Various Half-Eigenvalues of Scalar -Laplacian with Indefinite Integrable Weights
Consider the half-eigenvalue problem a.e. , where , , for , and and are indefinite integrable weights in the Lebesgue space . We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in , where denotes the weak topology in space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in , where is the norm of .
Fučik spectrum and half-eigenvalues are useful for solving problems with "jumping nonlinearities." Compared with Fučik spectrum, half-eigenvalues have not been paid much research. However, as half-eigenvalues are concerned with only one parameter, we think the theory on half-eigenvalues will lead to better knowledge for Fučik spectrum and jumping nonlinearities.
Given an exponent , let for which is used to define the scalar -Laplacian. Denote for and for any . For a pair of indefinite (sign-changing) integrable weights with , , we are concerned with the half-eigenvalue problem associated with the periodic boundary condition the antiperiodic boundary condition the Dirichlet boundary condition or the Neumann boundary condition respectively. Values of for which (1.1) has nontrivial solutions satisfying the boundary condition (1.2), (1.3), (1.4), or (1.5) will be called periodic, antiperiodic, Dirichlet, or Neumann half-eigenvalues, respectively, while the corresponding solutions will be called half-eigenfunctions. Subscripts , , , and will be used to indicate periodic, antiperiodic, Dirichlet and Neumann boundary conditions, respectively. The set of the half-eigenvalues under each of the preceding boundary conditions will be denoted by , , , and , respectively. Notice that these spectra are dependent on weights .
This article studies the half-eigenvalue problem (1.1) from two aspects. One is the structure of , , , and , and the other is the dependence of all these spectra on weights.
Generally speaking, for the -Laplacian eigenvalue problem, if separated boundary conditions (such as Dirichlet and Neumann) are imposed and if the weight is definite, the spectrum structure is similar to that of the classical linear case. However, compared with separated boundary conditions, definite weights, and weights (or potentials), more difficulty will always be brought on by nonseparated boundary conditions, indefinite weights, and () weights (or potentials), respectively. It has been proved very recently in  that with an indefinite weight , , the spectra structures of under both boundary conditions (1.4) and (1.5) are the same as when . Special case of (1.6) with indefinite weight is also studied by Anane et al. , Cuesta , and Eberhard and Elbert . Various spectra (including half-eigenvalues) of -Laplacian with potentials are studied by Binding and Rynne in . It is shown that compared with the linear case , some new phenomena occur in the periodic problems when . The existence of nonvariational periodic and antiperiodic eigenvalues of -Laplacian for some potentials in  can partially explain these new phenomena.
We will use the shooting method to obtain spectra structures. To this end, (1.1) is studied in -polar coordinates (2.5), and the associated argument function plays a crucial role. Careful analysis on the argument function leads to partial characterization of and and complete characterization of and , see Theorems 3.9, 4.2, and 5.2. Some quasimonotonicity of the argument function is proved by using the Fréchet derivatives and the boundary conditions, see Lemmas 3.3, 4.1, and 5.1. The uniformly asymptotical result is obtained from the compactness in the Banach-Alaoglu theorem as done in , see Lemma 2.3. Besides the properties of the argument functions, the Hamiltonian structure of the problem is essential for us to obtain variational periodic and antiperiodic half-eigenvalues, see Lemma 3.2. After introducing the rotation number function, we can easily obtain the ordering of these variational periodic half-eigenvalues, see (3.38).
For regular self-adjoint linear Sturm-Liouville problems, the continuous dependence of eigenvalues on weights or potentials in the usual topology is well understood, and so is the Fréchet differentiable dependence. Many of these results are summarized in . However, since the space of potentials (or weights) is infinite-dimensional, such a continuity result cannot answer many basic questions. For example, if potentials or weights are confined to a bounded set or a noncompact set, are the eigenvalues finite? To answer such kind of questions, a stronger continuity result is obtained in  for Sturm-Liouville operators and Hill's operators. That is, the eigenvalues are continuous in potentials in weak topology . Based on such a stronger continuity and the differentiability, variational method and singular integrals are applied in  to obtain the extremal value of smallest eigenvalues of Hill's operators with potentials confined to balls. The continuity result in weak topology are generalized to scalar -Laplacian for eigenvalues on potentials (see ), for (separated) eigenvalues on indefinite weights (see ), and for half-eigenvalues on potentials (see ). Some elementary applications are also presented in [1, 10].
