Relativistic equations with singular potentials

The first part of this paper concern with the study of the Lorentz force equation q′1-|q′|2′=E→(t,q)+q′×B→(t,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \frac{q'}{\sqrt{1-|q'|^2}}\right) '= \overrightarrow{E}(t,q)+q'\times \overrightarrow{B}(t,q) \end{aligned}$$\end{document}in the relevant physical configuration where the electric field E→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overrightarrow{E}$$\end{document} has a singularity in zero. By using Szulkin’s critical point theory, we prove the existence of T-periodic solutions provided that T and the electric and magnetic fields interact properly. In the last part, we employ both a variational and a topological argument to prove that the scalar relativistic pendulum-type equation q′1-(q′)2′+q=G′(q)+h(t),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \frac{q'}{\sqrt{1-(q')^2}}\right) ' +q = G^{\prime }(q) +h(t), \end{aligned}$$\end{document}admits at least a periodic solution when h∈L1(0,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in L^1 (0, T)$$\end{document} and G is singular at zero.


Introduction
The main scope of this paper is to investigate the existence of T -periodic solutions of the relativistic Lorentz force equation Here, − → E and − → B denote, respectively, the electric and magnetic fields and are given by where V : [0, T ] × (R 3 \{0}) → R and W : [0, T ] × R 3 → R 3 . By a solution of Eq. (1.1) we mean a function q = (q 1 , q 2 , q 3 ) ∈ C 2 satisfying (1.1) and such that |q (t)| < 1 for all t. Lorentz force equation (1.1) models the motion, in a relativistic regime, of a slowly accelerated charged particle under the influence of an electromagnetic field. The relativistic nature of Eq. (1.1) turns out in its left-hand side, which involves the relativistic momentum introduced by Poincaré in [11], with the velocity of light in the vacuum and the charge-to-mass ratio normalized to one, for simplicity. Instead, the presence of an electromagnetic field is emphasized by the Lorentz force − → E (t, q) + q × − → B (t, q) in its right-hand side. It represents one of the most significant equation of Mathematical Physics (see e.g., [9]). Nonetheless, a rigorous mathematical variational approach for its study was developed only recently in 91 Page 2 of 22 D. Arcoya and C. Sportelli ZAMP [3,4] (see also [5] for the case − → B ≡ 0) where the maps V and W are assumed to be of class C 1 , while the relevant cases including configurations of electric fields coming from the physical models and consisting of singular electric potential, had remained open.
An early result concerning these singular models was achieved recently in [10] via a topological method. In this work, the authors consider the case that the electric field − → E is a sufficiently large L 1 -pertubation of a Coulomb electric potential, or more specifically, − → E (t, q) = −∇V (q) − h(t) with h ∈ L 1 ([0, T ], R 3 ). The potential V is assumed to be singular at zero and, for some γ ≥ 1, c > 0, satisfies the inequality q · ∇V (q) ≤ −c/|q| γ if |q| is small enough. On the other hand, the magnetic field − → B is supposed to be bounded with a singularity at zero of lower order than the singularity |q| −γ−1 . Applying a global continuation theorem, the existence of a T periodic solution is guaranteed only when the mean value h of h is greater than the supremum of C(t) := lim sup |q|→∞ | − → B (t, q)|. In particular, this result clearly fails if e.g., h is identically zero. We emphasize in addition that the approach to the singular problem using variational methods had still maintained open.
The aim of this paper is to fill the observed gaps by developing the variational framework needed to address Eq. (1.1) and other relativistic singular problems as well, establishing the landmark for future investigations related on these topics. To be more precise, we show not only that Eq. (1.1) can be studied using a variational approach even when the electric field − → E is singular, widening the range of possible choices of V and covering the case untreated in [3,4]; but also, we prove that the topological argument carried on in [10] can be employed in such a way to handle other kinds of relativistic problems in which appears a singular term.
As a first step to study (1.1) variationally, we derive a new version of the Mountain Pass Theorem, which has its own interest and allows one to identify critical points of functionals which possess singularities (see Theorem 2.1). Our abstract result relies upon the idea developed in [1] to address the study of a relativistic spherical pendulum. In particular, it will be essential to impose for the action functional I that I(q n ) blows up when {q n } converges uniformly to a function which "touches" the singular set of I (see (2.1) below and compare with Lemma 5.1 in [1]). On this regard, we thank the anonymous referee who brought to our attention the paper [6], in which this condition is also used to provide the existence of solutions for another type of relativistic singular problem, namely, a relativistic Keplerian problem in the plane. Thus, by using our abstract result, we derive the existence of a T -periodic solution for Eq. (1.1).
To be more precise, we assume that V is dominated by the function −c/|q| (c is a positive constant) when q is located in a neighborhood of the origin, while the sum of the magnitudes of V (t, q) and ∇ q V (t, q) tend to 0 uniformly in t ∈ [0, T ] when q approaches infinity. Also, we suppose that W is bounded, its modulus and the sum of the magnitudes of the components of its gradient at q converge to 0 uniformly in t when q goes to infinity. Thus, if also there exists c 0 > 0 such that we prove that Eq. (1.1) admits a T -periodic solution. Observe that the periodic solution provided by the former result could be trivial provided that V and W depends only on the variable q , V is of class C 2 in R 3 \ {0} and there exists ξ ∈ R 3 \ {0} such that V (ξ) < 0, ∇V (ξ) = 0. In this case, we also prove that if the matrix is positive definite, then Eq. (1.1) has a periodic solution which is different from the constant solution ξ. Anyway, in order to not weigh this introduction down with too many details, we prefer to specify each hypothesis and to state our main results in Sect. 3. The remaining part of the paper is motivated by the study of the spherical pendulum in [1]. We study the existence of periodic Lipschitz solutions q(t) ∈ R of the scalar relativistic pendulum-type equation where h ∈ L 1 (0, T ), the singular function G dominates the function 1/|q| as q ∼ 0 and its first derivative is bounded when q is far away from the singularity at 0. It is worth noting that, unlike (1.

