Positive subharmonic solutions to nonlinear ODEs with indefinite weight

Positive subharmonic solutions
to nonlinear ODEs with indefinite weight

Abstract

We prove that the superlinear indefinite equation

where and is a -periodic sign-changing function satisfying the (sharp) mean value condition , has positive subharmonic solutions of order for any large integer , thus providing a further contribution to a problem raised by G. J. Butler in its pioneering paper [16]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré-Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it). 1 2

1 Introduction

In this paper, we study the existence of positive subharmonic solutions for nonlinear second order ODEs with indefinite weight. To describe our results, throughout the introduction we focus our attention to the superlinear indefinite equation

(1.1)

with a sign-changing -periodic function and , which has been indeed the main motivation for our investigation.

Boundary value problems associated with sign-indefinite equations are quite popular in the qualitative theory of nonlinear ODEs. The existence of oscillatory periodic solutions to superlinear indefinite equations like

(1.2)

where with , was first investigated by Butler in the pioneering paper [16]. Later on, along this line of research, several contributions followed (cf. [17, 38, 39, 42]) and it is nowadays well known that equation (1.2) possesses infinitely many periodic solutions (both harmonic and subharmonic) with a large number of zeros in the intervals where the weight function is positive, as well as globally defined solutions with chaotic-like oscillatory behavior.

On the other hand, the existence of positive periodic solutions to equations like (1.1), even if already raised by Butler as an open problem in [16, p. 477], was investigated only more recently.

In this regard, a first crucial observation is that a mean value condition on turns out to be necessary for the existence of positive -periodic solutions (with an integer number); indeed, dividing equation (1.1) by and integrating on , one readily obtains

so that (recalling that is -periodic)

(1.3)

This fact was first observed by Bandle, Pozio and Tesei in [2], showing that the condition is actually necessary and sufficient for the existence of a positive solution to the Neumann problem associated with the elliptic partial differential equation

in the sublinear case (notice, indeed, that the computation leading to (1.3) is valid for any , both for periodic and Neumann boundary conditions, and possibly in a PDE setting). A similar result was then proved in the superlinear case in [1, 4], using arguments from critical point theory.

To the best of our knowledge, periodic boundary conditions were explicitly taken into account only in the very recent paper [22]. Therein, a topological approach based on Mawhin’s coincidence degree was introduced to prove that the mean value condition (1.3) guarantees the existence of a positive -periodic solution for a large class of indefinite equations including (1.1). In such a way, a first affirmative answer to Butler’s question can be given.

On one hand, this result seems to be optimal (in the sense that no more than one -periodic solution can be expected for a general weight function with negative mean value); on the other hand, however, it is known that positive solutions to (1.1) can exhibit complex behavior for special choices of the weight function . More precisely, it was shown in [3, 24] (on the lines of previous results dealing with the Dirichlet and Neumann problems [5, 6, 23, 27, 30]) that, whenever has large negative part (that is, with ), equation (1.1) has infinitely many positive subharmonic solutions, as well as globally defined positive solutions with chaotic-like multibumb behavior.

It appears therefore a quite natural question if the sharp mean value condition (1.3) - besides implying the existence of a positive -periodic solution to (1.1) - also guarantees the existence of positive subharmonic solutions. Quite unexpectedly, as a corollary of our main results, we are able to show that the answer is always affirmative, thus providing a further contribution to Butler’s problem.

Theorem 1.1.

Assume that is a sign-changing continuous and -periodic function, having a finite number of zeros in and satisfying the mean value condition (1.3). Then, equation (1.1) has a positive -periodic solution, as well as positive subharmonic solutions of order , for any large integer number .

Actually, the assumptions on the weight function can be considerably weakened and the conclusion about the number of subharmonic solutions obtained can be made much more precise. We refer to Section 3 for more general statements.

Let us emphasize that investigating the existence of subharmonic solutions for time-periodic ODEs is often a quite delicate issue, the more difficult point being the proof of the minimality of the period. In Theorem 1.1, -periodic solutions are found (for large enough) oscillating around a -periodic solution and a precise information on the number of zeros of is the key point in showing that is the minimal period of . This approach, based on the celebrated Poincaré-Birkhoff fixed point theorem, was introduced (and then applied to Ambrosetti-Prodi type periodic problems) in the paper [12], to which we also refer for a quite complete bibliography about the theme of subharmonic solutions. It is worth noticing, however, that the application to equation (1.1) of the method described in [12] is not straightforward. First, due to the superlinear character of the nonlinearity, we cannot guarantee (as needed for the application of dynamical systems techniques) the global continuability of solutions to (1.1) (see [15]) and some careful a-priori bounds have to be performed. Second, due to the indefinite character of the equation, it seems impossible to perform explicit estimates on the solutions in order to prove the needed twist-condition of the Poincaré-Birkhoff theorem. To overcome this difficulty, we first use an idea from [10] to develop an abstract variant of the main result in [12], replacing an explicit estimate on the positive -periodic solution with an information about its Morse index. Using a clever trick by Brown and Hess (cf. [14]), such an information is then easily achieved. We emphasize this simple property here, since it is the crucial point for our arguments: any positive -periodic solution of (1.1) has non-zero Morse index.

