(s)χ[0,t](s)·f0(s)ds+∫01b(s)χ[0,t](s)·g0(s)ds-∫01χ[0,t](s)·h0(s)ds=∫0t{1+a(s)f0(s)+b(s)g0(s)-h0(s)}ds
for all t∈[0,1]. Thus
(2.36)1+a(t)f0(t)+b(t)g0(t)-h0(t)=0,a.e.t∈[0,1].
As hn=fn+gn 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
(2.37)h0(t)=f0(t)+g0(t)≠0,a.e.t∈[0,1].
Therefore,
(2.38)a(t)f0(t)+b(t)g0(t)=h0(t)-1≤0,a.e.t∈[0,1],
which contradicts the assumption (a,b)∈𝒲+γ.Remark 2.4.
Leta≻0 and b(t)=0, a.e. t∈[0,1] in (1.1). Then the equation for the corresponding argument function is
(2.39)θ′=λa(t)(Cp(θ))+p+(p-1)|Sp(θ)|p*
(cf. (2.7). The solution θ(t) takes a constant length of time, say T0>0, to start from πp/2+2mπp and to reach at 3πp/2+2mπp for any m∈ℤ. Consequently, Θ(-πp/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

Givena,b∈ℒγ, 1≤γ≤∞, by (2.9), (2.10) and Theorem 2.2(i), one sees that
(3.1)Θ̲(a,b)≔maxϑ0∈[0,2πp](Θ(ϑ0,a,b)-ϑ0)=maxϑ0∈ℝ(Θ(ϑ0,a,b)-ϑ0),(3.2)Θ¯(a,b)≔minϑ0∈[0,2πp](Θ(ϑ0,a,b)-ϑ0)=minϑ0∈ℝ(Θ(ϑ0,a,b)-ϑ0)
are well-defined. Moreover, we have the following lemma.Lemma 3.1.
Givenγ∈[1,∞], the functionals
(3.3)(ℒγ,wγ)2→ℝ,(a,b)↦Θ̲(a,b),(ℒγ,wγ)2→ℝ,(a,b)↦Θ¯(a,b)
are continuous.Given(a,b)∈𝒲+γ, 1≤γ≤∞, we write
(3.4)Θ̲(λ):=Θ̲(λa,λb),Θ¯(λ):=Θ¯(λa,λb)
for simplicity if there is no ambiguity. By Lemma 3.1, Θ̲(λ) and Θ¯(λ) are continuous in λ∈ℝ. By Lemma 2.3, one has
(3.5)limλ→∞Θ̲(λ)=limλ→∞Θ¯(λ)=∞.
Moreover, by setting a=b=0 in (2.7), we know that
(3.6)Θ̲(0)∈(0,πp),Θ¯(0)=0.
Now we are considering the following two sequences of equations for λ(3.7)Θ̲(λ)=mπp,m∈ℕ,(3.8)Θ¯(λ)=mπp,m∈ℤ+:=ℕ∪{0}.Lemma 3.2.
Given(a,b)∈𝒲+γ, 1≤γ≤∞.
(i)
For anym 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 Θ̲(λ) and Θ¯(λ) in λ∈ℝ+.
(ii) Supposeλ∈ℝ+ satisfies (3.7) or (3.8). Then there exists ϑ̂0∈[0,2πp] such that
(3.9)Θ(ϑ̂0,λa,λb)-ϑ̂0=mπp,(3.10)ddϑ0Θ(ϑ0,λa,λb)|ϑ0=ϑ̂0=1.
By (2.14), the latter equation is
(3.11)R(ϑ̂0,λa,λb)=1.
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,ϑ̂0) arrives at (r,θ)=(1,ϑ̂0+mπp) after one period. Thus λ is a periodic or an antiperiodic half-eigenvalue of (1.1) if m is even or odd, respectively.
Denote byx the half-eigenfunction corresponding to λ. Then x(t)=0 if and only if
(3.12)θ(t):=θ(t;ϑ̂0,λa,λb)=-πp2+kπp,k∈ℤ.
Notice that θ(0)=ϑ̂0, θ(1)=ϑ̂0+mπp (see (3.9), and θ(t) is quasimonotone in t (see (2.11). Consequently, if ϑ̂0=-πp/2+l0πp for some l0∈ℤ, then (3.12) holds only for k=l0,l0+1,l0+2,…,l0+m at (0=)t0<t1<t2<⋯<tm(=1). If -πp/2+l0πp<ϑ̂0<-πp/2+(l0+1)πp, then (3.12) holds only for k=l0+1,l0+2,…,l0+m at (0<)t1<t2<⋯<tm(<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)
IfΘ̲(μ)=mπp for some μ>0 and m∈ℕ, then there exists δ>0 such that
(3.13)Θ̲(λ)>mπp,∀λ∈(μ,μ+δ).(ii)
IfΘ¯(μ)=mπp for some μ>0 and m∈ℤ+, then there exists δ∈(0,μ) such that
(3.14)Θ¯(λ)<mπp,∀λ∈(μ-δ,μ).Proof.
We only prove (i) and the proof of (ii) is analogous.
Assume thatΘ̲(μ)=mπp. By Lemma 3.2, μ is a periodic or an antiperiodic half-eigenvalue of (1.1). Then there exists some ϑ0∈[0,2πp] such that
(3.15)Θ(ϑ0,μa,μb)-ϑ0=mπp,
and the corresponding half-eigenfunction X(t)=X(t;ϑ0,μa,μb) satisfies
(3.16)(ϕp(X′))′+μa(t)ϕp(X+)-μb(t)ϕp(X-)=0,a.e.t∈[0,1],
and the boundary condition (1.2) or (1.3). Applying (2.15) and (2.16), we have
(3.17)ddλΘ(ϑ0,λa,λb)|λ=μ=∫01(aX+p+bX-p)dt,
Multiplying (3.16) by X and integrating over [0,1], one has
(3.18)∫01(aX+p+bX-p)dt=1μ(∫01|X′|pdt-X(t)ϕp(X′(t))|t=01)=1μ∫01|X′|pdt>0,
and, therefore,
(3.19)ddλΘ(ϑ0,λa,λb)|λ=μ>0.
Then there exists δ>0 such that
(3.20)Θ(ϑ0,λa,λb)-ϑ0>mπp∀λ∈(μ,μ+δ).
Consequently, we have
(3.21)Θ̲(λ)≥Θ(ϑ0,λa,λb)-ϑ0>mπp,∀λ∈(μ,μ+δ).Definition 3.4.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, define
(3.22)λ̲mL≔min{λ>0∣Θ̲(λ)=mπp},m∈ℕ,(3.23)λ̲mR≔max{λ>0∣Θ̲(λ)=mπp},m∈ℕ,(3.24)λ¯mL≔min{λ≥0∣Θ¯(λ)=mπp},m∈ℤ+,(3.25)λ¯mR≔max{λ≥0∣Θ¯(λ)=mπp},m∈ℤ+.These valuesλ̲mL/R and λ¯mL/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
(3.26)Θ̲(λ){<mπp,if0≤λ<λ̲mL∈[mπp,mπp+πp),ifλ̲mL≤λ≤λ̲mR>mπp,ifλ>λ̲mR∀m∈ℕ,Θ¯(λ){<mπp,if0≤λ<λ¯mL∈(mπp-πp,mπp],ifλ¯mL≤λ≤λ¯mR>mπp,ifλ>λ¯mR.∀m∈ℤ+.Combining (3.5), (3.6), and (3.26), one has the following ordering for these half-eigenvalues:
(3.27)(0<)λ̲1L≤λ̲1R<λ̲2L≤λ̲2R<⋯<λ̲mL≤λ̲mR<⋯(→∞),(0=)λ¯0L≤λ¯0R<λ¯1L≤λ¯1R<⋯<λ¯mL≤λ¯mR<⋯(→∞).
Moreover, by the definition of λ̲mL/R and λ¯mL/R, one has
(3.28)λ̲mL≤λ¯mR,λ̲mR,λ¯mL∈[λ̲mL,λ¯mR],∀m∈ℕ.
