Abstract
In this paper, we study the structure of the Fučík spectrum of Dirichlet and Neumann problems for the scalar pLaplacian with indefinite weights . Besides the trivial horizontal lines and vertical lines, it will be shown that, confined to each quadrant of , 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 halfeigenvalues of the pLaplacian with weights a and b. The asymptotic lines will be estimated by using SturmLiouville eigenvalues of the pLaplacian with a weight a or b.
MSC: 34B09, 34B15, 34L05.
Keywords:
indefinite weights; pLaplacian; Fučík spectrum; spectral structure1 Introduction
Fučík spectrum was first introduced for the Laplacian on a bounded domain , , 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 pLaplacian, . See [3] and references therein.
In this paper, we are concerned with the Fučík spectrum of the scalar pLaplacian
Given , taking the notations , 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 and are defined as the sets of those such that equation (1.1) has nontrivial solutions satisfying the Dirichlet boundary condition
and the Neumann boundary condition
respectively. Similarly, for a pair of weights a and b, the Fučík spectra and are defined as the sets of those such that equation (1.2) has nontrivial solutions satisfying the corresponding boundary conditions (1.3) and (1.4), respectively.
The Fučík spectra and 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 (SturmLiouville) eigenvalues of the pLaplacian 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 and by means of ‘zero functions’, where the weights a and b were assumed to be signchanging (i.e., and ) 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 , consists of an odd number of or infinitely many hyperbolic like curves. The asymptotic behavior of the first nontrivial curves in each quadrant was also studied. It was observed that for instance the first curve of in 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 and , where the weights 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 secondorder ODE (1.2) into a system of firstorder 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 and are completely determined by this firstorder 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.
(i) Besides at most two vertical lines and two horizontal lines, confined to each quadrant of is made up of zero, an odd number of, or a double sequence of continuous, strictly monotonic, hyperbolic like curves.
(ii) The number of those trivial lines in is determined by the Dirichlet and Neumann eigenvalues of the pLaplacian.
(iii) The number of the hyperbolic like curves in is determined by the Dirichlet and Neumann halfeigenvalues of the pLaplacian.
(iv) All the hyperbolic like curves have vertical and horizontal asymptotic lines, and these asymptotic lines will be estimated by using (SturmLiouville) eigenvalues of the pLaplacian.
(v) If the weights a and b are positive, the structure of is comparable with that of , the case with potentials. More precisely, is composed of one horizontal line, one vertical line and a double sequence of differentiable, strictly decreasing, hyperbolic like curves in the quadrant . And all asymptotic lines of these hyperbolic like curves will be given by using (SturmLiouville) eigenvalues of the pLaplacian.
The paper is organized as follows. In Section 2, we will give some preliminary results. Section 3 is devoted to . We first decompose in Section 3.1, according to the number of zeroes of the eigenfunctions. Sections 3.2 and 3.3 are devoted to eigenvalues and halfeigenvalues of the pLaplacian, respectively. The results in these two subsections enables us to finally determine the structure of in Section 3.4. For a pair of positive weights a and b, we can get more information on and the results are given in Section 3.5. The Fučík spectrum can be studied by similar arguments and we just list the results in Section 4.
2 Preliminary results
Given an exponent , denote by the conjugate number of p, namely . The initial value problem
has a unique solution , . The functions and are the socalled pcosine and psine because they possess properties similar to those of the standard cosine and sine, as shown in the following lemma.
Thepcosine andpsine have the following properties.
(i) Bothandareperiodic, where
(ii) is even intandis odd int;
(iv) if and only if, , andif and only if, ;
Remark 2.1 For any , one has . In fact, if , then . If , then
Let . Via the ppolar 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 and , denote by , , the unique solution of system (2.3)(2.4) satisfying and . Let
The ppolar coordinates (2.2), one can verify that equation (2.1) has a nontrivial solution
One basic observation on equation (2.3) is that the vector field at those θ such that , i.e., , . Since and are only integrable, the derivative at any specific t is meaningless. However, one can still use such an observation to obtain the following property, called quasimonotonicity. We refer the readers to [[7], Lemma 2.3] for a detailed proof.