In this paper, we will prove that the variational periodic or antiperiodic half-eigenvalues and (defined by (3.22) and (3.25), resp.), and all the half-eigenvalues in and , are continuous in weights . See Theorems 3.12, 4.3, and 5.3. Moreover, the Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in weights , see Theorems 4.4 and 5.3. Due to the so-called parametric resonance  or the so-called coexistence of periodic and antiperiodic eigenvalues , periodic and antiperiodic half-eigenvalues are, in general, not differentiable in weights .
If is a half-eigenvalue of (1.1) corresponding to weights and satisfying the boundary condition (1.2), (1.3), (1.4), or (1.5), then is also a half-eigenvalue of (1.1) corresponding to weights and satisfying the same boundary condition. So we need only consider nonnegative half-eigenvalues of (1.1). Some preliminary results are given in Section 2. However, Sections 3, 4, and 5 are devoted to , , and , respectively.
2. Preliminary Results
Given , denote by the unique solution of the initial value problem The functions and are the so-called -cosine and -sine, because they possess some properties similar to those of cosine and sine functions, such as(i)both and are -periodic, where (ii) if and only if , , and if and only if , ;(iii)one has
Given , , let In the -polar coordinates the scalar equation is transformed into the following equations for and For any , denote by , , the unique solution of (2.7) + (2.8) satisfying and . Let As is independent of and is -periodic in , we have for all , , and .
An important property of the argument solution is the quasimonotonicity as in the following lemma.
Denote by the space of continuous functions from to . Some dependence results of solutions and on are collected in the following theorem.
Theorem 2.2 (see ). (i) The functional
is continuous. Here denotes the weak topology in .
(ii) The functional is continuously differentiable. The derivatives of at , at , and at (in the Fréchet sense), denoted, respectively, by , , and , are where is a solution of (2.6).
To characterize the spectrum structure of (1.1) via shooting method, we need careful analysis on the associated argument function . Given ,,,, write if and holds for in a subset of of positive measure. Write if and . Write if and both and hold for in a common subset of of positive measure. Denote The following asymptotical property of in plays a crucial role in the characterization of half-eigenvalues.
Lemma 2.3. Given , , one has uniformly in .
Proof. The proof is a slight extension of that in [1, Lemma ]. We write it out in detail for the convenience of the readers.
Besides (2.4), we introduce another transformation when . However, (1.1) becomes In the -polar coordinates of the -plane the equation for the new argument is Denote by the solution of (2.23) satisfying the initial condition . By (2.20), the argument functions of (1.1), and are related by an orientation-preserving homeomorphism , which fixes the points , in the following relation
If lemma is not true, then there exist and such that and is bounded from above. Combining (2.11), there exists some such that By (2.11) again, we have By (2.10), we may assume Hence,
Denote for simplicity. By the conjugacy (2.24) and the estimate (2.28), we have
Denote and define for any . By the Banach-Alaoglu theorem [15, pages 229-230], the unit ball is sequentially compact in by considering as the dual space of the (separable) Banach space . The order interval is a closed subset of . Hence, is also sequentially compact in . Consequently, passing to a subsequence, we may assume From (2.23), satisfies For any , we have Let , by (2.27) and (2.29), the left-hand side tends to . Since we can use (2.31) to deduce from the above equality for all . Thus As and the weak* limits satisfy a.e. , this equality implies that Therefore, which contradicts the assumption .
Remark 2.4. Let and , a.e. in (1.1). Then the equation for the corresponding argument function is (cf. (2.7). The solution takes a constant length of time, say , to start from and to reach at for any . Consequently, is bounded for and (2.19) does not hold in this case.
Notice that in the potential case , the associated argument function is strictly increasing in . However, as we are considering sign-changing weights , the monotonicity of in does not hold any more. We will develop some quasi-monotonicity of in by employing the Fréchet derivatives of and the boundary conditions, see Lemmas 3.3, 4.1, and 5.1.
3. Periodic and Antiperiodic Spectrum
3.1. Structure of Periodic and Antiperiodic Half-Eigenvalues
Lemma 3.1. Given , the functionals are continuous.
Given , , we write for simplicity if there is no ambiguity. By Lemma 3.1, and are continuous in . By Lemma 2.3, one has Moreover, by setting in (2.7), we know that Now we are considering the following two sequences of equations for
Lemma 3.2. Given , .