Notice that the ordering between λ̲mR and λ¯mL is not determinate.Till now, we still do not know the ordering betweenλ¯mR and λ̲nL with n>m. This will be partially settled by associating (1.1) with the rotation number function defined by
(3.29)ρ(λ)=ρ(λ,a,b):=limn→∞Θn(ϑ0,λa,λb)-ϑ02nπp,
where Θn is the nth iteration of Θ, namely,
(3.30)Θn(ϑ0,a,b)=Θ(Θn-1(ϑ0,a,b),a,b).
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)
limλ→∞ρ(λ)=∞;(iii)
λ∈[0,λ̲mL),m∈ℕ⇒ρ(λ)<m/2;(iv)
λ∈(λ¯mR,∞),m∈ℤ+,⇒ρ(λ)>m/2;(v)
λ∈[λ̲2mL,λ¯2mR],m∈ℕ⇒ρ(λ)=m;(vi)
λ=λ¯0R⇒ρ(λ)=0.Proof.
(i) By (2.11), one has Θn(0,λa,λb)>-πp/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)|Sp(θ)|p*, which has equilibria θ=kπp, k∈ℤ. Thus Θn(-πp/2,0·a,0·b)∈(-πp/2,0) for all n∈ℕ and consequently ρ(0)=0.
(ii) GivenM>0 arbitrarily large, it follows from (3.5) that there exists Λ>0 such that
(3.31)Θ(ϑ0,λa,λb)-ϑ0>2Mπp,∀λ>Λ,∀ϑ0∈ℝ.
Thus
(3.32)Θn(ϑ0,λa,λb)-ϑ0=[Θ(Θn-1(ϑ0,λa,λb),λa,λb)-Θ(Θn-2(ϑ0,λa,λb),λa,λb)]+[Θ(Θn-2(ϑ0,λa,λb),λa,λb)-Θ(Θn-3(ϑ0,λa,λb),λa,λb)]+⋯+[Θ(ϑ0,λa,λb)-ϑ0]≥2nMπp
for any λ>Λ and any ϑ0∈ℝ. Consequently, ρ(λ)≥M for any λ>Λ.
(iii) Givenλ∈[0,λ̲mL),m∈ℕ, it follows from (3.26) that there exists δ=δ(λ)>0 such that
(3.33)Θ̲(λ)<(m-δ)πp.
By the definition of Θ̲(λ)=Θ̲(λa,λb) (see (3.1), this implies that
(3.34)Θ(ϑ0,λa,λb)-ϑ0<(m-δ)πp,∀ϑ0∈ℝ.
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) Givenλ∈[λ̲2mL,λ¯2mR],m∈ℕ, it follows from (3.26) that
(3.35)Θ¯(λ)≤2mπp≤Θ̲(λ).
Then there exists ϑ0 such that Θ(ϑ0,λa,λb)-ϑ0=2mπp, and therefore ρ(λ)=m.
(vi) By (3.25), if λ=λ¯0R, 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
(3.36)λ̲2mL=min{λ>0∣ρ(λ)=m},∀m∈ℕ,(3.37)λ¯2mR=max{λ≥0∣ρ(λ)=m},∀m∈ℤ+,
and the ordering
(3.38)(0≤)λ¯0R<λ̲2L≤λ¯2R<⋯<λ̲2mL≤λ¯2mR<λ̲2m+2L≤λ¯2m+2R⋯(→∞).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¯2mR<λ̲2m+2L for any m∈ℕ. Assume on the contrary that λ¯2mR≥λ̲2m+2L for some m. Then
(3.39)λ̲2mL<λ̲2m+2L≤λ¯2mR.
Now Lemma 3.6(v) shows that m=ρ(λ̲2m+2L)=m+1, a contradiction.Corollary 3.8.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, one has
(3.40)λ∈(λ¯2mR,λ̲2m+2L),m∈ℕ⇒ρ(λ)∈(m,m+1).In summary, we can now partially characterize the set of periodic half-eigenvaluesΣP and the set of antiperiodic half-eigenvalues ΣA of (1.1). Denote ΣP+:=ΣP∩ℝ+ and ΣA+:=ΣA∩ℝ+. For any m∈ℤ+, denote by Σm+=Σm+(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
(3.41)ΣP+=⋃m=0∞Σ2m+,ΣA+=⋃m=0∞Σ2m+1+.Theorem 3.9.
Given(a,b)∈𝒲+γ, 1≤γ≤∞.
(i)
For periodic half-eigenvalues, one has(3.42){0,λ¯0R}⊂Σ0+⊂[0,λ¯0R],{λ̲2mL,λ̲2mR,λ¯2mL,λ¯2mR}⊂Σ2m+⊂[λ̲2mL,λ¯2mR],∀m∈ℕ.
Moreover, λ̲2mL/R and λ¯2mL/R satisfy (3.27), (3.28), and (3.38).(ii)
For antiperiodic half-eigenvalues, one has(3.43){λ̲2m+1L,λ̲2m+1R,λ¯2m+1L,λ¯2m+1R}⊂Σ2m+1+,∀m∈ℤ+.
Moreover, λ̲2m+1L/R and λ¯2m+1L/R satisfy (3.27) and (3.28).Proof.
By Lemma3.2 and the definitions of λ̲mL/R and λ¯mL/R (see (3.22)–(3.25), we need only prove
(3.44)Σ0+⊂[0,λ¯0R],Σ2m+⊂[λ̲2mL,λ¯2mR],∀m∈ℕ.
If λ∈Σ2m+, 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
(3.45)θ(ϑ0,λa,λb)=ϑ0+2mπp.
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 forλ¯0R>0.Theorem 3.10.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, then
(3.46)λ¯0R>0⟺∫01a(t)dt<0or∫01b(t)dt<0.Proof.
Assume thatλ¯0R>0. By Lemma 3.2, the half-eigenfunction X=X(t) corresponding to λ¯0R is nowhere vanishing and satisfies the periodic boundary condition (1.2). If X(t)>0 for any t∈[0,1], then
(3.47)(ϕp(X′))′+λ¯0Ra(t)ϕp(X)=0,X≢const.
Therefore,
(3.48)∫01a(t)dt=-1λ¯0R∫01(ϕp(X′))′ϕp(X)dt=-1λ¯0R∫01ϕp(X′)d(1ϕp(X))+ϕp(X′(t))λ¯0Rϕp(X(t))|t=01=-(p-1)λ¯0R∫01|X′|pXpdt<0.
Similarly, if X(t)<0 for any t∈[0,1], then ∫01b(t)dt<0.
On the other hand, assume that∫01a(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
(3.49)ddλΘ(0,λa,λb)|λ=0=∫01a(t)dt<0.
Thus there exists δ>0 such that
(3.50)Θ(0,λa,λb)<0,∀λ∈(0,δ).
By the definition of Θ¯(λ)=Θ¯(λa,λb) (see (3.2), we get
(3.51)Θ¯(λ)<0,∀λ∈(0,δ).
By (3.5), (3.6) and the continuity of Θ¯(λ) in λ∈ℝ+ (see Lemma 3.1), one has λ¯0R>0. Similarly one can obtain λ¯0R>0 if ∫01b(t)dt<0.Notice that for the antiperiodic half-eigenvalues, it does not hold(3.52)Σ2m+1⊂[λ̲2m+1L,λ¯2m+1R],∀m∈ℤ+,
for general a,b∈ℒγ. This is because A(t,θ;a,b) in (2.7) is 2πp-periodic in θ but not πp-periodic in θ. Hence (2.10) holds but one can not obtain
(3.53)θ(t;ϑ0+mπp,a,b)≡θ(t;ϑ0,a,b)+mπp,∀t∈[0,1],ϑ0∈ℝ,m∈ℤ.
However, if a=b, then A(t,θ;a,a) is πp-periodic in θ and
(3.54)θ(t;ϑ0+mπp,a,a)≡θ(t;ϑ0,a,a)+mπp,∀t∈[0,1],ϑ0∈ℝ,m∈ℤ.
The following theorem characterizes periodic and antiperiodic eigenvalues of (1.6). Now Σm+=Σm+(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.