Lemma 2.2Given, and, letbe the solution of equation (2.3). Iffor some, then
Denote by the weak topology in . By in , or , we mean that
Some important properties of , , and are collected in the following theorem.
Theorem 2.1 ([8])
Letandbe fixed. We have the following results.
(i) As mappings fromto, andare continuous. More precisely, ifand, then
(ii) The functional, is continuous. More precisely, ifand, thenas.
(iii) The functional, is continuously differentiable in the sense of Fréchet. The differentials ofataandb, denoted, respectively, byand, are the following mappings:
whereis the dual space of. Moreover, as mappings fromto, bothandare continuous.
Remark 2.2 Let and , . If and , then it follows from formulations (2.6) and (2.7) that
3 Fučík spectrum for Dirichlet problems:
3.1 Decomposition of
Given a pair of weights , the (Dirichlet type) Fučík spectrum is defined as the set of those such that system (1.2)(1.3) has nontrivial solutions. Let
In the ppolar 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 righthand side of equation (3.2) is periodic in θ, one has
for any , and . One can also check that
Suppose is an eigenfunction of system (1.2)(1.3) associated with . By equation (2.2), the corresponding solution of equation (3.2), , satisfies
for some . Due to equation (3.4), we may restrict . In other words, we may assume that and hence or . Moreover, it follows from the quasimonotonicity result in Lemma 2.2 that . We distinguish two cases: or . If , then it follows from equation (3.1) that . By equation (2.2), we have , and hence and . Let
Now equation (3.6) tells us that . In fact, the subscript k is related to the number of zeroes of on . By Lemma 2.2, the equation
has a solution if and only if , and
By equation (2.2), we see that has exactly zeroes in . Similarly, if and has exactly zeroes in , then , and , where
Till now, we have proved that
Conversely, let us show that
Suppose for some . Then satisfies
For this specific , take a nontrivial solution of equation (3.3). Then we can construct a function , which is a solution of equation (1.2) with exactly zeroes on . Particularly, . Thus , and hence . Furthermore, we have
and hence
Similarly, if for some , then and any associating eigenfunction satisfies and has exactly zeroes in .
Combining the previous arguments, we can conclude that.
Theorem 3.1Let. The Fučík spectrumcan be decomposed as
Moreover, the following characterization onandholds.
(i) , ⟺ any eigenfunctionassociated withsatisfies, andhas preciselyzeroes in.
(ii) , ⟺ any eigenfunctionassociated withsatisfies, andand has preciselyzeroes in.
By equation (3.5), the set defined as in equation (3.8) can be rewritten as
Thus
In other words, is symmetric to about the line . For this reason, essentially we need only to characterize those sets .
In Section 3.4, we will see that is made up of straight lines which are in connection with and , the Dirichlet eigenvalues of pLaplacian with the weight a. See Theorem 3.2.
For those sets , , it is easy to check that
Therefore we need only to focus our study on the subset
where . In Section 3.4, for each we will show that 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 halfeigenvalues of the pLaplacian with a pair of weights, we can determined whether is an empty set or not. Using eigenvalues of the pLaplacian with a weight, we can roughly locate the hyperbolic like curve . For these reasons, we will give in the successive two subsections some useful characterization on eigenvalues and halfeigenvalues of the pLaplacian with weights.
3.2 Eigenvalues of pLaplacian with an indefinite weight
Given , denote by the solution of
satisfying the initial value condition . Particularly, if , it follows from Lemma 2.1(vi) that equation (3.14) turns to be , and hence
Because the righthand side of equation (3.14) is periodic in θ, we have
for any , and . Since equation (3.14) can also be rewritten as
using the notations in Section 2, we have
By Lemma 2.2, we see that is also quasimonotonic in t.