(i)For any specified, (3.7) and (3.8) always have solutions in .(ii)All solutions of (3.7) and (3.8) are periodic half-eigenvalues or antiperiodic half-eigenvalues of (1.1) if is even or odd, respectively, while the corresponding half-eigenfunctions have precisely zeroes in the interval .
Proof. (i) The solvability of (3.7) and (3.8) follows immediately from (3.5), (3.6) and the continuity of and in .
(ii) Suppose satisfies (3.7) or (3.8). Then there exists such that By (2.14), the latter equation is Geometrically, (3.9) and (3.11) imply that in -polar coordinates (2.5), the solution of (1.1) starting at the point arrives at after one period. Thus is a periodic or an antiperiodic half-eigenvalue of (1.1) if is even or odd, respectively.
Denote by the half-eigenfunction corresponding to . Then if and only if Notice that , (see (3.9), and is quasimonotone in (see (2.11). Consequently, if for some , then (3.12) holds only for at . If , then (3.12) holds only for at . In both cases has precisely zeroes in .
Lemma 3.3. Given , .
(i)If for some and , then there exists such that (ii)If for some and , then there exists such that
Proof. We only prove (i) and the proof of (ii) is analogous.
Assume that . By Lemma 3.2, is a periodic or an antiperiodic half-eigenvalue of (1.1). Then there exists some such that and the corresponding half-eigenfunction satisfies and the boundary condition (1.2) or (1.3). Applying (2.15) and (2.16), we have Multiplying (3.16) by and integrating over , one has and, therefore, Then there exists such that Consequently, we have
Definition 3.4. Given , , define
Corollary 3.5. Given , , one has
Till now, we still do not know the ordering between and with . This will be partially settled by associating (1.1) with the rotation number function defined by where is the th iteration of , namely, It can be proved that the rotation number function in (3.29) is well defined and is independent of (cf. [16, Theorem ]).
Lemma 3.6. Given , , one has
(i) and ;(ii);(iii);(iv);(v);(vi).
Proof. (i) By (2.11), one has for any . Thus since defined by (3.29) is independent of . If , the equation for the argument function associated with (1.1) becomes , which has equilibria , . Thus for all and consequently .
(ii) Given arbitrarily large, it follows from (3.5) that there exists such that Thus for any and any . Consequently, for any .
(iii) Given , it follows from (3.26) that there exists such that By the definition of (see (3.1), this implies that Now similar arguments as in (ii) show that .
(iv) The proof is analogous to that of (iii).
(v) Given , it follows from (3.26) that Then there exists such that , and therefore .
(vi) By (3.25), if , then there exists such that . Hence .
Corollary 3.7. Given , , one has and the ordering
Proof. The characterization (3.36) follows from Lemma 3.6(iii) and (v), while (3.37) follows from Lemma 3.6(iv), (v), and (vi).
To prove (3.38), by (3.27) and (3.28), we need only prove for any . Assume on the contrary that for some . Then Now Lemma 3.6(v) shows that , a contradiction.
Corollary 3.8. Given , , one has
In summary, we can now partially characterize the set of periodic half-eigenvalues and the set of antiperiodic half-eigenvalues of (1.1). Denote and . For any , denote by the set of nonnegative periodic or antiperiodic half-eigenvalues for which the corresponding half-eigenfunctions have precisely zeroes in the interval . Then
Theorem 3.9. Given , .
(i)For periodic half-eigenvalues, one has Moreover, and satisfy (3.27), (3.28), and (3.38).(ii)For antiperiodic half-eigenvalues, one has Moreover, and satisfy (3.27) and (3.28).
Proof. By Lemma 3.2 and the definitions of and (see (3.22)–(3.25), we need only prove If , , then is a periodic half-eigenvalue of (1.1) and the corresponding half-eigenfunction has precisely zeroes . Consequently, there exists such that Hence . Combining (3.36), and the fact we obtain (3.44).
The following theorem gives the necessary and sufficient condition for .
Theorem 3.10. Given , , then
Proof. Assume that . By Lemma 3.2, the half-eigenfunction corresponding to is nowhere vanishing and satisfies the periodic boundary condition (1.2). If for any , then
Similarly, if for any , then .
On the other hand, assume that . Since is the periodic half-eigenfunction corresponding to , by (2.4), we have . Hence in the -polar coordinates (2.5) and the argument solution . Then it follows from (2.15) and (2.16) that Thus there exists such that