Let us finally recall that variational methods can be an alternative tool for the study of subharmonic solutions. In this case, information about the minimality of the period can be often achieved with careful level estimates (see, among others, [25, 41]). Maybe this technique can be successfully applied also to the superlinear indefinite equation (1.1); however, it has to be noticed that usually results obtained via a symplectic approach (namely, using the Poincaré-Birkhoff theorem) give sharper information (see [10, 12]).

The plan of the paper is the following. In Section 2 we present, on the lines of [10, 12], an auxiliary result ensuring, for a quite broad class of nonlinearities, the existence of subharmonic solutions oscillating around a -periodic solution with non-zero Morse index. In Section 3 we state our main results, dealing with equations of the type with satisfying (1.3) and defined on a (possibly bounded) interval of the type ; roughly speaking, we have that the existence of positive subharmonic solutions (oscillating around a positive -periodic solution) is always guaranteed whenever is superlinear at zero and strictly convex, with large enough near . Applications are given to equations superlinear at infinity (thus generalizing Theorem 1.1), to equations with a singularity as well as to parameter-dependent equations. In Section 4 we give the proof of these results; in more detail, we first prove (using a degree approach similarly as in [22], together with the trick in [14]) the existence of a positive -periodic solution with non-zero Morse index and we then apply the results of Section 2 to obtain the desired positive subharmonic solutions around it. Section 5 is devoted to some conclusive comments about our investigation.

2 Morse index, Poincaré-Birkhoff theorem, subharmonics

In this section, we present our auxiliary result for the search of subharmonic solutions to scalar second order ODEs of the type

(2.1)

where is a function -periodic in the first variable (for some ). Motivated by the applications to equations like (1.1) with , we set up our result in a Carathéodory setting. More precisely, we assume that the function is measurable in the -variable, continuously differentiable in the -variable and satisfies the following condition: for any , there exists such that for a.e  and for every with . Of course, in view of this assumption, solutions to (2.1) will be meant in the generalized sense, i.e. -functions satisfying equation (2.1) for a.e. .

Throughout the paper, we say that is a subharmonic solution of order of (2.1) (with an integer number) if is a -periodic solution of (2.1) which is not -periodic for any integer , that is, is the minimal period of in the class of the integer multiples of . This is the most natural definition of subharmonic solutions to equations like (2.1), when just the -periodicity of is assumed; on the lines of [36], if additional conditions on this time dependence are imposed, further information on the minimality of the period can be given (see Remark 3.1). Let us also notice that if is a subharmonic solution of order of (2.1), the functions , for , are subharmonic solutions of order of (2.1) too; these solutions, though distinct, have to be considered equivalent from the point of view of the counting of subharmonics. Accordingly, given subharmonic solutions of order of (2.1), we say that and are not in the same periodicity class if for any integer .

Finally, we introduce the following notation. For any , we denote by the principal eigenvalue of the linear problem

(2.2)

with -periodic boundary conditions. As well known (see, for instance, [18, ch. 8, Theorem 2.1] and [32, Theorem 2.1]) exists and is the unique real number such that the linear equation (2.2) admits one-signed -periodic solutions. Recalling that, by definition, the Morse index of the linear equation is the number of (strictly) negative -periodic eigenvalues of (2.2), we immediately see that if and only if .

We are now in position to state the following result.

Proposition 2.1.

Let be as above and assume that the global continuability for the solutions to (2.1) is guaranteed. Moreover, suppose that:

  • there exists a -periodic solution of (2.1) satisfying

    (2.3)
  • there exists a -periodic function satisfying

    (2.4)

    and

Then there exists such that for any integer there exists an integer such that, for any integer relatively prime with and such that , equation (2.1) has two subharmonic solutions , of order (not belonging to the same periodicity class), such that, for , has exactly zeros in the interval and

(2.5)

Incidentally, we observe that Proposition 2.1 in particular ensures that equation (2.1) has two subharmonic solutions of order (not belonging to the same periodicity class) for any large integer (just, take in the above statement).