Local mountain pass for singular non-smooth functionals
In [3] a Mountain Pass Theorem without compactness conditions is given for the Szulkin critical point theory [13]. We give here a generalization of it which will be useful to handle functionals having singularities.
Assume also that lim n→∞ I(q n ) = +∞, (2.1) for every sequence {q n } ⊂ Λ whose distance dist (q n , E\Λ) is converging to zero. Let also K be a compact metric space, K 0 ⊂ K a closed subset and γ 0 : K 0 → Λ a continuous map. Consider the set then, for every ε > 0 and γ ∈ Γ Λ such that there exist γ ε ∈ Γ Λ and q ε ∈ γ ε (K) ⊂ E satisfying Remark 2.2. Notice that the continuity of Ψ in its closed domain implies that Ψ is lower semicontinuous in E.
Proof. Let Γ be defined as which is a complete metric space endowed with the uniform distance Since Λ is open in E and K is compact, the set Γ Λ = {γ ∈ Γ : γ(t) ∈ Λ} is open in Γ. Consider the functional Υ : Γ Λ → (−∞, +∞] given by Observe that every γ in the domain of Υ, that is, verifying Υ(γ) < +∞, satisfies that γ(t) ∈ Dom Ψ for every t ∈ K. Hence, the continuity of Ψ in its closed domain implies that I • γ is continuous in the compact K and we have Υ(γ) = max t∈K I(γ(t)).
contradicting (2.4). The claim has been proved and thus there exists μ ∈ (0, μ 0 ) such that γ ε,μ ∈ • N μ . In the sequel we fix this constant μ and we denote γ ε,μ = γ ε . We conclude the proof by showing the existence of t ε ∈ T : Indeed, assume by contradiction that for every t ∈ T there exists ϕ t ∈ E\{γ ε (t)} such that We can repeat the argument in the proof of Theorem 1 in Section 2 of [3] to deduce for every sufficiently small δ > 0 the existence of γ * ∈ Γ such that The first inequality allows to choose δ > 0 such that γ * ∈ N μ (remind that γ ε ∈ • N μ ) and then the second inequality contradicts (2.5) and completes the proof.