Given , denote by , , and the sets of such that
has a nontrivial solution satisfying the Dirichlet boundary condition , the Neumann boundary condition , the DirichletNeumann boundary condition and the NeumannDirichlet boundary condition , respectively. Similar arguments as in Section 3.1 show that
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
See equation (2.5) for the definition of . Then is also a nontrivial solution of equation (3.17). Multiplying equation (3.20) by and integrating over , we have
Substituting this into equation (3.19), for any we have
If , then becomes the associated eigenfunction of equation (3.17) satisfying . In this case, the first item on the righthand side of equation (3.21) equals 0, and hence
where and . Similarly, we can obtain
For any , it follows from equations (3.18) and (3.22) that
has at most one positive solution and one negative solution, denoted by and , respectively, if they exist. In other words, we have
It has been proved in [9] that has at most one nonzero solution, called the principal Neumann eigenvalue and denoted by , if it exists. By equation (3.23) and the fact , we can deduce that , , has at most one positive solution and one negative solution, denoted by and respectively, if they exist. In other words, we have
For any , use the notation if for almost every and on a subset of of positive measure. Write if .
Lemma 3.1 ([9])
Let. Then it is necessary that.
(i) If, thencontains no negative eigenvalues, and it consists of a sequence of positive eigenvalues
(ii) If, thencontains no positive eigenvalues, and it consists of a sequence of negative eigenvalues
(iii) Ifand, thencontains 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, thencontains no negative eigenvalues, and it consists of a sequence of nonnegative eigenvalues
The principal eigenvaluedoes not exist in this case.
(ii) If, thencontains no positive eigenvalues, and it consists of a sequence of nonpositive eigenvalues
The principal eigenvaluedoes not exist in this case.
(iii) Ifand, thencontains both positive and negative eigenvalues, and it consists of a double sequence of eigenvalues
The principal eigenvalueis positive in this case.
(iv) Ifand, thenconsists of a double sequence of eigenvalues
The principal eigenvalueis negative in this case.
(v) Ifand, thenconsists of a double sequence of eigenvalues
The principal eigenvaluedoes not exist in this case.
The following lemma reveals, to some extent, the essential reason of the existence of positive eigenvalues.
Lemma 3.3Assume that, , , , and. Denote the indicator function of the subsetof the setby. Then
Proof We only prove equation (3.32), and equation (3.31) can be proved similarly.
If , by similar arguments as in [[9], Lemma 3.4] (see also Lemma 3.5) we have
Let in equation (3.2) and we get the equation
which has equilibria , . Because , we get
Therefore there must exist such that .
On the other hand, suppose that , namely, for almost every . If , then for almost every . Now it follows from the comparison theorem, equation (3.15), and Remark 2.1 that
completing the proof of equation (3.32). □
In the rest of this subsection, we aim to reveal some quasimonotonicity property of in λ, which will play an important role in analyzing the structure of the Fučík spectra .
Using equation (3.18), the characterization on , we can rewritten equation (3.22) more precisely as
Furthermore, we have
because it follows from equation (3.16) that
Though we have always been considering equations on the interval , similar results as in Theorem 2.1 still hold when the interval is replaced by any general interval. Thus equation (3.33) can also be generalized. In fact, for any , and , we have
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 quasimonotonicity of in λ.
(i) If there existand an integersuch that, then. Consequently, for anyandfor any.
(ii) If there existand an integersuch that, then. Consequently, for anyandfor any.
(iii) If there existand an integersuch that, then. Consequently, for anyandfor any.
(iv) If there existand an integersuch that, then. Consequently, for anyandfor any.
3.3 Halfeigenvalues of pLaplacian with a pair of indefinite weights
For any , denote by and the sets of halfeigenvalues of the scalar pLaplacian, namely, the sets of those such that
has a nontrivial solution satisfying the boundary conditions (1.3) and (1.4), respectively.
Based on the ppolar transformation (2.2) and the quasimonotonicity 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 is also quasimonotonic in λ. More precisely, we have
We also know that , because the equation
has equilibria , . Combining the quasimonotonicity results in Lemma 2.2, we have
and hence . It follows from equations (3.35) and (3.36) that for any , the equation
has at most one positive solution and one negative solution, denoted, respectively, by and , if they exist. More precisely, we have
By equations (3.26), (3.27), and Lemma 2.2, we have . Some immediate results are
Thus the Neumann type halfeigenvalues , , are defined as
And the existence of , , implies the existence of . By Lemma 2.2, the solution of is also that of . Thus there may exist at most one principal Neumann halfeigenvalue , which is defined as
and by equation (3.28). Note that may not exist even if exist.