Givenγ∈[1,∞] and a=b∈ℒγ 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(3.55){0,λ¯0R}⊂Σ0+⊂[0,λ¯0R],{λ̲mL,λ̲mR,λ¯mL,λ¯mR}⊂Σm+⊂[λ̲mL,λ¯mR],∀m∈ℕ.(iii)
One has the ordering(3.56)(0≤)λ¯0R<λ̲1L≤λ¯1R<λ̲2L≤λ¯2R<⋯<λ̲mL≤λ¯mR<⋯(→∞).(iv)
Finallyλ¯0R>0⇔∫01a(t)dt<0.Proof.
By Theorems3.9 and 3.10, we need only prove (3.52) and
(3.57)λ¯2mR<λ̲2m+1L,λ¯2m+1R<λ̲2m+2L,∀m∈ℤ+.
Ifλ∈Σm+, m∈ℕ, then there exists ϑ0∈ℝ such that Θ(ϑ0,λa,λa)-ϑ0=mπp. By (3.54) and the definition of ρ(·) in (3.29), one has
(3.58)λ∈Σm+⇒ρ(λ)=m2,
in particular
(3.59)ρ(λ̲2m+1L)=ρ(λ¯2m+1R)=2m+12,∀m∈ℤ+.
Then it follows from Lemma 3.6(iii) and (iv) that
(3.60)λ̲2m+1L=min{λ≥0∣ρ(λ)=m+12},m∈ℤ+,λ¯2m+1R=max{λ≥0∣ρ(λ)=m+12},m∈ℤ+.
Hence (3.52) follows immediately from (3.58) and (3.60).
Supposeλ¯2mR≥λ̲2m+1L for some m∈ℤ+. By Lemma 3.6(iii) and (v) we obtain ρ(λ̲2m+1L)≤m, which contradicts (3.59). Thus λ¯2mR>λ̲2m+1L for any m∈ℤ+. Similarly one can prove λ¯2m+1R<λ̲2m+2L, ∀m∈ℤ+, proving the desired results in (3.57).


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

Theorem 3.12.
Givenγ∈[1,∞], for any admissible m, the following functionals
(3.61)(𝒲+γ,wγ×wγ)→ℝ,(a,b)↦λ̲mL(a,b)(𝒲+γ,wγ×wγ)→ℝ,(a,b)↦λ¯mR(a,b)
are continuous. Here wγ denotes the weak topology in ℒγ.Proof.
We only prove the result forλ̲mL(a,b). The result for λ¯mR(a,b) can be attained analogously. Suppose (an,bn)→(a,b) in (𝒲+γ,wγ×wγ) as n→∞. Then an→a and bn→b in (ℒγ,wγ). Denote νn:=λ̲mL(an,bn). We aim to prove νn→λ̲mL(a,b).
Firstly, we show that the sequence{νn}n∈ℕ is bounded. If this is false, without loss of generality, we may assume that νn→∞ as n→∞. Then given any Λ>0, there exists N∈ℕ such that
(3.62)νn>Λ,∀n>N.
It follows from (3.26) that
(3.63)Θ̲(Λan,Λbn)<mπp,∀n>N.
Let n→∞, we obtain by Lemma 3.1(3.64)Θ̲(Λa,Λb)≤mπp,
which contradicts (3.5) since Λ>0 can be chosen arbitrarily large. Thus {νn}n∈ℕ is bounded.
Since{νn}n∈ℕ is bounded, passing to a subsequence, we assume that νn→ν0 as n→∞. By the definition of νn, we have
(3.65)Θ̲(νnan,νnbn)=mπp.
Let n→∞, by Lemma 3.1 again, we get
(3.66)Θ̲(ν0a,ν0b)=mπp.
Now it suffices to prove ν0=λ̲mL(a,b).
If this is not true, by the definition in (3.22) there holds ν0>λ̲mL(a,b). Then it follows from Lemma 3.3(i) that there exists
(3.67)λ∈(λ̲mL(a,b),ν0)
such that
(3.68)Θ̲(λa,λb)>mπp.
Since Θ̲(λan,λbn)→Θ̲(λa,λb) as n→∞, we may assume, without loss of generality, that
(3.69)Θ̲(λan,λbn)>mπp,∀n∈ℕ.
Therefore, by (3.26), we have
(3.70)λ>λ̲mL(an,bn)=νn,∀n∈ℕ.
Let n→∞, one has λ≥ν0, which contradicts the choice of λ in (3.67).


## 3.1. Structure of Periodic and Antiperiodic Half-Eigenvalues

Givena,b∈ℒγ, 1≤γ≤∞, by (2.9), (2.10) and Theorem 2.2(i), one sees that
(3.1)Θ̲(a,b)≔maxϑ0∈[0,2πp](Θ(ϑ0,a,b)-ϑ0)=maxϑ0∈ℝ(Θ(ϑ0,a,b)-ϑ0),(3.2)Θ¯(a,b)≔minϑ0∈[0,2πp](Θ(ϑ0,a,b)-ϑ0)=minϑ0∈ℝ(Θ(ϑ0,a,b)-ϑ0)
are well-defined. Moreover, we have the following lemma.Lemma 3.1.
Givenγ∈[1,∞], the functionals
(3.3)(ℒγ,wγ)2→ℝ,(a,b)↦Θ̲(a,b),(ℒγ,wγ)2→ℝ,(a,b)↦Θ¯(a,b)
are continuous.Given(a,b)∈𝒲+γ, 1≤γ≤∞, we write
(3.4)Θ̲(λ):=Θ̲(λa,λb),Θ¯(λ):=Θ¯(λa,λb)
for simplicity if there is no ambiguity. By Lemma 3.1, Θ̲(λ) and Θ¯(λ) are continuous in λ∈ℝ. By Lemma 2.3, one has
(3.5)limλ→∞Θ̲(λ)=limλ→∞Θ¯(λ)=∞.
Moreover, by setting a=b=0 in (2.7), we know that
(3.6)Θ̲(0)∈(0,πp),Θ¯(0)=0.
Now we are considering the following two sequences of equations for λ(3.7)Θ̲(λ)=mπp,m∈ℕ,(3.8)Θ¯(λ)=mπp,m∈ℤ+:=ℕ∪{0}.Lemma 3.2.
Given(a,b)∈𝒲+γ, 1≤γ≤∞.
(i)
For anym 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 Θ̲(λ) and Θ¯(λ) in λ∈ℝ+.
(ii) Supposeλ∈ℝ+ satisfies (3.7) or (3.8). Then there exists ϑ̂0∈[0,2πp] such that
(3.9)Θ(ϑ̂0,λa,λb)-ϑ̂0=mπp,(3.10)ddϑ0Θ(ϑ0,λa,λb)|ϑ0=ϑ̂0=1.
By (2.14), the latter equation is
(3.11)R(ϑ̂0,λa,λb)=1.
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,ϑ̂0) arrives at (r,θ)=(1,ϑ̂0+mπp) after one period. Thus λ is a periodic or an antiperiodic half-eigenvalue of (1.1) if m is even or odd, respectively.
Denote byx the half-eigenfunction corresponding to λ. Then x(t)=0 if and only if
(3.12)θ(t):=θ(t;ϑ̂0,λa,λb)=-πp2+kπp,k∈ℤ.
Notice that θ(0)=ϑ̂0, θ(1)=ϑ̂0+mπp (see (3.9), and θ(t) is quasimonotone in t (see (2.11). Consequently, if ϑ̂0=-πp/2+l0πp for some l0∈ℤ, then (3.12) holds only for k=l0,l0+1,l0+2,…,l0+m at (0=)t0<t1<t2<⋯<tm(=1). If -πp/2+l0πp<ϑ̂0<-πp/2+(l0+1)πp, then (3.12) holds only for k=l0+1,l0+2,…,l0+m at (0<)t1<t2<⋯<tm(<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)
IfΘ̲(μ)=mπp for some μ>0 and m∈ℕ, then there exists δ>0 such that
(3.13)Θ̲(λ)>mπp,∀λ∈(μ,μ+δ).(ii)
IfΘ¯(μ)=mπp for some μ>0 and m∈ℤ+, then there exists δ∈(0,μ) such that
(3.14)Θ¯(λ)<mπp,∀λ∈(μ-δ,μ).Proof.