The relativistic Lorentz force equation
Consider the relativistic Lorentz force equation (1.1) when the electric and magnetic fields, respectively In order to study it, we denote by W 1,∞ (0, T ) the space of all Lipschitz functions in [0, T ] (or equivalently the absolutely continuous functions in [0, T ] with bounded derivatives) and we consider the Banach space We consider also the subspace E of all T -periodic vector functions q ∈ W 1,∞ (i.e. q ∈ W 1,∞ such that q(0) = q(T )). Let also K be the convex and closed set given by Following [3] the Lagrangian action I : Λ → (−∞, +∞] associated to the problem of the existence of T -periodic solutions of the Lorentz force equation (1.1) is given by where the functionals Ψ and F are defined by Since Ψ is a proper convex function which is continuous in its domain K (similar proof to that of Lemma 2 in Section 3 of [3]) and F is a function of class C 1 in K Λ , Szulkin's critical point theory from [13] is applicable for I. Recall what is it understood by a critical point in this theory.
By a similar argument to this one in Theorem 2 of Section 3 in [3], we have that the critical points q ∈ K Λ of I are just the T -periodic solutions of (1.1).
The following lemma will be essential to control the singularity of V at q = 0.

Lemma 3.2.
Assume that W is bounded and V satisfies the following hypothesis: and W is bounded, the first and second integrals of are bounded. Hence to prove the lemma, it suffices to show that Let t 0 ∈ [0, T ] be such that q(t 0 ) = 0. Two cases can occur: either q ≡ 0, or (up to a change of the zero t 0 by other zero) we can assume that there exists In the first case, q ≡ 0, we have |q n (t)| ≤ ε 0 for every t ∈ [0, T ] provided that n is large enough. Thus, the above hypothesis implies dt, for n large enough.
By Fatou lemma, we deduce lim sup which shows that lim n→∞ T 0 V (t, q n ) dt = −∞ and the lemma is proved in this case.
In the second case, observe that is also bounded from above for all n. Therefore, to conclude the proof it suffices to show that lim n→∞ |qn|≤ε0 1 |q n | dt = ∞. In order to prove it, since q n ∞ ≤ 1, we note that q n q n |q n | 2 dt = log |q n (t 1 )| − log |q n (t 0 )|.
Consequently, using that q n (t 1 ) converges to q(t 1 ) = 0 and q n (t 0 ) to q(t 0 ) = 0, we deduce that and the proof is concluded.

Remark 3.4.
Observe that, as a consequence of the above lemma, the functional I satisfies the condition (2.1) of Theorem 2.1. Indeed, let {q n } ⊂ E be a sequence satisfying lim n→∞ dist (q n , E\Λ) = 0 and assume by contradiction that condition (2.1) does not hold true. Then, up to a subsequence, we can suppose that {I(q n )} is bounded from above. In particular, q n ∈ K Λ . By using this and choosing p n ∈ E\Λ such that q n − p n = dist (q n , E\Λ) we get the uniform boundedness of p n in [0, T ], which together to the fact that each p n vanishes at some point in In the incoming results we will use this direct sum decomposition  Proof. Let {q n } be a sequence satisfying (3.1) and (3.2). Since q n = q n + q n with q n = q n ∞ + q n ∞ = q n ∞ + q n ∞ ≤ T + 1, to deduce the boundedness of {q n } in E it suffices to show that {q n } is bounded. Suppose by contradiction that, up to a subsequence, |q n | converges to infinity. Choosing ϕ = q n in (3.2) we obtain Since |q n (t)| = |q n + q n (t)| converges to infinity, and Therefore, again by (V ∞ ) and (W ∞ ) we would obtain from (3.1) that a contradiction proving that the sequence {q n } is necessarily bounded. By the compact embedding of E into C([0, T ], R) we can assume, up to subsequences, that q n (t) → q(t) uniformly in [0, T ]. Since each q n is Lipschitz with Lipschitz constant equal q n ∞ ≤ 1, we deduce that q is also Lipschitz with Lipschitz constant smaller or equal to one; i.e., q ∈ K. By Lemma 3.   Proof. Recalling that π 2 T 2 is the second eigenvalue of the periodic problem associated to the operator −q (t), we deduce by its variational characterization that Thus, using this and the inequality 1− 1 − |p| 2 ≥ 1 2 |p| 2 for every |p| ≤ 1, we obtain for every q ∈ E ∩K Λ that The condition (3.3) implies then that On the other hand, by (V ∞ ), when q ∈ E converges to infinity we have and we can choose ρ > 0 such that the boundary ∂B ρ of the ball B ρ in E of center zero and radius ρ satisfies that sup that is, I verifies the geometry of the Rabinowitz's saddle point theorem [12]. Therefore, since dim E < ∞, we can apply Theorem 2.1 with K the closed ball B ρ in E of center zero and radius ρ, K 0 the boundary in E of this ball and γ 0 the identity function in K 0 to deduce the existence of a sequence {q n } ⊂ E such that Observe that the periodic solution given by Theorem 3.6 can be trivial. Indeed, if for instance, V and W does not depend on t, then q = ξ is a constant solution if and only if ∇V (ξ) = 0. In this section we show a sufficient condition in order to obtain a second solution of (1.1). Proof. As it has been mentioned, since V and W only depends on the variable q, the hypothesis ∇V (ξ) = 0 means that q = ξ is a constant (thus periodic) solution of (1.1). Moreover, the condition about the positive definiteness of the matrix given by (3.4) implies that the functional F presents a strict local minimum at q = ξ. Since any constant is trivially a local minimum of the function Ψ, we deduce that I = Ψ + F has a strict local minimum at q = ξ. In particular, there exists r 0 > 0 such that

