In this paper, we study the structure of the Fučík spectrum of Dirichlet and Neumann problems for the scalar p-Laplacian 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 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.
Keywords:indefinite weights; p-Laplacian; Fučík spectrum; spectral structure
Fučík spectrum was first introduced for the Laplacian on a bounded domain , , by Dancer  and by Fučík  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, . See  and references therein.
In this paper, we are concerned with the Fučík spectrum of the scalar p-Laplacian
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 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 and are defined as the sets of those such that equation (1.2) has non-trivial solutions satisfying the corresponding boundary conditions (1.3) and (1.4), respectively.
The Fučík spectra and have been comprehensively understood in : 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  studied and by means of ‘zero functions’, where the weights a and b were assumed to be sign-changing (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 non-trivial 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  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 and 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.
(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.
(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 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 (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 . 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 half-eigenvalues of the p-Laplacian, 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
Thep-cosine andp-sine have the following properties.
we can transform equation (2.1) into the following equations for r and θ:
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 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 quasi-monotonicity. We refer the readers to [, Lemma 2.3] for a detailed proof.
Theorem 2.1 ()
In the p-polar coordinates (2.2), equation (1.2) is equivalent to the following two equations:
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 quasi-monotonicity 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
Till now, we have proved that
Conversely, let us show that
For this specific , take a non-trivial 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
Combining the previous arguments, we can conclude that.
Therefore we need only to focus our study on the subset
With the help of half-eigenvalues of the p-Laplacian with a pair of weights, we can determined whether is an empty set or not. Using eigenvalues of the p-Laplacian 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 half-eigenvalues of the p-Laplacian with weights.
3.2 Eigenvalues of p-Laplacian with an indefinite weight
using the notations in Section 2, we have
has a non-trivial solution satisfying the Dirichlet boundary condition , the Neumann boundary condition , the Dirichlet-Neumann boundary condition and the Neumann-Dirichlet boundary condition , respectively. Similar arguments as in Section 3.1 show that
These spectra have been studied in . Consider the function of :
It follows from formulations (2.6) and (2.7) in Theorem 2.1 that
It has been proved in  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
Lemma 3.1 ()
Lemma 3.2 ()
The following lemma reveals, to some extent, the essential reason of the existence of positive eigenvalues.
Proof We only prove equation (3.32), and equation (3.31) can be proved similarly.
If , by similar arguments as in [, Lemma 3.4] (see also Lemma 3.5) we have
completing the proof of equation (3.32). □
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 quasi-monotonicity of in λ.
3.3 Half-eigenvalues of p-Laplacian with a pair of indefinite weights
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 is also quasi-monotonic in λ. More precisely, we have
It is easy to check that
Essentially we need only to concern and those positive half-eigenvalues , . 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 half-eigenvalues?
Proof This lemma can be proved by similar argument as in the proof of Lemma 2.3 in , thus we skip the details. □
and inequality (3.43) becomes an equality if and only if
Particularly, we get from equations (3.44) and (3.46)
It follows from Lemma 2.2 that
Particularly, we get
(ii) This result follows immediately from equations (3.35), (3.36), and Lemma 3.5.
We claim that
Let us take the notations
Particularly, we get
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.
In this subsection, we always use the notation
Proof We only prove (i), and (ii) can be proved by similar arguments.
In the following, we write
Furthermore, we can deduce that
Particularly, we get
To complete the proof of (i), we need only to prove
By the quasi-monotonicity results in Lemma 2.2, one has
It follows from Lemma 3.1 that the solutions to
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.
Take the following notations for simplicity:
By the quasi-monotonicity results in Lemma 2.2, we have
Inductively, we can show that
Similarly, applying equations (3.16), (3.61), (3.63), and Lemma 2.2, we can obtain
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 .
When the weight a or b is positive, we can improve the results about the asymptotic lines in Theorem 3.3.
By equation (3.16), we also have
Then it follows from equations (3.68) and (3.69) that
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 . □
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.
Then we get can the following results immediately from Property 3.4.
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 half-eigenvalues 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.
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 half-eigenvalues and , , exist; but none of the half-eigenvalues , , , , , , , exist. Then we have the following theorem.
4 Fučík spectrum for Neumann problems
Theorem 4.5One of the following three cases must occur.
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.
The authors declare that they have no competing interests.
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.
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. 111-2-01).
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