It is easy to check that
Essentially we need only to concern and those positive halfeigenvalues , . Now a natural question arises: for what kind of weights a and b do there exist no, finitely many, or infinitely many positive Dirichlet or Neumann type halfeigenvalues?
Lemma 3.5Assume thatand. If, then
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 3.6Suppose that, , and there exist, , and, , such that
and inequality (3.43) becomes an equality if and only if
Proof Let us write for simplicity.
Claim I: there exists such that . If this is false, then it follows from Lemma 2.2 that
and hence for any . Recall that satisfies the ODE
Then we can conclude that also satisfies
on the interval . Thus we have
Particularly, we get from equations (3.44) and (3.46)
On the other hand, let in equation (3.41), we get
Since , it follows from Lemma 3.4(i) that
a contradiction to equation (3.47). Thus there exists such that , proving Claim I.
If , we aim to show that . If this is not true, then , and one can check that equation (3.46) is still true. It follows from equations (3.46), (3.48), Lemma 3.4(i), and the fact that
Claim II: there exists such that . If this is not true, then the quasimonotonicity of in t shows that
on the interval , and the initial value condition . Therefore
If , similar arguments as in the proof of Claim I show the existence of such that , and Claim II is proved. Moreover, if , then .
If , then by Lemma 2.2. We can improve the result in Claim II as the existence of such that . If this result is not true, then
and we still have equations (3.49)(3.50). Now both and satisfy the same ODE (3.49) on the interval , while the initial values satisfy the condition
By case in equation (3.42) we have . Now the existence and uniqueness theorem for the firstorder ODEs shows that
Since , there exists such that .
Now we can conclude that Claim II is true. Moreover, if or , then .
Inductively, we can show that there exists such that . Moreover, we have , if for some , or for some . And if , it follows from Lemma 2.2 that .
Finally, if , , and , , then it follows from equations (3.41) and (3.42) that , completing the proof of the lemma. □
Property 3.1Given, we have the following results:
(i) if, then any positive halfeigenvalues, , does not exist;
(ii) if, then all positive halfeigenvalues, , exist;
(iv) bothandexist, ⟹ exists orexists.
Proof (i) Assume that . If there exists a positive halfeigenvalue , , then
and hence there must exist such that
It follows from Lemma 2.2 that
and hence
Particularly, we get
By Lemma 3.3, we have , a contradiction to the assumption . Consequently, there is not any positive halfeigenvalue , , if .
(ii) This result follows immediately from equations (3.35), (3.36), and Lemma 3.5.
(iii) Assume that exists and . The existence of has been given in equation (3.39). We need only to prove the existence of . Take the following notations for simplicity:
By the definition of , we have . By Lemma 2.2, there exist , such that
and
Therefore if and i is even (odd). Thus
We claim that
In fact, we have by Lemma 2.2, proving equation (3.52) for the case . To prove the case , we assume on the contrary that , then
and hence
Letting in equation (3.51), we see that and satisfy the same ODE,
Moreover, is also a solution of equation (3.53). Since
we obtain
thus the assumption is false, proving equation (3.52) for the case . Inductively, we can prove equation (3.52).
Let in equation (3.52); we get
Combining with equation (3.36), we conclude that there must exist such that
(iv) Suppose that both and exist and . We assume that , . The case can be proved similarly.
Let us take the notations
By the definition of halfeigenvalues in equation (3.37) and Lemma 2.2, there exist and , such that
and
Particularly, we get
Let in equations (3.54) and (3.56). We get from Lemma 3.3
Without loss of generality, we may assume that . We need only to distinguish three cases in the following.
