On the Fučík spectrum of the scalar p-Laplacian with indefinite integrable weights

In this paper, we study the structure of the Fučík spectrum {\mathrm{\Pi }}_p^{D/N}(a,b) of Dirichlet and Neumann problems for the scalar p-Laplacian with indefinite weights a,b\in L^1[0,1]. Besides the trivial horizontal lines and vertical lines, it will be shown that, confined to each quadrant of {\mathbb{R}}^2, {\mathrm{\Pi }}_p^{D/N}(a,b) is made up of zero, an odd number of, or a double sequence of hyperbolic like curves. These hyperbolic like curves are continuous and strictly monotonic, and they have horizontal and vertical asymptotic lines. The number of the hyperbolic like curves is determined by the Dirichlet and Neumann half-eigenvalues of the p-Laplacian with weights a and b. The asymptotic lines will be estimated by using Sturm-Liouville eigenvalues of the p-Laplacian with a weight a or b.

MSC:34B09, 34B15, 34L05.

Fučík spectrum was first introduced for the Laplacian on a bounded domain \mathrm{\Omega }\subset {\mathbb{R}}^N, N\ge 1, by Dancer [1] and by Fučík [2] in the 1970s, in connection with the study of semilinear elliptic boundary value problems with jumping nonlinearities. Thereafter this important concept was generalized to the p-Laplacian, p>1. See [3] and references therein.

In this paper, we are concerned with the Fučík spectrum of the scalar p-Laplacian

Given a,b\in L^1[0,1], taking the notations x_{\pm }=max\{\pm x,0\}, let us consider the ODE

in which a, b are called potentials, and the ODE

in which a, b are called weights. For a pair of potentials a and b, the Fučík spectra {\mathrm{\Sigma }}_p^D(a,b) and {\mathrm{\Sigma }}_p^N(a,b) are defined as the sets of those (\lambda ,\mu )\in {\mathbb{R}}^2 such that equation (1.1) has non-trivial solutions satisfying the Dirichlet boundary condition

and the Neumann boundary condition

respectively. Similarly, for a pair of weights a and b, the Fučík spectra {\mathrm{\Pi }}_p^D(a,b) and {\mathrm{\Pi }}_p^N(a,b) are defined as the sets of those (\lambda ,\mu )\in {\mathbb{R}}^2 such that equation (1.2) has non-trivial solutions satisfying the corresponding boundary conditions (1.3) and (1.4), respectively.

The Fučík spectra {\mathrm{\Sigma }}_p^D(a,b) and {\mathrm{\Sigma }}_p^N(a,b) have been comprehensively understood in [4]: each of them is composed of one horizontal line, one vertical line and a double sequence of differentiable, strictly decreasing, hyperbolic like curves; asymptotic lines of these hyperbolic like curves are given by using (Sturm-Liouville) eigenvalues of the p-Laplacian with a potential; moreover, these curves have a strong continuous dependence on the potentials.

Compared with potentials, indefinite weights will add difficulties to the study of the Fučík spectra. Alif [5] studied {\mathrm{\Pi }}_p^D(a,b) and {\mathrm{\Pi }}_p^N(a,b) by means of ‘zero functions’, where the weights a and b were assumed to be sign-changing (i.e., a_{\pm }\not\equiv 0 and b_{\pm }\not\equiv 0) continuous functions without ‘singular points’ (which is a technical hypothesis). Their main results are as follows. Besides the trivial horizontal lines and vertical lines, confined to each quadrant of {\mathbb{R}}^2, {\mathrm{\Pi }}_p^{D/N}(a,b) consists of an odd number of or infinitely many hyperbolic like curves. The asymptotic behavior of the first non-trivial curves in each quadrant was also studied. It was observed that for instance the first curve of {\mathrm{\Pi }}_p^N(a,b) in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+} is not asymptotic on any side to the trivial horizontal and vertical lines. In other words, there are always gaps between its asymptotic lines and the trivial horizontal and vertical lines. However, the exact asymptotic lines were not found in that paper.

