Existence and Multiplicity of Periodic Solutions to Indefinite Singular Equations Having a Non-monotone Term with Two Singularities

1 Introduction and Main Results

In a recent series of papers, see [12, 13, 14, 9, 10, 19, 20], Zanolin and his collaborators have studied the existence and multiplicity of solutions to Neumann and periodic boundary value problems for second order differential equations of the form

Here, h∈L⁢(ℝ/T⁢ℤ), σ>0 is a parameter, and g:(A,B)→(0,+∞) is a continuous function, with -∞≤A<B≤+∞. An elementary observation is that any solution to (1.1) satisfying the above-mentioned boundary conditions satisfies

and, consequently, the function h (sometimes called the weight function) has to change its sign. These types of equations with sign-changing weight function are called indefinite equations. The terminology “indefinite” was probably introduced by Hess and Kato in [27] under the framework of linear eigenvalue problems, meaning that h is not sign-constant. In addition to the above-mentioned papers, it is worth mentioning here that the qualitative study of the solutions to the indefinite equation (1.1) has also been treated by many other authors in both ODEs and PDEs settings, see, e.g., [6, 7, 11, 16, 4, 23, 8, 18, 17]. However, most of them considered equation (1.1) without singularities (i.e., A=-∞, B=+∞).

As far as singular nonlinearities are concerned (that is, either -∞<A or B<+∞), both periodic and Neumann problems become slightly more tricky. Indeed, the indefinite character of such problems increases the difficulty of finding a suitable region where a desired solution can be localized even for the non-singular case. However, in spite of this fact, in that case the nonlinearity g can be extended on the whole axis in such a way that a point of equilibrium exists. This procedure considerably simplifies the way of finding a region containing the above-mentioned solution, where the degree of a certain fixed point operator is not equal to zero. Moreover, in contrast to the singular case, the set of solutions can be included in an open bounded subset of continuous functions (some a priori bounds can be found). Thus, it seems to be more natural to use the adaptation of the newly developed method of the topological degree in the non-singular case than in the singular one. That is why, contrarily to the non-singular case, the existence and multiplicity of solutions to (1.1) with periodic or Neumann boundary conditions still remain investigated insufficiently, and only a few papers nowadays can be found that deal with the problem in question. We can recommend the reader some recently published papers concerning the indefinite equations with singularities for better understanding the actual state, see, e.g., [15, 31, 24, 25, 22] for the case with one singularity, and [14, 26], where the case of two singularities is investigated.

In this paper we investigate equation (1.1) with two singularities, i.e., we consider the case when A,B∈ℝ. The results established in the paper have a form of relation between multiplicities of zeros of the weight function h and orders of singularities of the nonlinear term. Therefore, as a mathematical model to illustrate the conditions obtained, we introduce the generalized Emden–Fowler equation with two negative exponents, that is,

where λ>0, μ>0, and σ>0 is a parameter.

In addition to the latter model, equations with two singularities appear frequently in models describing the problems in celestial mechanics. For instance, the dynamics of a charged particle moving on a line between two fixed charges can be mentioned as one of the simplest models in physics. Supposing that the distance between the fixed charges is equal to one, this model can be included into the following family of equations:

where λ>0, μ>0, and h1, h2 are related to the interaction forces between the charges with the free-particle. We point out that the latter equation can be viewed as an equation with two indefinite singularities. We refer to [1, 5, 3, 21, 30] for a review of works dealing with this type of models (most of them in larger dimensions).

An equation modeling the Kepler problem on the sphere is considered in the paper as a physical application. By the Kepler problem we understand a motion of a particle moving on a sphere of radius one subjected to the influence of an electric field created by a charge of a time-depending magnitude fixed in the north pole. This problem can be modeled by

where c corresponds to the angular momentum of the free-particle which is moving on the sphere, h is a kind of force associated with an electric field caused by charge distribution, σ is a positive parameter, and u is a real variable connected with the angular position of the particle (see, e.g., [2] and references therein). In the case when c=0, we obtain the equation

which is a particular case of (1.1), corresponding to the choice A=0, B=π, g⁢(x)=sin-2⁡x.