Case 1. and . In this case, it follows from Lemma 3.3 that there exist and such that
Let . Combining equations (3.54), (3.55), and the above two conditions, we get from Lemma 3.6
Case 2. and . In this case, we have . And it follows from the result (ii) in this property that both and exist.
Case 3. and . In this case, since , the condition (3.58) can be written as
Because , there must exist , such that
or
Then we can apply similar arguments as in Case 1 to get the existence of if equation (3.59) holds, and the existence of if equation (3.59) holds. □
Corollary 3.1Letand denote. One of the following three cases must occur.
(iii) There existssuch that either
or
Applying Lemmas 3.3 and 3.6, one can verify the following three examples.
The following property can be proved by similar arguments as used for Property 3.1.
Property 3.2Given, we have the following results:
(i) if, then any positive halfeigenvalues, , does not exist, namely
(ii) if, then those positive halfeigenvalues, , exist, but the existence of a positive principal halfeigenvalueis indefinite;
(iv) bothandexist, ⟹ exists orexists.
3.4 Structure of the Fučík spectrum
In this subsection, we always use the notation
for simplicity if there is no confusion. By Theorem 2.1, we see that is continuous in .
The following lemma tells us that is quasimonotonic in λ and in μ. This property is crucial for us to characterize the structure of the Fučík spectra .
Lemma 3.7Let. The following results hold:
Proof We only prove (i), and (ii) can be proved by similar arguments.
Suppose , and , . It follows from the definition of in equation (3.7) that
In the following, we write
for simplicity. By Lemma 2.2, there exist (0=) (=1), such that
Furthermore, we can deduce that
Particularly, we get
Then it follows from Lemma 3.6 that, for any (>0), we have
To complete the proof of (i), we need only to prove
If this is not true, then there exists such that . Write
for simplicity. Then there exist (0=) (≤1) such that
Since , we can use similar arguments as in the previous paragraph to get
Finally, it follows from the condition and Lemma 2.2 that
a contradiction to the fact . □
By the definition of in equation (3.7), one has
By the quasimonotonicity results in Lemma 2.2, one has
Then
It follows from Lemma 3.1 that the solutions to
are if , if , and if . Thus can be characterized as in the following theorem.
Theorem 3.2Letand assume that.
Remark 3.1 It follows from equation (3.9) and the above theorem that is made up of two horizontal lines and . If the eigenvalues and do not exist, then and should be understood as empty sets, respectively.
If , the set is more complicated than . As analyzed in Section 3.1, essentially we need only to discuss the subset as defined in equation (3.13). Finally we will show that is either an empty set, or a hyperbolic like curve. The following property helps us to locate roughly.
Proof We will only prove (i), and (ii) can be proved similarly.
If , then , , and there exist , such that
satisfies
Therefore satisfies equation (3.45) on each interval , , and it satisfies equation (3.49) on each interval , . In other words, we have
Take the following notations for simplicity:
Let and take in equation (3.62). Then it follows from equation (3.61) that
By the quasimonotonicity results in Lemma 2.2, we have
and hence
Inductively, we can show that
Similarly, applying equations (3.16), (3.61), (3.63), and Lemma 2.2, we can obtain
The case (≥1) in equation (3.65) implies that
By equation (3.36), we see that , and hence . By the definition of as in equation (3.26), one has
Then it follows from Lemma 3.4(i) that (>0).
Similarly, it can be deduced from equations (3.66), (3.26) and Lemma 3.4 that . □
Theorem 3.3If, , thenis a continuous, strictly decreasing, hyperbolic like curve
with the horizontal asymptotic lineand the vertical asymptotic line, where, , andis the floor function.
Proof Assume that , . We prove the theorem by seven steps.
Step 1. We aim to prove that is not a singlepoint set. Suppose that , then
By Property 3.3, and . Then it follows from Lemma 3.7 that
Since is continuous in , there must exist a point in the open line segment with endpoints and , and hence and , such that
Thus and , proving the claim. Furthermore, let
then is not a singlepoint set, because , and .