In this paper, we are interested in {\mathrm{\Pi }}_p^D(a,b) and {\mathrm{\Pi }}_p^N(a,b), where the weights a,b\in L^1[0,1] are assumed to be indefinite (i.e., a and b may or may not change sign). In this case, since the weights are integrable, the method employed in [5] does not work anymore. Using the Prüfer transformation, we convert the second-order ODE (1.2) into a system of first-order ODEs (3.2) and (3.3), for the argument θ and the radius r, respectively. The ODE (3.2) for θ turns to be independent of r, and the boundary conditions (1.3) and (1.4) can be characterized by the solutions of equation (3.2), therefore the Fučík spectra {\mathrm{\Pi }}_p^D(a,b) and {\mathrm{\Pi }}_p^N(a,b) are completely determined by this first-order ODE (3.2). The solutions of equation (3.2) admit (strong) continuity and Fréchet differentiability in the weights. Based on these properties, we will finally reveal the structure of the Fučík spectra. Our main results are as follows.

1. (i)

   Besides at most two vertical lines and two horizontal lines, {\mathrm{\Pi }}_p^{D/N}(a,b) confined to each quadrant of {\mathbb{R}}^2 is made up of zero, an odd number of, or a double sequence of continuous, strictly monotonic, hyperbolic like curves.

2. (ii)

   The number of those trivial lines in {\mathrm{\Pi }}_p^{D/N}(a,b) is determined by the Dirichlet and Neumann eigenvalues of the p-Laplacian.

3. (iii)

   The number of the hyperbolic like curves in {\mathrm{\Pi }}_p^{D/N}(a,b) is determined by the Dirichlet and Neumann half-eigenvalues of the p-Laplacian.

4. (iv)

   All the hyperbolic like curves have vertical and horizontal asymptotic lines, and these asymptotic lines will be estimated by using (Sturm-Liouville) eigenvalues of the p-Laplacian.

5. (v)

   If the weights a and b are positive, the structure of {\mathrm{\Pi }}_p^{D/N}(a,b) is comparable with that of {\mathrm{\Sigma }}_p^{D/N}(a,b), the case with potentials. More precisely, {\mathrm{\Pi }}_p^{D/N}(a,b) is composed of one horizontal line, one vertical line and a double sequence of differentiable, strictly decreasing, hyperbolic like curves in the quadrant {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}. And all asymptotic lines of these hyperbolic like curves will be given by using (Sturm-Liouville) eigenvalues of the p-Laplacian.

The paper is organized as follows. In Section 2, we will give some preliminary results. Section 3 is devoted to {\mathrm{\Pi }}_p^D(a,b). We first decompose {\mathrm{\Pi }}_p^D(a,b) in Section 3.1, according to the number of zeroes of the eigenfunctions. Sections 3.2 and 3.3 are devoted to eigenvalues and half-eigenvalues of the p-Laplacian, respectively. The results in these two subsections enables us to finally determine the structure of {\mathrm{\Pi }}_p^D(a,b) in Section 3.4. For a pair of positive weights a and b, we can get more information on {\mathrm{\Pi }}_p^D(a,b) and the results are given in Section 3.5. The Fučík spectrum {\mathrm{\Pi }}_p^N(a,b) can be studied by similar arguments and we just list the results in Section 4.

Given an exponent p\in (1,\mathrm{\infty }), denote by p^{\ast } the conjugate number of p, namely p^{\ast }=\frac{p}{p-1}. The initial value problem

has a unique solution ({cos}_{p}t,{sin}_{p}t), t\in \mathbb{R}. The functions {cos}_{p}t and {sin}_{p}t are the so-called p-cosine and p-sine because they possess properties similar to those of the standard cosine and sine, as shown in the following lemma.

Lemma 2.1 ([6, 7])

The p-cosine and p-sine have the following properties.

