### Abstract

In this paper, we study the structure of the Fučík spectrum
*p*-Laplacian with indefinite weights
*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.

##### Keywords:

indefinite weights;*p*-Laplacian; Fučík spectrum; spectral structure

### 1 Introduction

Fučík spectrum was first introduced for the Laplacian on a bounded domain
*p*-Laplacian,

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

Given

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

and the Neumann boundary condition

respectively. Similarly, for a pair of weights *a* and *b*, the Fučík spectra

The Fučík spectra
*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
*a* and *b* were assumed to be sign-changing (*i.e.*,

In this paper, we are interested in
*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

(i) Besides at most two vertical lines and two horizontal lines,

(ii) The number of those trivial lines in
*p*-Laplacian.

(iii) The number of the hyperbolic like curves in
*p*-Laplacian.

(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.

(v) If the weights *a* and *b* are positive, the structure of
*p*-Laplacian.

The paper is organized as follows. In Section 2, we will give some preliminary results.
Section 3 is devoted to
*p*-Laplacian, respectively. The results in these two subsections enables us to finally
determine the structure of
*a* and *b*, we can get more information on

### 2 Preliminary results

Given an exponent
*p*, namely

has a unique solution
*p*-cosine and *p*-sine because they possess properties similar to those of the standard cosine and
sine, as shown in the following lemma.

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

(i) *Both*
*and*
*are*
*periodic*, *where*

(ii)
*is even in**t**and*
*is odd in**t*;

(iii)
*and*
*for all**t*;

(iv)
*if and only if*
*and*
*if and only if*

(v)
*and*
*and*

(vi)

**Remark 2.1** For any

If

Given

Let
*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

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
*θ* such that
*i.e.*,
*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*
*and*
*let*
*be the solution of equation* (2.3). *If*
*for some*
*then*

Denote by

Some important properties of

**Theorem 2.1** ([8])

*Let*
*and*
*be fixed*. *We have the following results*.

(i) *As mappings from*
*to*
*and*
*are continuous*. *More precisely*, *if*
*and*
*then*

*as*

(ii) *The functional*
*is continuous*. *More precisely*, *if*
*and*
*then*
*as*

(iii) *The functional*
*is continuously differentiable in the sense of Fréchet*. *The differentials of*
*at**a**and**b*, *denoted*, *respectively*, *by*
*and*
*are the following mappings*:

*where*
*is the dual space of*
*Moreover*, *as mappings from*
*to*
*both*
*and*
*are continuous*.

**Remark 2.2** Let

### 3 Fučík spectrum for Dirichlet problems:
Π
p
D
(
a
,
b
)

#### 3.1 Decomposition of
Π
p
D
(
a
,
b
)

Given a pair of weights

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
*θ*, one has

for any

Suppose

for some

Now equation (3.6) tells us that
*k* is related to the number of zeroes of

has a solution

By equation (2.2), we see that

Till now, we have proved that

Conversely, let us show that

Suppose

For this specific

and hence

Similarly, if

Combining the previous arguments, we can conclude that.

**Theorem 3.1***Let*
*The Fučík spectrum*
*can be decomposed as*

*Moreover*, *the following characterization on*
*and*
*holds*.

(i)
*any eigenfunction*
*associated with*
*satisfies*
*and*
*has precisely*
*zeroes in*

(ii)
*any eigenfunction*
*associated with*
*satisfies*
*and*
*and has precisely*
*zeroes in*

By equation (3.5), the set

Thus

In other words,

In Section 3.4, we will see that
*p*-Laplacian with the weight *a*. See Theorem 3.2.

For those sets

Therefore we need only to focus our study on the subset

where

With the help of half-eigenvalues of the *p*-Laplacian with a pair of weights, we can determined whether
*p*-Laplacian with a weight, we can roughly locate the hyperbolic like curve
*p*-Laplacian with weights.

#### 3.2 Eigenvalues of *p*-Laplacian with an indefinite weight

Given

satisfying the initial value condition

Because the right-hand side of equation (3.14) is
*θ*, we have

for any

using the notations in Section 2, we have

By Lemma 2.2, we see that
*t*.

Given

has a non-trivial solution satisfying the Dirichlet boundary condition

These spectra have been studied in [9]. Consider the function of

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

where

See equation (2.5) for the definition of

Substituting this into equation (3.19), for any

If

where

For any

has at most one positive solution and one negative solution, denoted by

It has been proved in [9] that

For any

**Lemma 3.1** ([9])

*Let*
*Then it is necessary that*

(i) *If*
*then*
*contains no negative eigenvalues*, *and it consists of a sequence of positive eigenvalues*

(ii) *If*
*then*
*contains no positive eigenvalues*, *and it consists of a sequence of negative eigenvalues*

(iii) *If*
*and*
*then*
*contains both positive and negative eigenvalues*, *and it consists of a double sequence of eigenvalues*

**Lemma 3.2** ([9])

*Let*
*Then it is necessary that*

(i) *If*
*then*
*contains no negative eigenvalues*, *and it consists of a sequence of non*-*negative eigenvalues*

*The principal eigenvalue*
*does not exist in this case*.