Step 2. It follows from Lemma 3.7 that every horizontal line intersects at one time at most, so does every vertical line. Therefore is the image of some function
Suppose and . Let , . If , then it follows from Lemma 3.7 that
which is a contradiction. Thus . Therefore is a strictly decreasing function on . Let
and
Step 3. We claim that if and , then for any . Let , . By the monotonicity result proved in Step 2, we have . By Lemma 3.7 again, for any , one has
Then the continuity of in implies the existence of such that
and hence , completing the proof of the claim.
Step 4. We aim to prove that . If , then . Thus
and . Similar arguments as in Step 1 show that there exists some point in the open line segment with endpoints and , such that , and hence and , a contradiction. Therefore . Similarly, we can prove that . Combining the results in Step 3, we see that .
Step 5. We aim to prove that . Since and the function is strictly decreasing, we see that is not a singlepoint set. Then similar arguments as in Step 3 and Step 4 show that .
Step 6. We aim to show that is a continuous function on . In fact, we have
Fix any . By the monotonicity results proved in Step 2, we see that both
exist, and . Furthermore, we can show that . Thus . Let and in equation (3.67), then the continuity of in guarantees that
Similarly, let and in equation (3.67), then
Now Lemma 3.7 implies that . Thus is continuous at the point . Since can be chosen arbitrarily, is continuous in .
Step 7. In the above six steps, we have shown that the continuous and strictly decreasing function maps onto . Then it is necessary that
Therefore is a hyperbolic like curve with the horizontal asymptotic line and the vertical asymptotic line . □
When the weight a or b is positive, we can improve the results about the asymptotic lines in Theorem 3.3.
Theorem 3.4Let. The following results hold.
(i) Ifand, thenfor any, and the vertical asymptotic line ofis.
(ii) Ifand, thenfor any, and the horizontal asymptotic line ofis.
Proof We only prove (i) for the case , . Other cases can be proved similarly.
Since , it follows from Lemma 3.1 that the eigenvalue exists. By the definition of as in equation (3.26), we have . By Lemma 2.2, there exist such that
and hence
By equation (3.16), we also have
Given any , it follows from Lemma 3.4(i) that
and hence there exist , such that
We have because . By Lemma 3.3, there exist , such that
Define a weight q on the interval as
Then it follows from equations (3.68) and (3.69) that
Let and . One has on . Now Remark 2.2 implies that
By the continuity of in , there exists , and hence , such that . Thus and .
Now it follows from Theorem 3.3 that is a hyperbolic like curve and its vertical asymptotic line satisfies . On the other hand, since , we get . Furthermore, because can be chosen arbitrarily. Therefore the vertical asymptotic line of is . □
If exists and , then it follows from equation (3.37) that
and hence . Thus is a hyperbolic like curve by Theorem 3.3. On the other hand, suppose that , , is a hyperbolic like curve. Theorem 3.3 tells us it has a horizontal asymptotic line and a vertical one. Then it must intersect the diagonal at a unique point . Furthermore, we can deduce that . In conclusion, we have the following property.
Property 3.4Letand. The following results hold.
(i) if and only ifdoes not exist.
(ii) is a hyperbolic like curve if and only ifexists.
(iii) is either an empty set or a hyperbolic like curve in the quadrantemanating from the point.
Remark 3.2 If , , then it follows from Properties 3.1 and 3.4 that . Furthermore, we can deduce by Lemma 3.7 that the higherorder hyperbolic like curve always lies above the lowerorder curve .
By equations (3.9) and (3.13), we know that is asymmetric to about the diagonal . Namely,
Then we get can the following results immediately from Property 3.4.
Property 3.5Letand. Thenis either an empty set or a hyperbolic like curve in the quadrantemanating from, andif and only ifexists.
By Properties 3.4 and 3.5, we see that the existence of those hyperbolic like curves and , , is determined by the existence of those halfeigenvalues and , respectively. By Corollary 3.1, we can conclude that besides those trivial lines, the Fučík spectrum confined to the quadrant is an empty set, or made up of an odd number of hyperbolic like curves, or made up of a double sequence of hyperbolic like curves. Taking the relations (3.10)(3.12) into consideration, we obtain the following theorem.