We only prove (i) and the proof of (ii) is analogous.
Assume thatΘ̲(μ)=mπp. By Lemma 3.2, μ is a periodic or an antiperiodic half-eigenvalue of (1.1). Then there exists some ϑ0∈[0,2πp] such that
(3.15)Θ(ϑ0,μa,μb)-ϑ0=mπp,
and the corresponding half-eigenfunction X(t)=X(t;ϑ0,μa,μb) satisfies
(3.16)(ϕp(X′))′+μa(t)ϕp(X+)-μb(t)ϕp(X-)=0,a.e.t∈[0,1],
and the boundary condition (1.2) or (1.3). Applying (2.15) and (2.16), we have
(3.17)ddλΘ(ϑ0,λa,λb)|λ=μ=∫01(aX+p+bX-p)dt,
Multiplying (3.16) by X and integrating over [0,1], one has
(3.18)∫01(aX+p+bX-p)dt=1μ(∫01|X′|pdt-X(t)ϕp(X′(t))|t=01)=1μ∫01|X′|pdt>0,
and, therefore,
(3.19)ddλΘ(ϑ0,λa,λb)|λ=μ>0.
Then there exists δ>0 such that
(3.20)Θ(ϑ0,λa,λb)-ϑ0>mπp∀λ∈(μ,μ+δ).
Consequently, we have
(3.21)Θ̲(λ)≥Θ(ϑ0,λa,λb)-ϑ0>mπp,∀λ∈(μ,μ+δ).Definition 3.4.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, define
(3.22)λ̲mL≔min{λ>0∣Θ̲(λ)=mπp},m∈ℕ,(3.23)λ̲mR≔max{λ>0∣Θ̲(λ)=mπp},m∈ℕ,(3.24)λ¯mL≔min{λ≥0∣Θ¯(λ)=mπp},m∈ℤ+,(3.25)λ¯mR≔max{λ≥0∣Θ¯(λ)=mπp},m∈ℤ+.These valuesλ̲mL/R and λ¯mL/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
(3.26)Θ̲(λ){<mπp,if0≤λ<λ̲mL∈[mπp,mπp+πp),ifλ̲mL≤λ≤λ̲mR>mπp,ifλ>λ̲mR∀m∈ℕ,Θ¯(λ){<mπp,if0≤λ<λ¯mL∈(mπp-πp,mπp],ifλ¯mL≤λ≤λ¯mR>mπp,ifλ>λ¯mR.∀m∈ℤ+.Combining (3.5), (3.6), and (3.26), one has the following ordering for these half-eigenvalues:
(3.27)(0<)λ̲1L≤λ̲1R<λ̲2L≤λ̲2R<⋯<λ̲mL≤λ̲mR<⋯(→∞),(0=)λ¯0L≤λ¯0R<λ¯1L≤λ¯1R<⋯<λ¯mL≤λ¯mR<⋯(→∞).
Moreover, by the definition of λ̲mL/R and λ¯mL/R, one has
(3.28)λ̲mL≤λ¯mR,λ̲mR,λ¯mL∈[λ̲mL,λ¯mR],∀m∈ℕ.
Notice that the ordering between λ̲mR and λ¯mL is not determinate.Till now, we still do not know the ordering betweenλ¯mR and λ̲nL with n>m. This will be partially settled by associating (1.1) with the rotation number function defined by
(3.29)ρ(λ)=ρ(λ,a,b):=limn→∞Θn(ϑ0,λa,λb)-ϑ02nπp,
where Θn is the nth iteration of Θ, namely,
(3.30)Θn(ϑ0,a,b)=Θ(Θn-1(ϑ0,a,b),a,b).
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)
limλ→∞ρ(λ)=∞;(iii)
λ∈[0,λ̲mL),m∈ℕ⇒ρ(λ)<m/2;(iv)
λ∈(λ¯mR,∞),m∈ℤ+,⇒ρ(λ)>m/2;(v)
λ∈[λ̲2mL,λ¯2mR],m∈ℕ⇒ρ(λ)=m;(vi)
λ=λ¯0R⇒ρ(λ)=0.Proof.
(i) By (2.11), one has Θn(0,λa,λb)>-πp/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)|Sp(θ)|p*, which has equilibria θ=kπp, k∈ℤ. Thus Θn(-πp/2,0·a,0·b)∈(-πp/2,0) for all n∈ℕ and consequently ρ(0)=0.
(ii) GivenM>0 arbitrarily large, it follows from (3.5) that there exists Λ>0 such that
(3.31)Θ(ϑ0,λa,λb)-ϑ0>2Mπp,∀λ>Λ,∀ϑ0∈ℝ.
Thus
(3.32)Θn(ϑ0,λa,λb)-ϑ0=[Θ(Θn-1(ϑ0,λa,λb),λa,λb)-Θ(Θn-2(ϑ0,λa,λb),λa,λb)]+[Θ(Θn-2(ϑ0,λa,λb),λa,λb)-Θ(Θn-3(ϑ0,λa,λb),λa,λb)]+⋯+[Θ(ϑ0,λa,λb)-ϑ0]≥2nMπp
for any λ>Λ and any ϑ0∈ℝ. Consequently, ρ(λ)≥M for any λ>Λ.
(iii) Givenλ∈[0,λ̲mL),m∈ℕ, it follows from (3.26) that there exists δ=δ(λ)>0 such that
(3.33)Θ̲(λ)<(m-δ)πp.
By the definition of Θ̲(λ)=Θ̲(λa,λb) (see (3.1), this implies that
(3.34)Θ(ϑ0,λa,λb)-ϑ0<(m-δ)πp,∀ϑ0∈ℝ.
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) Givenλ∈[λ̲2mL,λ¯2mR],m∈ℕ, it follows from (3.26) that
(3.35)Θ¯(λ)≤2mπp≤Θ̲(λ).
Then there exists ϑ0 such that Θ(ϑ0,λa,λb)-ϑ0=2mπp, and therefore ρ(λ)=m.
(vi) By (3.25), if λ=λ¯0R, 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
(3.36)λ̲2mL=min{λ>0∣ρ(λ)=m},∀m∈ℕ,(3.37)λ¯2mR=max{λ≥0∣ρ(λ)=m},∀m∈ℤ+,
and the ordering
(3.38)(0≤)λ¯0R<λ̲2L≤λ¯2R<⋯<λ̲2mL≤λ¯2mR<λ̲2m+2L≤λ¯2m+2R⋯(→∞).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¯2mR<λ̲2m+2L for any m∈ℕ. Assume on the contrary that λ¯2mR≥λ̲2m+2L for some m. Then
(3.39)λ̲2mL<λ̲2m+2L≤λ¯2mR.
Now Lemma 3.6(v) shows that m=ρ(λ̲2m+2L)=m+1, a contradiction.Corollary 3.8.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, one has
(3.40)λ∈(λ¯2mR,λ̲2m+2L),m∈ℕ⇒ρ(λ)∈(m,m+1).In summary, we can now partially characterize the set of periodic half-eigenvaluesΣP and the set of antiperiodic half-eigenvalues ΣA of (1.1). Denote ΣP+:=ΣP∩ℝ+ and ΣA+:=ΣA∩ℝ+. For any m∈ℤ+, denote by Σm+=Σm+(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
(3.41)ΣP+=⋃m=0∞Σ2m+,ΣA+=⋃m=0∞Σ2m+1+.Theorem 3.9.
Given(a,b)∈𝒲+γ, 1≤γ≤∞.
(i)
For periodic half-eigenvalues, one has(3.42){0,λ¯0R}⊂Σ0+⊂[0,λ¯0R],{λ̲2mL,λ̲2mR,λ¯2mL,λ¯2mR}⊂Σ2m+⊂[λ̲2mL,λ¯2mR],∀m∈ℕ.