Up to our knowledge, the periodic problem for equation (1.4) has been considered only under very restrictive assumptions on the weight function (h is a piecewise constant function with two pieces of different signs, see [26]). However, a general setting may cover more realistic situations (e.g., the case when h is a continuous function).

It is worth mentioning here that indefinite equations with one singularity were considered with a monotone function g in the above-mentioned papers. Obviously, this is not the case anymore for equations with two singularities. The appearing of another singularity plays an important role in the dynamics of solutions. Roughly speaking, we could name two aspects that substantially change the attitude to the problem:

1. the nonlinear term is not monotone,

2. the right-hand side possesses two singularities.

Now we describe our setting in more detail. Throughout the paper, we will assume that -∞<A<B<+∞, the function g:(A,B)→(0,+∞) is continuously differentiable, and there exists P∈(A,B) such that

By h¯ we understand the mean value of h, i.e., h¯=1T⁢∫0Th⁢(s)⁢𝑑s. Moreover, we set

where h+⁢(s)=max⁡{0,h⁢(s)} and h-⁢(s)=max⁡{0,-h⁢(s)}. Obviously, from (1.2) it follows that H+⁢H-≠0 is a necessary condition for the solvability of a periodic problem for (1.1).

From (1.6) it follows, in particular, that g has singularities at the points x=A and x=B. We are interested in solutions that avoid these singularities. More precisely, we are looking for T-periodic functions u:ℝ→(A,B), which are continuous together with their first derivative on [0,T] and satisfy (1.1) almost everywhere on [0,T]. The main result of this paper is the following.

For the particular cases introduced above, Theorem 1.1 can be reformulated as follows.

Explicit corollaries of the above-mentioned results are the following statements.

The paper is structured as follows. In Section 2, by passing to an auxiliary equation, we introduce a functional analytical setting to apply the results established in [25] in the classical framework of the Leray–Schauder degree. Then we prove the existence of at least one T-periodic solution, provided that σ is sufficiently small. In Section 3 we prove the nonexistence of a T-periodic solution to equation (1.1) when the parameter σ is large enough. In Section 4 we consider a family of operators Mσ whose fixed points correspond to T-periodic solutions to (1.1). Then some a priori estimates are established in order to prove that the above-mentioned homotopy is admissible on some adequate open set of continuous functions. Consequently, the degree of the operator Mσ is equal to zero in the above-mentioned open set, according to the results of Section 3. Finally, we prove the multiplicity result using the excision property of the degree. The paper ends with Section 5, where we discuss the applicability of our results, we give an estimation of the parameter σ*>0, and in the end some open problems are introduced.

Before passing to the mathematical details of the paper, we observe that it is sufficient to assume that h¯<0 in the proofs of the main results. Indeed, if h¯>0, then the transformation

leads to the equation

where the mean value of the function h~ is negative.

Furthermore, without loss of generality, we can assume that g⁢(P)<1. Indeed, if this is not the case, we can pass to the equation

where h~⁢(t)=(g⁢(P)+1)⁢h⁢(t) for a.e. t∈[0,T] and g~⁢(x)=g⁢(x)/(g⁢(P)+1) for x∈(A,B).

Finally, it is worth noting that the change of variable v=u-A allows us to assume, without loss of generality, that A=0 and B>0. This will be assumed in the rest of the paper.

2 Existence of a Solution and Topological Degree

Let us momentarily consider the truncated nonlinearity g0:(0,+∞)→ℝ, defined by

and the modified equation

We are about to rewrite the problem of finding T-periodic solutions to (2.1) as a fixed-point problem for an operator equation. For this purpose we need a suitable functional framework that is described briefly in what follows.