(ii) *If*
*then*
*contains no positive eigenvalues*, *and it consists of a sequence of non*-*positive eigenvalues*

*The principal eigenvalue*
*does not exist in this case*.

(iii) *If*
*and*
*then*
*contains both positive and negative eigenvalues*, *and it consists of a double sequence of eigenvalues*

*The principal eigenvalue*
*is positive in this case*.

(iv) *If*
*and*
*then*
*consists of a double sequence of eigenvalues*

*The principal eigenvalue*
*is negative in this case*.

(v) *If*
*and*
*then*
*consists of a double sequence of eigenvalues*

*The principal eigenvalue*
*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*
*and*
*Denote the indicator function of the subset*
*of the set*
*by*
*Then*

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

Write

If

Let

which has equilibria

Therefore there must exist

On the other hand, suppose that

completing the proof of equation (3.32). □

In the rest of this subsection, we aim to reveal some quasi-monotonicity property
of
*λ*, which will play an important role in analyzing the structure of the Fučík spectra

Using equation (3.18), the characterization on

Furthermore, we have

because it follows from equation (3.16) that

Though we have always been considering equations on the interval

Similar arguments can be applied to (3.23)-(3.25) to obtain results analogous to
equation (3.34). We skip the proof and collect these results in the following lemma,
which can be understood as the quasi-monotonicity of
*λ*.

**Lemma 3.4***Given*
*and*
*let*

(i) *If there exist*
*and an integer*
*such that*
*then*
*Consequently*,
*for any*
*and*
*for any*

(ii) *If there exist*
*and an integer*
*such that*
*then*
*Consequently*,
*for any*
*and*
*for any*

(iii) *If there exist*
*and an integer*
*such that*
*then*
*Consequently*,
*for any*
*and*
*for any*

(iv) *If there exist*
*and an integer*
*such that*
*then*
*Consequently*,
*for any*
*and*
*for any*

#### 3.3 Half-eigenvalues of *p*-Laplacian with a pair of indefinite weights

For any
*p*-Laplacian, namely, the sets of those

has a non-trivial solution satisfying the boundary conditions (1.3) and (1.4), respectively.

Based on the *p*-polar transformation (2.2) and the quasi-monotonicity results in Lemma 2.2, by similar
arguments as in Section 3.1 we can show that

Applying the differentiability results (2.6) and (2.7) in Theorem 2.1, together with
the Dirichlet boundary condition (1.3), by similar arguments as in Section 3.2 we
can show that
*λ*. More precisely, we have

We also know that

has equilibria

and hence

has at most one positive solution and one negative solution, denoted, respectively,
by

By equations (3.26), (3.27), and Lemma 2.2, we have

Similarly, we have

Thus the Neumann type half-eigenvalues

And the existence of

and

It is easy to check that

Essentially we need only to concern
*a* and *b* do there exist no, finitely many, or infinitely many positive Dirichlet or Neumann
type half-eigenvalues?

**Lemma 3.5***Assume that*
*and*
*If*
*then*

*uniformly in*

*Proof* This lemma can be proved by similar argument as in the proof of Lemma 2.3 in [10], thus we skip the details. □

**Lemma 3.6***Suppose that*
*and there exist*
*and*
*such that*

*Let*
*and*
*Then*

*and inequality* (3.43) *becomes an equality if and only if*

*Proof* Let us write

Claim I: there exists

and hence

Then we can conclude that

on the interval

Particularly, we get from equations (3.44) and (3.46)

On the other hand, let

Since

a contradiction to equation (3.47). Thus there exists

If

a contradiction to

Claim II: there exists
*t* shows that

Thus

on the interval

If

If

and we still have equations (3.49)-(3.50). Now both

By case

Since

Now we can conclude that Claim II is true. Moreover, if

Inductively, we can show that there exists

Finally, if

**Property 3.1***Given*
*we have the following results*:

(i) *if*
*then any positive half*-*eigenvalues*
*does not exist*;

(ii) *if*
*then all positive half*-*eigenvalues*
*exist*;

(iii)
*exists*,
*both*
*and*
*exist*;

(iv) *both*
*and*
*exist*,
*exists or*
*exists*.

*Proof* (i) Assume that

and hence there must exist