Moreover, λ̲2mL/R and λ¯2mL/R satisfy (3.27), (3.28), and (3.38).(ii)
For antiperiodic half-eigenvalues, one has(3.43){λ̲2m+1L,λ̲2m+1R,λ¯2m+1L,λ¯2m+1R}⊂Σ2m+1+,∀m∈ℤ+.
Moreover, λ̲2m+1L/R and λ¯2m+1L/R satisfy (3.27) and (3.28).Proof.
By Lemma3.2 and the definitions of λ̲mL/R and λ¯mL/R (see (3.22)–(3.25), we need only prove
(3.44)Σ0+⊂[0,λ¯0R],Σ2m+⊂[λ̲2mL,λ¯2mR],∀m∈ℕ.
If λ∈Σ2m+, 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
(3.45)θ(ϑ0,λa,λb)=ϑ0+2mπp.
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 forλ¯0R>0.Theorem 3.10.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, then
(3.46)λ¯0R>0⟺∫01a(t)dt<0or∫01b(t)dt<0.Proof.
Assume thatλ¯0R>0. By Lemma 3.2, the half-eigenfunction X=X(t) corresponding to λ¯0R is nowhere vanishing and satisfies the periodic boundary condition (1.2). If X(t)>0 for any t∈[0,1], then
(3.47)(ϕp(X′))′+λ¯0Ra(t)ϕp(X)=0,X≢const.
Therefore,
(3.48)∫01a(t)dt=-1λ¯0R∫01(ϕp(X′))′ϕp(X)dt=-1λ¯0R∫01ϕp(X′)d(1ϕp(X))+ϕp(X′(t))λ¯0Rϕp(X(t))|t=01=-(p-1)λ¯0R∫01|X′|pXpdt<0.
Similarly, if X(t)<0 for any t∈[0,1], then ∫01b(t)dt<0.
On the other hand, assume that∫01a(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
(3.49)ddλΘ(0,λa,λb)|λ=0=∫01a(t)dt<0.
Thus there exists δ>0 such that
(3.50)Θ(0,λa,λb)<0,∀λ∈(0,δ).
By the definition of Θ¯(λ)=Θ¯(λa,λb) (see (3.2), we get
(3.51)Θ¯(λ)<0,∀λ∈(0,δ).
By (3.5), (3.6) and the continuity of Θ¯(λ) in λ∈ℝ+ (see Lemma 3.1), one has λ¯0R>0. Similarly one can obtain λ¯0R>0 if ∫01b(t)dt<0.Notice that for the antiperiodic half-eigenvalues, it does not hold(3.52)Σ2m+1⊂[λ̲2m+1L,λ¯2m+1R],∀m∈ℤ+,
for general a,b∈ℒγ. This is because A(t,θ;a,b) in (2.7) is 2πp-periodic in θ but not πp-periodic in θ. Hence (2.10) holds but one can not obtain
(3.53)θ(t;ϑ0+mπp,a,b)≡θ(t;ϑ0,a,b)+mπp,∀t∈[0,1],ϑ0∈ℝ,m∈ℤ.
However, if a=b, then A(t,θ;a,a) is πp-periodic in θ and
(3.54)θ(t;ϑ0+mπp,a,a)≡θ(t;ϑ0,a,a)+mπp,∀t∈[0,1],ϑ0∈ℝ,m∈ℤ.
The following theorem characterizes periodic and antiperiodic eigenvalues of (1.6). Now Σm+=Σm+(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.
Givenγ∈[1,∞] and a=b∈ℒγ 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(3.55){0,λ¯0R}⊂Σ0+⊂[0,λ¯0R],{λ̲mL,λ̲mR,λ¯mL,λ¯mR}⊂Σm+⊂[λ̲mL,λ¯mR],∀m∈ℕ.(iii)
One has the ordering(3.56)(0≤)λ¯0R<λ̲1L≤λ¯1R<λ̲2L≤λ¯2R<⋯<λ̲mL≤λ¯mR<⋯(→∞).(iv)
Finallyλ¯0R>0⇔∫01a(t)dt<0.Proof.
By Theorems3.9 and 3.10, we need only prove (3.52) and
(3.57)λ¯2mR<λ̲2m+1L,λ¯2m+1R<λ̲2m+2L,∀m∈ℤ+.
Ifλ∈Σm+, m∈ℕ, then there exists ϑ0∈ℝ such that Θ(ϑ0,λa,λa)-ϑ0=mπp. By (3.54) and the definition of ρ(·) in (3.29), one has
(3.58)λ∈Σm+⇒ρ(λ)=m2,
in particular
(3.59)ρ(λ̲2m+1L)=ρ(λ¯2m+1R)=2m+12,∀m∈ℤ+.
Then it follows from Lemma 3.6(iii) and (iv) that
(3.60)λ̲2m+1L=min{λ≥0∣ρ(λ)=m+12},m∈ℤ+,λ¯2m+1R=max{λ≥0∣ρ(λ)=m+12},m∈ℤ+.
Hence (3.52) follows immediately from (3.58) and (3.60).
Supposeλ¯2mR≥λ̲2m+1L for some m∈ℤ+. By Lemma 3.6(iii) and (v) we obtain ρ(λ̲2m+1L)≤m, which contradicts (3.59). Thus λ¯2mR>λ̲2m+1L for any m∈ℤ+. Similarly one can prove λ¯2m+1R<λ̲2m+2L, ∀m∈ℤ+, proving the desired results in (3.57).


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

Theorem 3.12.
Givenγ∈[1,∞], for any admissible m, the following functionals
(3.61)(𝒲+γ,wγ×wγ)→ℝ,(a,b)↦λ̲mL(a,b)(𝒲+γ,wγ×wγ)→ℝ,(a,b)↦λ¯mR(a,b)
are continuous. Here wγ denotes the weak topology in ℒγ.Proof.
We only prove the result forλ̲mL(a,b). The result for λ¯mR(a,b) can be attained analogously. Suppose (an,bn)→(a,b) in (𝒲+γ,wγ×wγ) as n→∞. Then an→a and bn→b in (ℒγ,wγ). Denote νn:=λ̲mL(an,bn). We aim to prove νn→λ̲mL(a,b).
Firstly, we show that the sequence{νn}n∈ℕ is bounded. If this is false, without loss of generality, we may assume that νn→∞ as n→∞. Then given any Λ>0, there exists N∈ℕ such that
(3.62)νn>Λ,∀n>N.
It follows from (3.26) that
(3.63)Θ̲(Λan,Λbn)<mπp,∀n>N.
Let n→∞, we obtain by Lemma 3.1(3.64)Θ̲(Λa,Λb)≤mπp,
which contradicts (3.5) since Λ>0 can be chosen arbitrarily large. Thus {νn}n∈ℕ is bounded.
Since{νn}n∈ℕ is bounded, passing to a subsequence, we assume that νn→ν0 as n→∞. By the definition of νn, we have
(3.65)Θ̲(νnan,νnbn)=mπp.
Let n→∞, by Lemma 3.1 again, we get
(3.66)Θ̲(ν0a,ν0b)=mπp.
Now it suffices to prove ν0=λ̲mL(a,b).
If this is not true, by the definition in (3.22) there holds ν0>λ̲mL(a,b). Then it follows from Lemma 3.3(i) that there exists
(3.67)λ∈(λ̲mL(a,b),ν0)
such that
(3.68)Θ̲(λa,λb)>mπp.
Since Θ̲(λan,λbn)→Θ̲(λa,λb) as n→∞, we may assume, without loss of generality, that
(3.69)Θ̲(λan,λbn)>mπp,∀n∈ℕ.
Therefore, by (3.26), we have
(3.70)λ>λ̲mL(an,bn)=νn,∀n∈ℕ.
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
(4.1)Θ(λ)=Θ(λa,λb)=Θ(-πp2,λa,λb),X(t)=X(t;a,b)=X(t;-πp2,a,b).Fora=b=0, (2.7) is θ′=(p-1)|Sp(θ)|p*, which has equilibria θ=kπp, k∈ℤ. Hence, we have
(4.2)Θ(0)∈(-πp2,0).