1. (i)

   Both {cos}_{p}t and {sin}_{p}t are 2{\pi }_p-periodic, where

   {\pi }_p=2{\int }_0^{{(p-1)}^{1/p}}\frac{ds}{{(1-s^p(p-1))}^{1/p}}=\frac{2\pi {(p-1)}^{1/p}}{psin(\pi /p)};
2. (ii)

   {cos}_{p}t is even in t and {sin}_{p}t is odd in t;

3. (iii)

   {cos}_p(t+{\pi }_p)=-{cos}_{p}t and {sin}_p(t+{\pi }_p)=-{sin}_{p}t for all t;

4. (iv)

   {cos}_{p}t=0 if and only if t={\pi }_p/2+n{\pi }_p, n\in \mathbb{Z}, and {sin}_{p}t=0 if and only if t=n{\pi }_p, n\in \mathbb{Z};

5. (v)

   {cos}_p^{\prime }t=-{\varphi }_{p^{\ast }}({sin}_{p}t) and {sin}_p^{\prime }t={\varphi }_p({cos}_{p}t); and

6. (vi)

   {|{cos}_{p}t|}^p+(p-1){|{sin}_{p}t|}^{p^{\ast }}\equiv 1.

Remark 2.1 For any p>1, one has {\pi }_p>2. In fact, if p=2, then {\pi }_p=\pi >2. If p>2, then

If , then

Given a,b\in L^1[0,1], consider the equation

Let y=-{\varphi }_p(x^{\prime }). Via the p-polar coordinates (or Prüfer transformation)

we can transform equation (2.1) into the following equations for r and θ:

Note that equation (2.3) for θ is independent of r. Given t_0\in [0,1] and {\theta }_0\in \mathbb{R}, denote by (\theta (t;t_0,{\theta }_0,a,b),r(t;t_0,{\theta }_0,a,b)), t\in [0,1], the unique solution of system (2.3)-(2.4) satisfying \theta (t_0;t_0,{\theta }_0,a,b)={\theta }_0 and r(1;t_0,{\theta }_0,a,b)=1. Let

The p-polar coordinates (2.2), one can verify that equation (2.1) has a non-trivial solution

One basic observation on equation (2.3) is that the vector field A(t,\theta ;a,b)=1>0 at those θ such that {cos}_p\theta =0, i.e., \theta =-\frac{{\pi }_p}{2}+m{\pi }_p, m\in \mathbb{Z}. Since a(t) and b(t) are only integrable, the derivative {\theta }^{\prime }(t) at any specific t is meaningless. However, one can still use such an observation to obtain the following property, called quasi-monotonicity. We refer the readers to [[7], Lemma 2.3] for a detailed proof.

Lemma 2.2 Given a,b\in L^1[0,1], t_0\in [0,1) and {\theta }_0\in \mathbb{R}, let \theta (t)=\theta (t;t_0,{\theta }_0,a,b) be the solution of equation (2.3). If \theta (t_0)\ge -\frac{{\pi }_p}{2}+m{\pi }_p for some m\in \mathbb{Z}, then

Denote by w_1 the weak topology in L^1[0,1]. By g_n\to g_0 in (L^1[0,1],w_1), or g_n\stackrel{w_1}{\to }{g}_0, we mean that

Some important properties of \theta (\cdot ;t_0,{\theta }_0,a,b), r(\cdot ;t_0,{\theta }_0,a,b), \mathrm{\Theta }({\theta }_0,a,b) and X(\cdot ;{\theta }_0,a,b) are collected in the following theorem.

Theorem 2.1 ([8])

Let t_0\in [0,1] and {\theta }_0\in \mathbb{R} be fixed. We have the following results.

1. (i)

   As mappings from {(L^1[0,1],w_1)}^2 to (C[0,1],{\parallel \cdot \parallel }_{\mathrm{\infty }}), \theta (\cdot ;t_0,{\theta }_0,a,b) and r(\cdot ;t_0,{\theta }_0,a,b) are continuous. More precisely, if a_n\stackrel{w_1}{\to }{a}_0 and b_n\stackrel{w_1}{\to }{b}_0, then

   \begin{array}{c}{\parallel \theta (\cdot ;t_0,{\theta }_0,a_n,b_n)-\theta (\cdot ;t_0,{\theta }_0,a_0,b_0)\parallel }_{\mathrm{\infty }}\to 0,\hfill \\ {\parallel r(\cdot ;t_0,{\theta }_0,a_n,b_n)-r(\cdot ;t_0,{\theta }_0,a_0,b_0)\parallel }_{\mathrm{\infty }}\to 0,\hfill \end{array}

   as n\to \mathrm{\infty }.