Let C denote the Banach space of continuous functions x:[0,T]→ℝ endowed with a standard norm ∥x∥∞=max⁡{|x⁢(t)|:t∈[0,T]}. We consider the closed subspace

The open ball centered at zero and with radius r>0 is denoted by B⁢(0,r). We denote by P,Q:C→C the continuous projectors

Furthermore, let K:C→C be a continuous linear operator, given by

All T-periodic solutions to (2.1) belong to the open set

Thus, for each σ>0, the operator M~σ:Λ→CT is defined by

where N~σ is the Nemitsky operator associated with (2.1), i.e., N~σ⁢[u]⁢(t)=σ⁢h⁢(t)⁢g⁢(u⁢(t)). The operator M~σ is continuous and maps bounded subsets of Λ, whose closure is contained in Λ, into relatively compact sets of CT (it is completely continuous on Λ). Now the periodic boundary value problem for (2.1) becomes equivalent to the fixed-point problem for the operator equation

Our starting point will be the computation of the degree of the operator I-M~σ on a certain bounded open set (here I denotes the identity map defined on the space CT).

A result on the existence of a periodic solution to (2.1) was recently established in [25, Theorem 1.1] using the coincidence degree. More details on this theory can be found in [29] (see also [28] for an extension for p-Laplacian operators). Using the techniques applied in [25] one can obtain the following assertion, which will be useful in the proof of our main result.

Now we establish two auxiliary lemmas dealing with a priori estimates of solutions to (2.1).

A key step towards the proof of the existence of a solution to (1.1) is the result formulated below. Under the assumption that our parameter σ is small enough, we have proven (see Lemma 2.3) that any T-periodic solution to (2.1) lies in the unmodified region of the nonlinearity g (i.e., (2.6) holds). Hence, in view of (2.3) and an excision property of the degree, the following assertion immediately follows from Proposition 2.1.

To prove that the assumptions of Theorem 1.1 guarantee the existence of a T-periodic solution to (1.1), it remains only to observe that the conditions of Theorem 1.1 imply, in particular, the conditions of Proposition 2.1. Then, by applying Lemma 2.4, the existence of such a solution becomes clear. More precisely, the following assertion holds.

3 Nonexistence of Solutions for a Large Parameter

Now we turn our attention to equation (1.1) in order to investigate whether or not there are T-periodic solutions, provided that σ is large enough. In particular, we will prove that there is no T-periodic solution, and hence the degree of the corresponding operator is equal to zero. To be more precise, we will introduce a suitable abstract framework in the same way as it was done in the previous section. The first elementary observation is that all solutions to (1.1) are located in the open set

So, for every σ>0, the operator Mσ:Δ→CT defined by

where Nσ is the Nemitsky operator associated with equation (1.1) (P, Q and K were defined in Section 2), is completely continuous in the sense that it is continuous and maps subsets of Δ whose closure is contained in Δ into relatively compact sets in CT. Now the periodic problem associated with (1.1) may be seen as a fixed-point problem for the operator equation

for every σ>0. In order to compute the degree of the operator I-Mσ for σ sufficiently large, we will prove a stronger result.

The assertion below is an elementary consequence of the degree theory and Lemma 3.1.

Actually, Proposition 3.2 can be proven assuming only that condition (3.3) holds (that can be derived from the assumptions of Theorem 1.1). It is worth noting here that the appearance of two singularities (i.e., the boundedness of the interval (0,B)) plays an important role.

4 The Homotopy. Proof of the Main Result

In this section we complete the proof of Theorem 1.1. We will begin our study considering the “homotopy” operator Mσ defined in (3.1). The degree of the operator I-Mσ for σ>0 large enough was derived in Proposition 3.2. Then, by virtue of the generalized homotopy invariance of the degree, we can calculate the degree of I-Mσ for every σ>0, provided that we are able to find a suitable open set whose boundary does not contain solutions to equation (3.2) (that is, T-periodic solutions to (1.1)). In other words, before passing to the proof of Theorem 1.1, we take advantage of the structure of our nonlinear term in order to find a priori bounds for all possible T-periodic solutions to (1.1) (note that this is possible due to the fact that g possesses two singularities; otherwise, [31, Corollary 3.4] shows that it cannot be done in general).