Therefore, λ=0 is not a Dirichlet half-eigenvalue of (1.1), namely, 0∉ΣD. Due to the jumping nonlinearity of (1.1), λ∈ΣD+:=ΣD∩ℝ+ if and only if λ>0 is determined by one of the following two equations
(4.3)Θ(-πp2,λa,λb)=-πp2+mπp,(4.4)Θ(πp2,λa,λb)=πp2+mπp,
for some m∈ℤ. The range of m will be proved to be ℕ.The following lemma shows thatΘ(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
(4.5)Θ(μ)=-πp2+mπp,
then
(4.6)ddλΘ(λ)|λ=μ>0,(4.7)Θ(λ)>(<)-πp2+mπp∀λ>(<)μ.Proof.
Ifμ>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),
(4.8)ddλΘ(λ)|λ=u=∫01(aX+p+bX-p)dt=1μ(∫01|X′(t)|pdt-X(t)ϕp(X′(t))|t=01)=1μ∫01|X′(t)|pdt>0,
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 λm+(a,b). Moreover, one has
(4.9)(0<)λ1+(a,b)<λ2+(a,b)<⋯<λm+(a,b)⋯(→∞).
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 λm-(a,b). Moreover, one has
(4.10)(0<)λ1-(a,b)<λ2-(a,b)<⋯<λm-(a,b)⋯(→∞).
Let z=-x in (1.1). Then z+=x-,  z-=x+, and
(4.11)(ϕp(z′))′+λb(t)ϕp(z+)-λa(t)ϕp(z-)=0.
Consequently, we have
(4.12)λm-(a,b)=λm+(b,a)∀m∈ℕ.In summary, we can completely characterize the structure ofΣD+.Theorem 4.2.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, then ΣD+ consists of two sequences (4.9) and (4.10), which are related to each other by (4.12). Moreover, the half-eigenfunctions corresponding to λm±(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 λm+(a,b) on weights (a,b). Analogous results hold for λm-(a,b).Theorem 4.3.
Givenγ∈[1,∞] and m∈ℕ, the functional
(4.13)(𝒲+γ,wγ×wγ)→ℝ,(a,b)↦λm+(a,b),
is continuous. Here wγ denotes the weak topology in ℒγ.Proof.
Suppose(an,bn)→(a,b) in (𝒲+γ,wγ×wγ). Denote νn:=λm+(an,bn). We aim to prove νn→λm+(a,b) as n→∞.
We claim that the sequence{νn}n∈ℕ is bounded. If this is false, we may assume that νn→∞. Then for any Λ>0, there exists N∈ℕ such that νn>Λ for any n>N. By the definition of νn and Lemma 4.1, we have
(4.14)Θ(Λan,Λbn)<-πp2+mπp,∀n∈ℕ.
Let n→∞, it follows from Theorem 2.2(i) that
(4.15)Θ(Λa,Λb)≤-πp2+mπp,
which contradicts with Lemma 2.3, because Λ can be chosen arbitrarily large. Thus, the claim is true.
Since{νn}n∈ℕ is bounded and νn>0, without loss of generality, we may assume that νn→ν0≥0. By the definition of νn, one has
(4.16)Θ(νnan,νnbn)=-πp2+mπp,∀n∈ℕ.
Let n→∞, by Theorem 2.2(i) again, one has
(4.17)Θ(ν0a,ν0b)=-πp2+mπp.
Therefore, ν0=λm+(a,b)(>0), proving the desired result.Given(a,b)∈𝒲γ, 1≤γ≤∞, and m∈ℕ, denote
(4.18)Em(t)=Em(t;a,b):=X(t){∫01(aX+p+bX-p)dt}1/p,
where X(t)=X(t;λm+a,λm+b) (see (2.17) is the half-eigenfunction of (1.1) associated with the half-eigenvalue λm+=λm+(a,b). Then Em is the half-eigenfunction associated with λm+ and satisfying the normalized condition
(4.19)∫01(a(Em)+p+b(Em)-p)dt=1.Theorem 4.4.
Givenγ∈[1,∞] and m∈ℕ, the functional
(4.20)(𝒲+γ,∥·∥γ×∥·∥γ)→ℝ,(a,b)↦λm+=λm+(a,b)
is continuously differentiable. Moreover, the Fréchet derivatives of λm+ at a and at b, denoted by ∂aλm+ and ∂bλm+, respectively, are given by the following functionals:
(4.21)∂aλm+=-λm+(Em)+p∈𝒞0⊂(ℒγ,∥·∥γ)*,(4.22)∂bλm+=-λm+(Em)-p∈𝒞0⊂(ℒγ,∥·∥γ)*,
where Em(·)=Em(·;a,b) is the normalized half-eigenfunction associated with λm+(a,b) as defined in (4.18).Proof.
Consider (4.3) and  let F(λ,a,b):=Θ(-πp/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
(4.23)∂F∂λ(λm+(a,b),a,b)>0.
Thus the Implicit Function Theorem implies that the functional λ=λm+=λm+(a,b)(>0) determined by (4.3) is continuously differentiable in (a,b)∈𝒲+γ. Moreover, for any h∈ℒγ, one has
(4.24)∂aΘ∘(λ+a∂aλ)∘h+∂bΘ∘(b∂aλ)∘h=0.
By (2.15) and (2.16), this reads
(4.25)∫01{X+p[λh+(∂aλ)ha]+X-p[(∂aλ)hb]}dt=0,
where X(·)=X(·;λm+a,λm+b) is the half-eigenfunction of (1.1) associated with the half-eigenvalue λm+. Since ∂aλ:ℒγ→ℝ and (∂aλ)h∈ℝ, one has
(4.26)(∂aλ)h∫01(aX+p+bX-p)dt=-λ∫01X+phdt,(∂aλ)h=-λ∫01X+phdt∫01(aX+p+bX-p)dt=-λ∫01(Em)+phdt,
which proves (4.21). One can obtain (4.22) analogously.Corollary 4.5.
Givenγ∈[1,∞] and (ai,bi)∈𝒲+γ, i=0,1, one has
(4.27)(a0,b0)≻(≥)(a1,b1)⇒λm+(a0,b0)<(≤)λm+(a1,b1),∀m∈ℕ.Proof.
Letaτ=τa0+(1-τ)a1 and bτ=τb0+(1-τ)b1 for any τ∈[0,1]. Although 𝒲+γ is not convex, if (a0,b0)≥(a1,b1), we still have (aτ,bτ)∈𝒲+γ for any τ∈[0,1]. Therefore, for any given m∈ℕ, the function
(4.28)M:[0,1]→(0,∞),τ↦M(τ)=λm+(aτ,bτ),
is well-defined (by Theorem 4.2) and is continuously differentiable (by Theorem 4.4). Moreover, by (4.21) and (4.22), one has
(4.29)M′(τ)=-M(τ)∫01{(Em)+p(a0-a1)+(Em)-p(b0-b1)}dt,∀τ∈[0,1],
where Em(·)=Em(·;aτ,bτ) is the normalized half-eigenfunction of (1.1) associated with λm+(aτ,bτ) (cf. (4.18). Consequently, one has M(τ)<(≤)0 if only (a0,b0)≻(≥)(a1,b1), proving the desired results.


## 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
(4.1)Θ(λ)=Θ(λa,λb)=Θ(-πp2,λa,λb),X(t)=X(t;a,b)=X(t;-πp2,a,b).Fora=b=0, (2.7) is θ′=(p-1)|Sp(θ)|p*, which has equilibria θ=kπp, k∈ℤ. Hence, we have
(4.2)Θ(0)∈(-πp2,0).
Therefore, λ=0 is not a Dirichlet half-eigenvalue of (1.1), namely, 0∉ΣD. Due to the jumping nonlinearity of (1.1), λ∈ΣD+:=ΣD∩ℝ+ if and only if λ>0 is determined by one of the following two equations
(4.3)Θ(-πp2,λa,λb)=-πp2+mπp,(4.4)Θ(πp2,λa,λb)=πp2+mπp,
for some m∈ℤ. The range of m will be proved to be ℕ.The following lemma shows thatΘ(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
(4.5)Θ(μ)=-πp2+mπp,
then
(4.6)ddλΘ(λ)|λ=μ>0,(4.7)Θ(λ)>(<)-πp2+mπp∀λ>(<)μ.Proof.
