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 non-trivial solutions. Let
In the p-polar coordinates (2.2), equation (1.2) is equivalent to the following two equations:
(3.2)
(3.3)
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 -periodic in θ, one has
(3.4)
for any , and . One can also check that
(3.5)
Suppose is an eigenfunction of system (1.2)-(1.3) associated with . By equation (2.2), the corresponding solution of equation (3.2), , satisfies
(3.6)
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
(3.7)
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
(3.8)
Till now, we have proved that
Conversely, let us show that
Suppose for some . Then satisfies
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
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 3.1 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 defined as in equation (3.8) can be rewritten as
Thus
(3.9)
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 p-Laplacian with the weight a. See Theorem 3.2.
For those sets , , it is easy to check that
(3.10)
(3.11)
(3.12)
Therefore we need only to focus our study on the subset
(3.13)
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 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
Given , denote by the solution of
(3.14)
satisfying the initial value condition . Particularly, if , it follows from Lemma 2.1(vi) that equation (3.14) turns to be , and hence
(3.15)
Because the right-hand side of equation (3.14) is -periodic in θ, we have
(3.16)
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 quasi-monotonic in t.
Given , denote by , , and the sets of such that
(3.17)
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
(3.18)
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
(3.19)
where satisfies
(3.20)
See equation (2.5) for the definition of . Then is also a non-trivial 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
(3.21)
If , then becomes the associated eigenfunction of equation (3.17) satisfying . In this case, the first item on the right-hand side of equation (3.21) equals 0, and hence
(3.22)
where and . Similarly, we can obtain
(3.23)
(3.24)
(3.25)
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
(3.26)
(3.27)
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
(3.28)
(3.29)
(3.30)
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 , 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
(3.31)
(3.32)
Proof We only prove equation (3.32), and equation (3.31) can be proved similarly.
Write for simplicity.
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 quasi-monotonicity 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
(3.33)
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
(3.34)
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.
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 , denote by and the sets of half-eigenvalues of the scalar p-Laplacian, namely, the sets of those such that
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
(3.35)
We also know that , because the equation
has equilibria , . Combining the quasi-monotonicity results in Lemma 2.2, we have
(3.36)
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
(3.37)
(3.38)
By equations (3.26), (3.27), and Lemma 2.2, we have . Some immediate results are
(3.39)
Similarly, we have and
Thus the Neumann type half-eigenvalues , , 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 half-eigenvalue , 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 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?
Lemma 3.5 Assume that and . If , then
(3.40)
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
(3.41)
(3.42)
Let and . Then
(3.43)
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
(3.44)
and hence for any . Recall that satisfies the ODE
Then we can conclude that also satisfies
(3.45)
on the interval . Thus we have
(3.46)
Particularly, we get from equations (3.44) and (3.46)
(3.47)
On the other hand, let in equation (3.41), we get
(3.48)
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
a contradiction to .
Claim II: there exists such that . If this is not true, then the quasi-monotonicity of in t shows that
Thus satisfies
(3.49)
on the interval , and the initial value condition . Therefore
(3.50)
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 first-order 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 3.1 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 . If there exists a positive half-eigenvalue , , 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 half-eigenvalue , , 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
(3.51)
We claim that
(3.52)
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,
(3.53)
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 half-eigenvalues in equation (3.37) and Lemma 2.2, there exist and , such that
and
Particularly, we get
(3.54)
(3.55)
(3.56)
(3.57)
Let in equations (3.54) and (3.56). We get from Lemma 3.3
(3.58)
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
and hence exists.
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
(3.59)
or
(3.60)
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 3.1 Let and denote . One of the following three cases must occur.
-
(i)
and .
-
(ii)
and .
-
(iii)
There exists
such that either
or
Applying Lemmas 3.3 and 3.6, one can verify the following three examples.
Example 3.1 Suppose , , , and
Then and .
Example 3.2 Suppose , , , and
Then and .
Example 3.3 Suppose , , , and
Then and .
The following property can be proved by similar arguments as used for Property 3.1.
Property 3.2 Given , we have the following results:
-
(i)
if , then any positive half-eigenvalues , , does not exist, namely
-
(ii)
if , then those positive half-eigenvalues , , exist, but the existence of a positive principal half-eigenvalue is indefinite;
-
(iii)
exists, ⟹ both and exist;
-
(iv)
both and exist, ⟹ exists or exists.
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 quasi-monotonic in λ and in μ. This property is crucial for us to characterize the structure of the Fučík spectra .
Lemma 3.7 Let . The following results hold:
-
(i)
if , and , , then
-
(ii)
if , and , , then
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 quasi-monotonicity 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 3.2 Let and assume that .
-
(i)
If and , then .
-
(ii)
If , then .
-
(iii)
If , then .
Remark 3.1 It follows from equation (3.9) and the above theorem that is made up of two horizontal lines