If one of the halfeigenvaluesordoes not exist, the corresponding straight lineorshould be understood as an empty set. Let
thenconsists of zero, an odd number of, or a double sequence of hyperbolic like curves:
Ifordoes not exist, orshould be understood as an empty set, respectively. Ifandexist, andare continuous, strictly decreasing, hyperbolic like curves. Moreover,
(i) is asymmetric toabout the vertical line;
(ii) is asymmetric toabout the origin;
(iii) is asymmetric toabout the horizontal line.
3.5 Fučík spectrum with positive weights
Assume that , and . Then it follows from Lemma 3.1 that and exist, but and do not exist. By Property 3.1 and Example 3.1, all halfeigenvalues and , , exist; but none of the halfeigenvalues , , , , , , , exist. Then we have the following theorem.
Theorem 3.6Let, and. Thenis made up of one vertical line, one horizontal line and a double sequence of differentiable, strictly decreasing, hyperbolic like curves in:
For each, has the vertical asymptotic lineand the horizontal asymptotic line.
Proof We need only to prove the differentiability of . Recall from Theorem 3.3 that for any , the curve , is determined by
Given , , denote the associated eigenfunction by . Then because . Since for almost every , it follows from formulation (2.7) that
Thus the Implicit Function Theorem can be applied to equation (3.70), and we see that the hyperbolic like curve is differentiable and
□
4 Fučík spectrum for Neumann problems
Given a pair of indefinite weights , the (Neumann type) Fučík spectrum is defined as the set of those such that system (1.2)(1.4) has nontrivial solutions.
Via similar arguments as in the previous sections, can also be characterized. We list the results in the following but omit the detailed proof.
Theorem 4.1The Neumann Fučík spectrumcan be decomposed as
and hence
Moreover, for any set, , one has
By the relations (4.1)(4.3), we need only to consider , and
Theorem 4.2The setis made up of two vertical linesand. Andis made up of two horizontal linesand. Ifordoes not exist, the corresponding straight lineorshould be understood as an empty set.
Theorem 4.3If, , thenis a continuous, strictly decreasing, hyperbolic like curve
with the horizontal asymptotic lineand the vertical asymptotic line. Moreover, the lower bound ofandcan be estimated as follows:
(i) if and only ifexists. Andis either an empty set or a hyperbolic like curve emanating from.
(ii) if and only ifexists. Andis either an empty set or a hyperbolic like curve emanating from.
From the relation between and as stated in Property 3.2, we obtain the following spectral structure of .
Theorem 4.5One of the following three cases must occur.
(iii) There existsuch that either
or
From the relations (4.1)(4.3) and the above theorems, the structure of the Neumann Fučík spectrum becomes clear.
Theorem 4.6Let. Thenis composed of (at most) four trivial lines, , , (if one of the involved principal eigenvalues does not exist, the corresponding straight line is understood as an empty set), and in each quadrant ofzero, a finite odd number of, or a double sequence of hyperbolic like curves.
Finally, if the weights a and b are positive, then neither nor exists, and we have the following results.
Theorem 4.7Let, and. Thenis made up of one vertical line, one horizontal line and a double sequence of differentiable, strictly decreasing, hyperbolic like curves in the quadrant:
If, , the hyperbolic like curvehas the asymptotic lines
If, , the hyperbolic like curvehas the asymptotic lines
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
PY gave the idea of this article and drafted the manuscript. All authors discussed the methods of proving the main results. All authors read and approved the final manuscript.
Acknowledgements
Jifeng Chu was supported by the National Natural Science Foundation of China (Grant No. 11171090, No. 11271078 and No. 11271333), China Postdoctoral Science Foundation funded project (Grant No. 2012T50431) and the Alexander von Humboldt Foundation of Germany. Ping Yan was supported by the National Natural Science Foundation of China (Grant No. 10901089, No. 11171090 and No. 11371213). Meirong Zhang was supported by the National Natural Science Foundation of China (Grant No. 1123001) and the National 111 Project of China (Station No. 111201).
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