Ifμ>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),
(4.8)ddλΘ(λ)|λ=u=∫01(aX+p+bX-p)dt=1μ(∫01|X′(t)|pdt-X(t)ϕp(X′(t))|t=01)=1μ∫01|X′(t)|pdt>0,
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 λm+(a,b). Moreover, one has
(4.9)(0<)λ1+(a,b)<λ2+(a,b)<⋯<λm+(a,b)⋯(→∞).
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 λm-(a,b). Moreover, one has
(4.10)(0<)λ1-(a,b)<λ2-(a,b)<⋯<λm-(a,b)⋯(→∞).
Let z=-x in (1.1). Then z+=x-,  z-=x+, and
(4.11)(ϕp(z′))′+λb(t)ϕp(z+)-λa(t)ϕp(z-)=0.
Consequently, we have
(4.12)λm-(a,b)=λm+(b,a)∀m∈ℕ.In summary, we can completely characterize the structure ofΣD+.Theorem 4.2.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, then ΣD+ consists of two sequences (4.9) and (4.10), which are related to each other by (4.12). Moreover, the half-eigenfunctions corresponding to λm±(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 λm+(a,b) on weights (a,b). Analogous results hold for λm-(a,b).Theorem 4.3.
Givenγ∈[1,∞] and m∈ℕ, the functional
(4.13)(𝒲+γ,wγ×wγ)→ℝ,(a,b)↦λm+(a,b),
is continuous. Here wγ denotes the weak topology in ℒγ.Proof.
Suppose(an,bn)→(a,b) in (𝒲+γ,wγ×wγ). Denote νn:=λm+(an,bn). We aim to prove νn→λm+(a,b) as n→∞.
We claim that the sequence{νn}n∈ℕ is bounded. If this is false, we may assume that νn→∞. Then for any Λ>0, there exists N∈ℕ such that νn>Λ for any n>N. By the definition of νn and Lemma 4.1, we have
(4.14)Θ(Λan,Λbn)<-πp2+mπp,∀n∈ℕ.
Let n→∞, it follows from Theorem 2.2(i) that
(4.15)Θ(Λa,Λb)≤-πp2+mπp,
which contradicts with Lemma 2.3, because Λ can be chosen arbitrarily large. Thus, the claim is true.
Since{νn}n∈ℕ is bounded and νn>0, without loss of generality, we may assume that νn→ν0≥0. By the definition of νn, one has
(4.16)Θ(νnan,νnbn)=-πp2+mπp,∀n∈ℕ.
Let n→∞, by Theorem 2.2(i) again, one has
(4.17)Θ(ν0a,ν0b)=-πp2+mπp.
Therefore, ν0=λm+(a,b)(>0), proving the desired result.Given(a,b)∈𝒲γ, 1≤γ≤∞, and m∈ℕ, denote
(4.18)Em(t)=Em(t;a,b):=X(t){∫01(aX+p+bX-p)dt}1/p,
where X(t)=X(t;λm+a,λm+b) (see (2.17) is the half-eigenfunction of (1.1) associated with the half-eigenvalue λm+=λm+(a,b). Then Em is the half-eigenfunction associated with λm+ and satisfying the normalized condition
(4.19)∫01(a(Em)+p+b(Em)-p)dt=1.Theorem 4.4.
Givenγ∈[1,∞] and m∈ℕ, the functional
(4.20)(𝒲+γ,∥·∥γ×∥·∥γ)→ℝ,(a,b)↦λm+=λm+(a,b)
is continuously differentiable. Moreover, the Fréchet derivatives of λm+ at a and at b, denoted by ∂aλm+ and ∂bλm+, respectively, are given by the following functionals:
(4.21)∂aλm+=-λm+(Em)+p∈𝒞0⊂(ℒγ,∥·∥γ)*,(4.22)∂bλm+=-λm+(Em)-p∈𝒞0⊂(ℒγ,∥·∥γ)*,
where Em(·)=Em(·;a,b) is the normalized half-eigenfunction associated with λm+(a,b) as defined in (4.18).Proof.
Consider (4.3) and  let F(λ,a,b):=Θ(-πp/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
(4.23)∂F∂λ(λm+(a,b),a,b)>0.
Thus the Implicit Function Theorem implies that the functional λ=λm+=λm+(a,b)(>0) determined by (4.3) is continuously differentiable in (a,b)∈𝒲+γ. Moreover, for any h∈ℒγ, one has
(4.24)∂aΘ∘(λ+a∂aλ)∘h+∂bΘ∘(b∂aλ)∘h=0.
By (2.15) and (2.16), this reads
(4.25)∫01{X+p[λh+(∂aλ)ha]+X-p[(∂aλ)hb]}dt=0,
where X(·)=X(·;λm+a,λm+b) is the half-eigenfunction of (1.1) associated with the half-eigenvalue λm+. Since ∂aλ:ℒγ→ℝ and (∂aλ)h∈ℝ, one has
(4.26)(∂aλ)h∫01(aX+p+bX-p)dt=-λ∫01X+phdt,(∂aλ)h=-λ∫01X+phdt∫01(aX+p+bX-p)dt=-λ∫01(Em)+phdt,
which proves (4.21). One can obtain (4.22) analogously.Corollary 4.5.
Givenγ∈[1,∞] and (ai,bi)∈𝒲+γ, i=0,1, one has
(4.27)(a0,b0)≻(≥)(a1,b1)⇒λm+(a0,b0)<(≤)λm+(a1,b1),∀m∈ℕ.Proof.
Letaτ=τa0+(1-τ)a1 and bτ=τb0+(1-τ)b1 for any τ∈[0,1]. Although 𝒲+γ is not convex, if (a0,b0)≥(a1,b1), we still have (aτ,bτ)∈𝒲+γ for any τ∈[0,1]. Therefore, for any given m∈ℕ, the function
(4.28)M:[0,1]→(0,∞),τ↦M(τ)=λm+(aτ,bτ),
is well-defined (by Theorem 4.2) and is continuously differentiable (by Theorem 4.4). Moreover, by (4.21) and (4.22), one has
(4.29)M′(τ)=-M(τ)∫01{(Em)+p(a0-a1)+(Em)-p(b0-b1)}dt,∀τ∈[0,1],
where Em(·)=Em(·;aτ,bτ) is the normalized half-eigenfunction of (1.1) associated with λm+(aτ,bτ) (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
(5.1)Θ̃(λ)=Θ̃(λa,λb)≔Θ(0,λa,λb),X̃(t)=X̃(t;a,b)≔X(t;0,a,b).
Notice that 0∈ΣN with the constant half-eigenfunctions. Due to the jumping nonlinearity of (1.1), λ∈ΣN+:=ΣN∩ℝ+ if and only if λ≥0 is determined by one of the following two equations:
(5.2)Θ(0,λa,λb)=mπp,(5.3)Θ(πp,λa,λb)=πp+mπp,
for some m∈ℤ. We still need to specify the range of m.The following lemma shows thatΘ̃(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
(5.4)Θ̃(μ)=mπp,
then
(5.5)ddλΘ̃(λ)|λ=μ>0,Θ̃(λ)>(<)-πp2+mπp∀λ>(<)μ.Notice thatΘ̃(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=λ̃m+(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=λ̃0+(a,b). Moreover, one has
(5.6)0<λ̃0+(a,b)<λ̃1+(a,b)<λ̃2+(a,b)<⋯<λ̃m+(a,b)⋯(→∞).
Similar arguments as in the proof of Theorem 3.10 show that
(5.7)λ̃0+(a,b)(>0)exists⟺∫01a(t)dt<0.
By Lemma 5.1, if the principal half-eigenvalue λ̃0+ exists, then
(5.8)Θ̃(λ)<0,∀λ∈(0,λ̃0+).Similar results hold for (5.3). Denote the positive solution of (5.3) by λ̃m-(a,b) for any m∈ℤ+. One has
(5.9)0<λ̃0-(a,b)<λ̃1-(a,b)<λ̃2-(a,b)<⋯<λ̃m-(a,b)⋯(→∞),
where λ̃0-(a,b) is also called positive principal half-eigenvalue and
(5.10)λ̃0-(a,b)(>0)exists⟺∫01b(t)dt<0.