Theorem 3.7. Assume that V and W only depends on the variable q, satisfies conditions
On the other hand, by (V ∞ ) when the constant η ∈ R 3 converges to infinity we have

Periodic oscillations of a relativistic type-pendulum
This section is devoted to the study of the scalar equation (1.3). Our main existence result is achieved by means of both variational and topological arguments. We divide the section in two subsections.

Existence via a variational approach
Following [1], in this subsection we study the existence of T -periodic solutions of (1.3) where h ∈ L 1 (0, T ) and the singular function G : R\{0} → R satisfies the hypothesis Since the function G has a singularity at q = 0, we work in the subset Taking the functional I related to problem (1.3) is given for every q ∈ Λ by Similarly to the previous section, the critical points q of I in E are just the T -periodic solutions of the Eq.  Proof. Since {q n } is bounded in C([0, T ])R), the first, second and fourth integrals of I(q n ) (I is given by (4.1)) are bounded. Hence to prove the lemma, it suffices to show that T 0 G(q n ) dt converges to infinity. By the hypothesis (G 0 ), there exists ε 0 > 0 such that Let t 0 ∈ [0, T ] be such that q(t 0 ) = 0. Two cases can occur: either q ≡ 0, or (up to a change of the zero t 0 by other zero) we can assume that there exists t 1 ∈ (t 0 , T ] such that q(t 0 ) = 0 < |q(t)| ≤ ε 0 , for every t ∈ (t 0 , t 1 ]. In the first case, q ≡ 0, we have |q n (t)| ≤ ε 0 for every t ∈ [0, T ] provided that n is large enough. Thus, the above hypothesis implies By Fatou lemma, we deduce lim inf which shows that lim n→∞ T 0 G(q n ) dt = +∞ and the lemma is proved in this case.
In the second case, we have Using that G(s) is bounded from below for 0 ≤ t ≤ T and ε 0 < |s| ≤ sup n q n ∞ < ∞, we have |qn|>ε0 G(q n ) dt is also bounded from below for all n. Therefore, to conclude the proof it suffices to show that lim n→∞ |qn|≤ε0 In order to prove it, since q n ∞ ≤ 1, we note that q n q n q 2 n dt = log |q n (t 1 )| − log |q n (t 0 )|.
Consequently, using that, as n tends to infinity, q n (t 1 ) converges to q(t 1 ) = 0 and q n (t 0 ) to q(t 0 ) = 0, we deduce that lim n→∞ |qn|≤ε0 and the proof is concluded.