2. (ii)

   The functional {(L^1[0,1],w_1)}^2\to \mathbb{R}, (a,b)\mapsto \mathrm{\Theta }({\theta }_0,a,b) is continuous. More precisely, if a_n\stackrel{w_1}{\to }{a}_0 and b_n\stackrel{w_1}{\to }{b}_0, then \mathrm{\Theta }({\theta }_0,a_n,b_n)\to \mathrm{\Theta }({\theta }_0,a_0,b_0) as n\to \mathrm{\infty }.

3. (iii)

   The functional {(L^1[0,1],{\parallel \cdot \parallel }_1)}^2\to \mathbb{R}, (a,b)\mapsto \mathrm{\Theta }({\theta }_0,a,b) is continuously differentiable in the sense of Fréchet. The differentials of \mathrm{\Theta }({\theta }_0,a,b) at a and b, denoted, respectively, by {\partial }_a\mathrm{\Theta }({\theta }_0,a,b) and {\partial }_b\mathrm{\Theta }({\theta }_0,a,b), are the following mappings:

   {\partial }_a\mathrm{\Theta }({\theta }_0,a,b)=X_{+}^p(\cdot ;{\theta }_0,a,b)\in (C[0,1],{\parallel \cdot \parallel }_{\mathrm{\infty }})\subset {(L^1,{\parallel \cdot \parallel }_1)}^{\ast },

   (2.6)
   {\partial }_b\mathrm{\Theta }({\theta }_0,a,b)=X_{-}^p(\cdot ;{\theta }_0,a,b)\in (C[0,1],{\parallel \cdot \parallel }_{\mathrm{\infty }})\subset {(L^1,{\parallel \cdot \parallel }_1)}^{\ast },

   (2.7)

   where {(L^1,{\parallel \cdot \parallel }_1)}^{\ast } is the dual space of (L^1,{\parallel \cdot \parallel }_1). Moreover, as mappings from {(L^1[0,1],w_1)}^2 to (C[0,1],{\parallel \cdot \parallel }_{\mathrm{\infty }}), both {\partial }_a\mathrm{\Theta }({\theta }_0,a,b) and {\partial }_b\mathrm{\Theta }({\theta }_0,a,b) are continuous.

Remark 2.2 Let {\theta }_0\in \mathbb{R} and a_i,b_i\in L^1[0,1], i=1,2. If a_1\le a_2 and b_1\le b_2, then it follows from formulations (2.6) and (2.7) that

3.1 Decomposition of {\mathrm{\Pi }}_p^D(a,b)

Given a pair of weights a,b\in L^1[0,1], the (Dirichlet type) Fučík spectrum {\mathrm{\Pi }}_p^D(a,b) is defined as the set of those (\lambda ,\mu )\in {\mathbb{R}}^2 such that system (1.2)-(1.3) has non-trivial solutions. Let

In the p-polar coordinates (2.2), equation (1.2) is equivalent to the following two equations:

Compared with equations (2.3) and (2.4), the pair of weights a and b are now replaced by λa and μb, respectively. Since the right-hand side of equation (3.2) is 2{\pi }_p-periodic in θ, one has

for any t_0\in [0,1], {\theta }_0\in \mathbb{R} and l\in \mathbb{Z}. One can also check that

Suppose x(t) is an eigenfunction of system (1.2)-(1.3) associated with (\lambda ,\mu )\in {\mathrm{\Pi }}_p^D(a,b). By equation (2.2), the corresponding solution of equation (3.2), \theta (t):=\theta (t;0,-\frac{{\pi }_p}{2},\lambda a,\mu b), satisfies

for some l,k\in \mathbb{Z}. Due to equation (3.4), we may restrict \theta (0)\in [-\frac{{\pi }_p}{2},\frac{3{\pi }_p}{2}). In other words, we may assume that l\in \{0,1\} and hence \theta (0)=-\frac{{\pi }_p}{2} or \theta (0)=\frac{{\pi }_p}{2}. Moreover, it follows from the quasi-monotonicity result in Lemma 2.2 that k>0. We distinguish two cases: x^{\prime }(0)>0 or . If x^{\prime }(0)>0, then it follows from equation (3.1) that . By equation (2.2), we have , and hence l=0 and \theta (0)=-\frac{{\pi }_p}{2}. Let