Remark 2.1.

Let us recall that a -periodic function satisfying (2.4) is a lower solution for the -periodic problem associated with (2.1) (weaker notions of lower/upper solutions could be introduced in the Carathéodory setting, see [19]). Clearly, if is a -periodic solution of (2.1), then is a -periodic lower solution; in this case, due to the uniqueness for the Cauchy problems, (2.5) implies for any .

Proposition 2.1 is a variant of [12, Theorem 2.2]3. However, some care is needed in comparing the two results. First, [12, Theorem 2.2] is stated for smooth; the generalization to the Carathéodory setting in this case is not completely straightforward. Second, the assumption corresponding to in [12, Theorem 2.2] reads as

(2.6)

The possibility of replacing this explicit condition with the abstract assumption has been discussed in [10, Theorem 2.1]4. Actually, in that paper the assumption is used, where is the Moser rotation number (see [37]) of the linear equation . However, it is very well known in the theory of the Hill’s equation (see, for instance, [26, Proposition 2.1]) that if and only if (that is, if and only if the equation is not disconjugate).

Related results, yielding the existence and multiplicity of harmonic (i.e. -periodic) solutions according to the interaction of the nonlinearity with (non-principal) eigenvalues, can be found in [33, 34, 45].

The complete proof of Proposition 2.1, based on the Poincaré-Birkhoff fixed point theorem, is quite long. For this reason, we provide just a sketch of it, referring to previous papers (in particular, to [10, 12]) for the most standard steps.

Sketch of the proof of Proposition 2.1.

We define the truncated function

and we set

Then, we consider the equation

(2.7)

The following fact is easily proved, using maximum principle-type arguments (see [12, p. 95]).

  • If is a sign-changing -period solutions of (2.7) (for some integer ) then for any .

Now, we observe that both uniqueness and global continuability for the solutions to the Cauchy problems associated with (2.7) are ensured; moreover, since , the constant function is a solution of (2.7). We can therefore transform (2.7) into an equivalent first order system in , passing to clockwise polar coordinates , .

We claim that:

  • there exists an integer such that, for any integer , there exist an integer and such that any solution with satisfies ;

  • for any integer there exists such that any solution with satisfies .

From the above facts, it follows that the Poincaré-Birkhoff theorem (in the generalized version for non-invariant annuli, see [20, 40]) can be applied, giving, for any and any , the existence of two -periodic solutions () to equation (2.7) having exactly zeros on . Using , it is then immediate to see that is a -periodic solution of (2.1), satisfying (2.5) and such that has exactly zeros in the interval . The fact that, for and relatively prime, is a subharmonic solution of order is also easily verified, while and are not in the same periodicity class due to a standard corollary of the Poincaré-Birkhoff theorem for the iterates of a map. For more details on the application of this method, we refer to [7, 34, 45].

To conclude the proof, we then have to verify the claims and . As for the first one, it can be proved exactly as in [10, Proof of Theorem 2.1] (see also [10, Remark 2.2]). The fact that we are working in a Carathéodory setting does not cause here serious difficulties, since the dominated convergence theorem easily yields in for , and this is enough to use continuous dependence arguments as in [10]. On the other hand, the proof of is more delicate (especially when dealing with Carathéodory functions) and we prefer to give some more details. We are going to use a trick based on modified polar coordinates, introduced in [21] (see also [7]). More precisely, for any , we write

for further convenience we also compute

(2.8)

The angular coordinates and are in general different. However, the angular width of any quadrant of the plane is also if measured using the angle . As a consequence, recalling (2.8) we can write the formula

(2.9)

valid whenever are such that , (or viceversa) and belongs to the same quadrant for . We stress that (2.9) holds for any .

We can now give the proof. Preliminarily, we observe that, using the Carathéodory condition together with the definition of , we can obtain

(2.10)

where . We now fix an integer and take so small that

(2.11)

In view of the global continuability of the solutions, there exists large enough such that implies that

(2.12)

At this point, assume by contradiction that for a solution with . Then it is not difficult to see that there exist with and such that either and belongs to the third quadrant for or and belongs to the fourth quadrant for . As a consequence, on one hand (2.9) holds true; on the other hand, since for we can use (2.10) so as to obtain

Combining these two facts, we find

Using (2.11) and (2.12), we finally find , a contradiction. ∎

Remark 2.2.