Remark 4.4.
As in the Remark 3.4, the above lemma implies that the functional I given by (4.1) satisfies the condition (2.1) required in Theorem 2.1.
As in the previous section, we will use the direct sum decomposition  Proof. Let {q n } ⊂ E be a sequence satisfying (4.3) and (4.4). Observe that for every q ∈ K Λ we have q ∞ = q ∞ ≤ 1, which together to the existence of a zero of q in [0, T ] (consequence of the zero mean value of q) implies that Hence, in order to prove that {q n } is bounded in E, it suffices to show that {q n } is bounded. To this aim, choosing w = q n + q n in (4.4), it follows that This implies that the sequence {q n } is bounded. Indeed, otherwise we can assume, up to subsequences, that |q n | converges to ∞. Thus, by (4.5), q n = q n + q n is away from zero for large n; that is, s 0 , n 0 0 exist such that |q n | ≥ s 0 for every n ≥ n 0 . By assumption (G ∞ ), there exists η > 0 such that By (4.6) we infer for every n ≥ n 0 that q 2 n T ≤ ηT |q n | + |q n | h L 1 + ε n |q n | i.e. |q n | ≤ η + T −1 h L 1 + ε n T −1 , (for every n ≥ n 0 ) contradicting the convergence of |q n | to ∞ and proving that the sequence {q n }, and thus the sequence {q n }, is bounded in E.
By the compact embedding of E into C([0, T ])R we can assume, up to subsequences, that q n (t) → q(t) uniformly in [0, T ].
Since each q n is Lipschitz with Lipschitz constant equal q n ∞ ≤ 1, we deduce that q is also Lipschitz with Lipschitz constant smaller or equal to one; i.e., q ∈ K. By Lemma 4.3 and (4.3), we have concluding the proof.  Proof. Taking into account the decomposition given by (4.2) and using (4.5) we have By this and since 1 − √ 1 − s 2 ≥ 1 2 s 2 ≥ 0 we deduce that the functional I defined by (4.1) satisfies We claim that this inequality implies that I is bounded from below over E. Indeed, if we take a minimizing sequence { q n } in E ∩ K Λ of I; i.e., such that . This means that there exists ε > 0 such that q n ∞ ≥ ε, for every n. Therefore, by using again that q n ∞ ≤ T , we obtain from (4.7) that This implies that inf E I ∈ R and the claim is proved.
On the other hand, for every q ∈ E we have and thus, by (G ∞ ), we have lim q →∞ In consequence, we can choose ρ > 0 such that the boundary ∂B ρ of the ball B ρ in E of center zero and radius ρ satisfies that sup that is, I verifies the geometry of the Rabinowitz's saddle point theorem [12]. By Lemma 4.5 (instead of Lemma 3.5) we can repeat the argument in the proof of Theorem 3.6 to deduce the existence a subsequence (q n k ) of (q n ) converging in C([0, T ], R) to a critical point q ∈ K Λ of I with critical level I(q) = c.
Now we look for T -periodic solutions q ∈ E of the Eq. (1.3) with h ≡ 0, i.e., Firstly, observe that every constant ξ ∈ R\{0} verifying ξ = G (ξ) is a trivial (constant) T -periodic solution of (4.8). In this case, in order to find another solution we apply the Mountain Pass Theorem to the functional I given by (4.1) with h ≡ 0; that is, Observe that it is of class C 2 (because G ∈ C 2 ) with the first and second derivatives given for every q, w 1 , w 2 ∈ E by In particular, the condition ξ = G (ξ) implies q = ξ is a critical point of F and the hypothesis G (ξ) > 1 means that the second derivative of F at q = ξ is positive definite. Thus, F has a strict local minimum at q = ξ. Taking into account that we deduce that q = ξ is also a strict local minimum of I. Hence, there exist δ, r > 0 such that I(q) ≥ I(ξ) + δ, when q − ξ = r.
As in the proof of Theorem 4.6, by assumption (G ∞ ), for every q ∈ E we have
Setting x := (q, p) and denoting by N f λ the Nemitskii operator associated to the function f λ (t, x), the previous problem can be written as the first order ordinary differential equation If P : X → X is the projection given by Note that, if X := Ker P = x ∈ X : 1 T T 0 x(t)dt = 0 , we can consider the operator K : L 1 ([0, T ], R 2 ) → X defining for each g ∈ L 1 ([0, T ], R 2 ), the function Kg as the unique solution x ∈ X of the equation Thus, using the previous notations, (4.9) turns into where • P has a finite range, • N f λ is continuous with N f λ (Ω) bounded in X, • and K| X : X → C 1 ([0, T ], R 2 ) is linear and continuous. Thus, by the compact embedding of C 1 ([0, T ]) into C([0, T ], R 2 ) (due to the Ascoli-Arzelà theorem) we have that T λ : X → X is compact and we can employ [7, Theorem 2] to address problem (4.10). For the sake of completeness, we recall it here in our particular case.