Now equation (3.6) tells us that (\lambda ,\mu )\in W_k^D(a,b). In fact, the subscript k is related to the number of zeroes of x(t) on [0,1]. By Lemma 2.2, the equation

has a solution t_m\in [0,1] if and only if 0\le m\le k, and

By equation (2.2), we see that x(t) has exactly k+1 zeroes in [0,1]. Similarly, if and x(t) has exactly k+1 zeroes in [0,1], then l=1, \theta (0)=\frac{{\pi }_p}{2} and (\lambda ,\mu )\in {\tilde{W}}_k^D(a,b), where

Till now, we have proved that

Conversely, let us show that

Suppose (\lambda ,\mu )\in W_k^D(a,b) for some k\ge 1. Then \theta (t)=\theta (t;0,-\frac{{\pi }_p}{2},\lambda a,\mu b) satisfies

For this specific \theta (t), take a non-trivial solution r(t) of equation (3.3). Then we can construct a function x(t)=r{(t)}^{2/p}{cos}_p\theta (t), which is a solution of equation (1.2) with exactly k+1 zeroes on [0,1]. Particularly, x(0)=x(1)=0. Thus (\lambda ,\mu )\in {\mathrm{\Pi }}_p^D(a,b), and hence W_k^D(a,b)\subset {\mathrm{\Pi }}_p^D(a,b). Furthermore, we have

and hence

Similarly, if (\lambda ,\mu )\in {\tilde{W}}_k^D(a,b) for some k\ge 1, then (\lambda ,\mu )\in {\mathrm{\Pi }}_p^D(a,b) and any associating eigenfunction x(t) satisfies and has exactly k+1 zeroes in [0,1].

Combining the previous arguments, we can conclude that.

Theorem 3.1 Let a,b\in L^1[0,1]. The Fučík spectrum {\mathrm{\Pi }}_p^D(a,b) can be decomposed as

Moreover, the following characterization on W_k^D(a,b) and {\tilde{W}}_k^D(a,b) holds.

1. (i)

   (\lambda ,\mu )\in W_k^D(a,b), k\ge 1 ⟺ any eigenfunction x(t) associated with (\lambda ,\mu ) satisfies x^{\prime }(0)>0, x(0)=x(1)=1 and x(t) has precisely k-1 zeroes in (0,1).

2. (ii)

   (\lambda ,\mu )\in {\tilde{W}}_k^D(a,b), k\ge 1 ⟺ any eigenfunction x(t) associated with (\lambda ,\mu ) satisfies , x(0)=x(1)=1 and x(t) and has precisely k-1 zeroes in (0,1).

By equation (3.5), the set {\tilde{W}}_k^D(a,b) defined as in equation (3.8) can be rewritten as

Thus

In other words, {\tilde{W}}_k^D(a,b) is symmetric to W_k^D(b,a) about the line \lambda =\mu. For this reason, essentially we need only to characterize those sets W_k^D(a,b).

In Section 3.4, we will see that W_1^D(a,b) is made up of straight lines which are in connection with {\lambda }_1^D(a) and {\lambda }_{-1}^D(a), the Dirichlet eigenvalues of p-Laplacian with the weight a. See Theorem 3.2.

For those sets W_k^D(a,b), k\ge 2, it is easy to check that

Therefore we need only to focus our study on the subset

where {\mathbb{R}}_{+}=[0,+\mathrm{\infty }). In Section 3.4, for each k\ge 2 we will show that {\mathrm{\Gamma }}_k^D(a,b) is either an empty set or a continuous, strictly decreasing, hyperbolic like curve with a horizontal asymptotic line and a vertical asymptotic line.