It can also be proved that
(5.11)λ̃m-(a,b)=λ̃m+(b,a),∀m∈ℤ+
if only both λ̃m+(b,a) and λ̃m-(a,b) exist.In summary, we can completely characterize the structure ofΣN+.Theorem 5.2.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, then ΣN+ consists of two sequences (5.6) and (5.9), which satisfy (5.7), (5.10), and (5.11). The half-eigenfunctions corresponding to λ̃m±(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 λ̃m+(a,b) on weights (a,b). Analogous results hold for λ̃m-(a,b). Given (a,b)∈𝒲+γ and m∈ℤ+, denote
(5.12)Ẽm(t)=Ẽm(t;a,b):=X̃(t){∫01(aX̃+p+bX̃-p)dt}1/p,
where X̃(t)=X̃(t;λ̃m+a,λ̃m+b) is the half-eigenfunction of (1.1) associated with the half-eigenvalue λ̃m+=λ̃m+(a,b). Then Ẽm is also a half-eigenfunction, and it satisfies the normalized condition
(5.13)∫01(a(Ẽm)+p+b(Ẽm)-p)dt=1.
Denote
(5.14)𝒲+-γ:={(a,b)∈𝒲+γ∣∫01a(t)dt<0}.Theorem 5.3.
Givenγ∈[1,∞], the functionals
(5.15)𝒲+γ→ℝ,(a,b)↦λ̃m+=λ̃m+(a,b)(m∈ℕ),𝒲+-γ→ℝ,(a,b)↦λ̃0+=λ̃0+(a,b),
are continuous with respect to the weak topology wγ×wγ, and are continuously Fréchet differentiable with respect to ∥·∥γ×∥·∥γ. Moreover, for any m∈ℤ+, the derivatives of λ̃m+ at a and at b, denoted by ∂aλ̃m+ and ∂bλ̃m+, respectively, are given by
(5.16)∂aλ̃m+=-λ̃m+(Ẽm)+p∈𝒞0⊂(ℒγ,∥·∥γ)*,∂bλ̃m+=-λ̃m+(Ẽm)-p∈𝒞0⊂(ℒγ,∥·∥γ)*,
where Ẽm=Ẽm(t;a,b) is the normalized half-eigenfunction associated with λ̃m+(a,b) as defined in (5.12).Proof.
By checking the proof of Theorems4.3 and 4.4, the only matter to be proved is the continuity of
(5.17)λ̃0+:(𝒲+-γ,wγ)→(0,∞),
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(ai,bi)∈𝒲+γ,i=0,1, and (a0,b0)≻(≥)(a1,b1), then
(5.18)λ̃m+(a0,b0)<(≤)λ̃m+(a1,b1),∀m∈ℕ.
(ii) If(ai,bi)∈𝒲+-γ,i=0,1, and (a0,b0)≻(≥)(a1,b1), then
(5.19)λ̃m+(a0,b0)<(≤)λ̃m+(a1,b1),∀m∈ℤ+.


## 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
(5.1)Θ̃(λ)=Θ̃(λa,λb)≔Θ(0,λa,λb),X̃(t)=X̃(t;a,b)≔X(t;0,a,b).
Notice that 0∈ΣN with the constant half-eigenfunctions. Due to the jumping nonlinearity of (1.1), λ∈ΣN+:=ΣN∩ℝ+ if and only if λ≥0 is determined by one of the following two equations:
(5.2)Θ(0,λa,λb)=mπp,(5.3)Θ(πp,λa,λb)=πp+mπp,
for some m∈ℤ. We still need to specify the range of m.The following lemma shows thatΘ̃(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
(5.4)Θ̃(μ)=mπp,
then
(5.5)ddλΘ̃(λ)|λ=μ>0,Θ̃(λ)>(<)-πp2+mπp∀λ>(<)μ.Notice thatΘ̃(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=λ̃m+(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=λ̃0+(a,b). Moreover, one has
(5.6)0<λ̃0+(a,b)<λ̃1+(a,b)<λ̃2+(a,b)<⋯<λ̃m+(a,b)⋯(→∞).
Similar arguments as in the proof of Theorem 3.10 show that
(5.7)λ̃0+(a,b)(>0)exists⟺∫01a(t)dt<0.
By Lemma 5.1, if the principal half-eigenvalue λ̃0+ exists, then
(5.8)Θ̃(λ)<0,∀λ∈(0,λ̃0+).Similar results hold for (5.3). Denote the positive solution of (5.3) by λ̃m-(a,b) for any m∈ℤ+. One has
(5.9)0<λ̃0-(a,b)<λ̃1-(a,b)<λ̃2-(a,b)<⋯<λ̃m-(a,b)⋯(→∞),
where λ̃0-(a,b) is also called positive principal half-eigenvalue and
(5.10)λ̃0-(a,b)(>0)exists⟺∫01b(t)dt<0.
It can also be proved that
(5.11)λ̃m-(a,b)=λ̃m+(b,a),∀m∈ℤ+
if only both λ̃m+(b,a) and λ̃m-(a,b) exist.In summary, we can completely characterize the structure ofΣN+.Theorem 5.2.
Given(a,b)∈𝒲+γ, 1≤γ≤∞, then ΣN+ consists of two sequences (5.6) and (5.9), which satisfy (5.7), (5.10), and (5.11). The half-eigenfunctions corresponding to λ̃m±(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 λ̃m+(a,b) on weights (a,b). Analogous results hold for λ̃m-(a,b). Given (a,b)∈𝒲+γ and m∈ℤ+, denote
(5.12)Ẽm(t)=Ẽm(t;a,b):=X̃(t){∫01(aX̃+p+bX̃-p)dt}1/p,
where X̃(t)=X̃(t;λ̃m+a,λ̃m+b) is the half-eigenfunction of (1.1) associated with the half-eigenvalue λ̃m+=λ̃m+(a,b). Then Ẽm is also a half-eigenfunction, and it satisfies the normalized condition
(5.13)∫01(a(Ẽm)+p+b(Ẽm)-p)dt=1.
Denote
(5.14)𝒲+-γ:={(a,b)∈𝒲+γ∣∫01a(t)dt<0}.Theorem 5.3.
Givenγ∈[1,∞], the functionals
(5.15)𝒲+γ→ℝ,(a,b)↦λ̃m+=λ̃m+(a,b)(m∈ℕ),𝒲+-γ→ℝ,(a,b)↦λ̃0+=λ̃0+(a,b),
are continuous with respect to the weak topology wγ×wγ, and are continuously Fréchet differentiable with respect to ∥·∥γ×∥·∥γ. Moreover, for any m∈ℤ+, the derivatives of λ̃m+ at a and at b, denoted by ∂aλ̃m+ and ∂bλ̃m+, respectively, are given by
(5.16)∂aλ̃m+=-λ̃m+(Ẽm)+p∈𝒞0⊂(ℒγ,∥·∥γ)*,∂bλ̃m+=-λ̃m+(Ẽm)-p∈𝒞0⊂(ℒγ,∥·∥γ)*,
where Ẽm=Ẽm(t;a,b) is the normalized half-eigenfunction associated with λ̃m+(a,b) as defined in (5.12).Proof.
By checking the proof of Theorems4.3 and 4.4, the only matter to be proved is the continuity of
(5.17)λ̃0+:(𝒲+-γ,wγ)→(0,∞),
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(ai,bi)∈𝒲+γ,i=0,1, and (a0,b0)≻(≥)(a1,b1), then
(5.18)λ̃m+(a0,b0)<(≤)λ̃m+(a1,b1),∀m∈ℕ.
(ii) If(ai,bi)∈𝒲+-γ,i=0,1, and (a0,b0)≻(≥)(a1,b1), then
(5.19)λ̃m+(a0,b0)<(≤)λ̃m+(a1,b1),∀m∈ℤ+.



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*Source: 109757-2009-09-13.xml*