It is worth noticing that, although related, conditions (2.3) and (2.6) are not equivalent. More precisely, given a general weight ,

(2.13)

as an easy consequence of the variational characterization of the principal eigenvalue (see [32, Theorem 4.2])

(2.14)

(just, take in the above formula; denotes the Sobolev space of -periodic -functions). Of course, (2.14) also implies that

but there exist (sign-changing) weights such that and , showing that the converse of (2.13) is not true. Explicit examples can be constructed, for instance, as in [7, Remark 3.5]. An even more interesting example will be given later (see Remark 4.3), showing that the possibility of replacing (2.6) with the weaker assumption (2.3) is crucial for our purposes.

3 Statement of the main results

In this section, we state our main results, dealing with positive solutions to equations of the type

(3.1)

We always assume that , with a right neighborhood of , and satisfies the following conditions:

Hence, is superlinear at zero and strictly convex. Incidentally, notice that from , and it follows that is strictly increasing; in particular

implying that the only constant solution to (3.1) is the trivial one, i.e. .

As for the weight function, we suppose that is a -periodic and locally integrable function satisfying the following condition:

  • there exist intervals , closed and pairwise disjoint in the quotient space , such that

Moreover, motivated by the discussion in the introduction, we suppose that the mean value condition

holds true.

Of course, by a solution to equation (3.1) we mean a function , with for any and solving (3.1) for a.e. . Notice that, since , any solution is a non-negative function; we say that a solution is positive if for any .

As a first result, we provide a statement generalizing the one given in the introduction for equation (1.1). More precisely, we show that the existence of positive subharmonic solutions (in the sense clarified at the beginning of Section 2, see also Remark 3.1) to (3.1) is ensured for any function which satisfies , , for and which is superlinear at infinity. Needless to say, this is the case for the model nonlinearity with .

Theorem 3.1.

Let be a -periodic locally integrable function satisfying and . Let satisfy , and , as well as

Then, there exists a positive -periodic solution of (3.1); moreover, there exists such that for any integer there exists an integer such that, for any integer relatively prime with and such that , equation (3.1) has two positive subharmonic solutions () of order (not belonging to the same periodicity class), such that has exactly zeros in the interval .

In our second result, we deal with the case , with finite, assuming a singular behavior for when .

Theorem 3.2.

Let be a -periodic locally integrable function satisfying and . Let (for some finite) satisfy , and , as well as

Then, there exists a positive -periodic solution of (3.1); moreover, there exists such that for any integer there exists an integer such that, for any integer relatively prime with and such that , equation (3.1) has two positive subharmonic solutions () of order (not belonging to the same periodicity class), such that has exactly zeros in the interval .

We mention that singular equations with indefinite weight were considered in [13, 43, 44]. More precisely, these papers deal with equations like , where . Our setting is different and Theorem 3.2 applies for instance to the equation

(3.2)

for and . To the best of our knowledge, even the mere existence of a positive -periodic solution to (3.2) is a fact which has never been noticed.

Finally, we give a purely local result. More precisely, we just assume , and in a bounded interval , with finite; on the other hand, we deal with an equation depending on a real parameter and we manage to obtain the result by varying it.

Theorem 3.3.

Let be a -periodic locally integrable function satisfying and . Let (for some ) satisfy , and . Then, there exists such that for any there exists a positive -periodic solution of the parameter-dependent equation

(3.3)

satisfying . Moreover, there exists such that for any integer there exists an integer such that, for any integer relatively prime with and such that , equation (3.3) has two positive subharmonic solutions () of order (not belonging to the same periodicity class), with and such that has exactly zeros in the interval .

Of course, in the above statement may be defined also for , but no assumptions on its behavior are made. For instance, we can apply Theorem 3.3 to parameter-dependent equations like

(3.4)

with , obtaining the following: for any small enough, there exists such that for any equation (3.4) has a positive -periodic solution as well as positive subharmonic solutions of any large order; all these periodic solutions, moreover, have maximum less than . In such a way, we can complement - in the direction of proving the existence of positive subharmonics - recent results dealing with positive harmonic solutions in the asymptotically linear case (see [22, Corollary 3.7]) and in the sublinear one (see [9, 11]). It is worth noticing that, according to [9, Theorem 4.3], in this latter case a further positive -periodic solution (having maximum greater than ) to (3.4) appears. This second solution is expected to have typically zero Morse index, and no positive subharmonic solutions oscillating around it.

Remark 3.1.

We notice that all the positive subharmonic solutions of order found in this section actually have minimal period if we further assume that is the mimimal period of . This is easily seen, by writing (3.1) in the equivalent form .