With the help of half-eigenvalues of the p-Laplacian with a pair of weights, we can determined whether {\mathrm{\Gamma }}_k^D(a,b) is an empty set or not. Using eigenvalues of the p-Laplacian with a weight, we can roughly locate the hyperbolic like curve {\mathrm{\Gamma }}_k^D(a,b). For these reasons, we will give in the successive two subsections some useful characterization on eigenvalues and half-eigenvalues of the p-Laplacian with weights.

3.2 Eigenvalues of p-Laplacian with an indefinite weight

Given a\in L^1[0,1], denote by \vartheta (t)=\vartheta (t;t_0,{\vartheta }_0,a) the solution of

satisfying the initial value condition \vartheta (t_0)={\vartheta }_0. Particularly, if a(t)\equiv 1, it follows from Lemma 2.1(vi) that equation (3.14) turns to be {\theta }^{\prime }\equiv 1, and hence

Because the right-hand side of equation (3.14) is {\pi }_p-periodic in θ, we have

for any t_0\in [0,1], {\vartheta }_0\in \mathbb{R} and k\in \mathbb{Z}. Since equation (3.14) can also be rewritten as

using the notations in Section 2, we have

By Lemma 2.2, we see that \vartheta (t;t_0,{\vartheta }_0,a) is also quasi-monotonic in t.

Given a\in L^1[0,1], denote by {\mathrm{\Sigma }}_p^D(a), {\mathrm{\Sigma }}_p^N(a), {\mathrm{\Sigma }}_p^{DN}(a) and {\mathrm{\Sigma }}_p^{ND}(a) the sets of \lambda \in \mathbb{R} such that

has a non-trivial solution satisfying the Dirichlet boundary condition x(0)=x(1)=0, the Neumann boundary condition x^{\prime }(0)=x^{\prime }(1)=0, the Dirichlet-Neumann boundary condition x(0)=x^{\prime }(1)=0 and the Neumann-Dirichlet boundary condition x^{\prime }(0)=x(1)=0, respectively. Similar arguments as in Section 3.1 show that

These spectra have been studied in [9]. Consider the function of \lambda \in \mathbb{R}:

It follows from formulations (2.6) and (2.7) in Theorem 2.1 that

where X(t)=X(t;-\frac{{\pi }_p}{2},\lambda a,\lambda a)\not\equiv 0 satisfies

See equation (2.5) for the definition of X(t). Then X(t) is also a non-trivial solution of equation (3.17). Multiplying equation (3.20) by X(t) and integrating over [0,1], we have

Substituting this into equation (3.19), for any \lambda \ne 0 we have

If \lambda \in {\mathrm{\Sigma }}_p^D(a), then X(t) becomes the associated eigenfunction of equation (3.17) satisfying X(0)=X(1)=0. In this case, the first item on the right-hand side of equation (3.21) equals 0, and hence

where {\mathbb{R}}^{+}=(0,+\mathrm{\infty }) and {\mathbb{R}}^{-}=(-\mathrm{\infty },0). Similarly, we can obtain

For any k\ge 1, it follows from equations (3.18) and (3.22) that

has at most one positive solution and one negative solution, denoted by {\lambda }_k^D(a) and {\lambda }_{-k}^D(a), respectively, if they exist. In other words, we have

It has been proved in [9] that \vartheta (1;0,0,\lambda a)=0 has at most one nonzero solution, called the principal Neumann eigenvalue and denoted by {\lambda }_0^N(a), if it exists. By equation (3.23) and the fact \vartheta (1;0,0;0\cdot a)=0, we can deduce that \vartheta (1;0,0,\lambda a)=k{\pi }_p, k\ge 1, has at most one positive solution and one negative solution, denoted by {\lambda }_k^N(a) and {\lambda }_{-k}^N(a) respectively, if they exist. In other words, we have

For any a\in L^1[0,1], use the notation a\succ 0 if a(t)\ge 0 for almost every t\in [0,1] and a(t)>0 on a subset of [0,1] of positive measure. Write a\prec 0 if -a\succ 0.