4 Proof of the main results

In this section we provide the proof of the results presented in Section 3. Actually, we are going to give and prove a further statement, which looks slightly more technical but has the advantage of unifying all the situations considered in Theorem 3.1, Theorem 3.2 and Theorem 3.3.

Henceforth, we deal with the equation

(4.1)

where , for some finite, and satisfies:

Accordingly, by a solution to equation (4.1) we mean a function , with for any and solving (4.1) in the Carathéodory sense; a solution is said to be positive if for any .

In this setting, the following result can be given.

Theorem 4.1.

Let be a -periodic locally integrable function satisfying and . Then there exist two real constants and such that, for any and for any satisfying , , and

the following holds true: there exists a positive -periodic solution of (4.1) with ; moreover, there exists such that for any integer there exists an integer such that, for any integer relatively prime with and such that , equation (4.1) has two positive subharmonic solutions () of order (not belonging to the same periodicity class), with and such that has exactly zeros in the interval .

It is clear that all the theorems in Section 3 follows from Theorem 4.1. More precisely:

  • in order to obtain Theorem 3.1, we take and large enough: then , , correspond to , , , while comes from ;

  • in order to obtain Theorem 3.2, we take and with small enough: then , , correspond to , , , while comes from ;

  • in order to obtain Theorem 3.3, we take : then , , correspond to , , (independently on ), while is certainly satisfied for large enough.

Now we are going to prove Theorem 4.1. Wishing to apply Proposition 2.1, we proceed as follows. First, we define an extension of for , having linear growth at infinity and thus ensuring the global continuability of the (positive) solutions of ; in doing this, we need to check that any periodic solution of this modified equation is actually smaller than , thus solving the original equation . This is the most technical part of the proof (producing the constants appearing in assumption ) and is developed in Section 4.1. Second, in Section 4.2, using a degree theoretic approach (and taking advantage of the a-priori bound given in the previous section), we prove the existence of a positive -periodic solution of . Third, in Section 4.3 we provide the desired Morse index information. The easy conclusion of the proof is finally given in Section 4.4 (we just notice here that the existence of a lower solution is straightforward, since we can take ).

It is worth noticing that condition (requiring in particular that ) will be essential only in Section 4.3. For this reason, we carry out the discussion in Section 4.1 and Section 4.2 (containing results which may have some independent interests) in a slightly more general setting than the one in Theorem 4.1.

4.1 The a-priori bound

In this section, we prove an a-priori bound valid for periodic solutions of (4.1) as well as for periodic solutions of a related equation (see (4.3) below). This will be useful both for the application of Proposition 2.1 (requiring a globally defined nonlinearity) and for the degree approach discussed in the Section 4.2.

As already anticipated, in this section we do not assume all the conditions on required in Theorem 4.1. More precisely, we are going to deal with continuously differentiable functions satisfying and the following condition

Moreover, instead of we just suppose that is a convex function, namely

For further convenience, we observe that from the above conditions it follows that is non-decreasing and such that is a non-decreasing map in . Indeed, let and let be such that . Then, we have

and thus the map is non-decreasing in . Consequently, we immediately obtain that is a non-decreasing map in , since it is the product of two non-decreasing positive maps in .

We also recall that a function is convex if and only if lies above all of its tangents, hence

Using , and the above inequality (with and ), we immediately obtain

With this in mind, we introduce the extension defined as

(4.2)

It is easily seen that the map is continuously differentiable, convex, non-decreasing and such that for all . Then, arguing as above, we immediately obtain that the map is non-decreasing as well.

We are now in a position to state our technical result (whose proof benefits from some arguments developed in [28, p. 421] and in [8, Lemma 4.1]) giving a-priori bounds for periodic solutions of the equation

(4.3)

where and denotes the indicator function of the set . Incidentally, notice that neither the mean value condition nor the superlinearity assumption at zero are required in the statement below.

Lemma 4.1.

Let be a -periodic locally integrable function satisfying . Then there exist two real constants and such that, for every , for every convex function satisfying , and , for every and for every integer , any -periodic solution to (4.3) satisfies .

Proof.

According to condition , we can find points

such that

We then fix such that

so that the constant

is well defined and positive. Next, we define the constants

Notice that , since , for all . We stress that and depend only on the weight function .

Let us consider an arbitrary and an arbitrary convex function satisfying , and . By contradiction, we suppose that is a -periodic solution of (4.3) such that

Setting (for and ), the convexity of on ensures that the maximum is attained in some . Accordingly, we can suppose that there is an index and such that

Up to a relabeling of the intervals , we can also suppose (notice that the constants and do not change since is -periodic). From now on, we therefore assume that