Lemma 3.1 ([9])

Let a\in L^1[0,1]. Then it is necessary that 0\notin {\mathrm{\Sigma }}_p^D(a).

1. (i)

   If a\succ 0, then {\mathrm{\Sigma }}_p^D(a) contains no negative eigenvalues, and it consists of a sequence of positive eigenvalues
2. (ii)

   If a\prec 0, then {\mathrm{\Sigma }}_p^D(a) contains no positive eigenvalues, and it consists of a sequence of negative eigenvalues

   (0>){\lambda }_{-1}^D(a)>{\lambda }_{-2}^D(a)>\cdots >{\lambda }_{-k}^D(a)>\cdots (\to -\mathrm{\infty }).
3. (iii)

   If a_{+}\succ 0 and a_{-}\succ 0, then {\mathrm{\Sigma }}_p^D(a) contains both positive and negative eigenvalues, and it consists of a double sequence of eigenvalues

Lemma 3.2 ([9])

Let a\in L^1[0,1]. Then it is necessary that 0\in {\mathrm{\Sigma }}_p^N(a).

1. (i)

   If a\succ 0, then {\mathrm{\Sigma }}_p^N(a) contains no negative eigenvalues, and it consists of a sequence of non-negative eigenvalues

   The principal eigenvalue {\lambda }_0^N(a) does not exist in this case.

2. (ii)

   If a\prec 0, then {\mathrm{\Sigma }}_p^N(a) contains no positive eigenvalues, and it consists of a sequence of non-positive eigenvalues

   0>{\lambda }_{-1}^N(a)>{\lambda }_{-2}^N(a)>\cdots >{\lambda }_{-k}^N(a)>\cdots (\to -\mathrm{\infty }).

   The principal eigenvalue {\lambda }_0^N(a) does not exist in this case.

3. (iii)

   If a_{\pm }\succ 0 and , then {\mathrm{\Sigma }}_p^N(a) contains both positive and negative eigenvalues, and it consists of a double sequence of eigenvalues

   The principal eigenvalue {\lambda }_0^N(a) is positive in this case.

4. (iv)

   If a_{\pm }\succ 0 and {\int }_0^{1}a(t)dt>0, then {\mathrm{\Sigma }}_p^N(a) consists of a double sequence of eigenvalues

   The principal eigenvalue {\lambda }_0^N(a) is negative in this case.

5. (v)

   If a_{\pm }\succ 0 and {\int }_0^{1}a(t)dt=0, then {\mathrm{\Sigma }}_p^N(a) consists of a double sequence of eigenvalues

   The principal eigenvalue {\lambda }_0^N(a) does not exist in this case.

The following lemma reveals, to some extent, the essential reason of the existence of positive eigenvalues.

Lemma 3.3 Assume that a\in L^1[0,1], , {\vartheta }_0\le k{\pi }_p, {\theta }_0\le k{\pi }_p-\frac{{\pi }_p}{2}, and k\in \mathbb{Z}. Denote the indicator function of the subset [t_1,t_2] of the set [0,1] by 1_{[t_1,t_2]}. Then

Proof We only prove equation (3.32), and equation (3.31) can be proved similarly.

Write f(\lambda )=\vartheta (t_2;t_1,{\theta }_0,\lambda a) for simplicity.

If a_{+}\cdot 1_{[t_1,t_2]}\succ 0, by similar arguments as in [[9], Lemma 3.4] (see also Lemma 3.5) we have

Let \lambda =0 in equation (3.2) and we get the equation

which has equilibria \theta =k{\pi }_p, k\in \mathbb{Z}. Because , we get

Therefore there must exist {\lambda }^{\ast }>0 such that f({\lambda }^{\ast })=k{\pi }_p.

On the other hand, suppose that a_{+}\cdot 1_{[t_1,t_2]}=0, namely, a(t)\le 0 for almost every t\in [t_1,t_2]. If \lambda >0, then for almost every t\in [t_1,t_2]. Now it follows from the comparison theorem, equation (3.15), and Remark 2.1 that