Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential

We consider the Lorentz force equation $$ \frac{d}{dt}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right) = q \left(E(t,x) + \dot x \times B(t,x)\right), \qquad x \in \mathbb{R}^3, $$ in the physically relevant case of a singular electric field $E$. Assuming that $E$ and $B$ are $T$-periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many $T$-periodic solutions. The proof is based on a min-max principle of Lusternik-Schrelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Li\'enard-Wiechert potential and to the relativistic forced Kepler problem.


Introduction
According to the principles of electrodynamics [16], the motion of a slowly accelerated charged particle under the influence of an electromagnetic field is ruled by the Lorentz force equation where m is the mass of the particle, q is its charge and c is the speed of light; moreover, the electric and magnetic fields E and B are provided by the potentials V and A via the usual relations As well known (see, for instance, [13]) equation (1.1) is formally the Euler-Lagrange equation of the action functional In spite of this, and probably due to the lack of smoothness of the kinetic part of the above functional, a systematic investigation of equation (1.1) with the tools of critical point theory has been initiated only very recently.More precisely, in [3,4] a rigorous variational formulation in the space W 1,∞ is introduced, allowing for the use of non-smooth critical point theory in the version developed by Skulzkin [21], and, as a consequence, several existence and multiplicity results are given for solutions of equation (1.1) with either Dirichlet or periodic boundary conditions (see also [14] for the use of topological techniques).However, in both the papers the physically relevant case of singular electric and magnetic fields is not taken into account.
The aim of the present paper is to provide a contribution in this direction.To this end, we take advantage of a recent research [6] dealing with the equation which is a version of (1.1) in the plane with A ≡ 0 and q = 1.More precisely, in [6] equation (1.3) with a singular potential V given by V (t, x) = −α/|x| − U (t, x) (with α > 0), is considered, namely x ∈ R 2 . (1.4) Let us point out that the motivation given in [6] for the above equation was not coming from electrodynamics, but rather from relativistic celestial mechanics: indeed, equation (1.4) is interpreted as a simple model, in special relativity, for the motion of a particle in a forced Kepler potential (see, for instance, [2] as well as the references in [5]).Of course, however, this can be of interest also in the context of electromagnetism and, actually, this interpretation is even more natural, since a rigorous treatment of the theory of gravitation should require the framework of general relativity: we refer to [15,Problem 34.3] for an interesting discussion and comparison about Kepler and Coulomb problems from the relativistic point of view.By using minimization and min-max arguments in the framework of non-smooth critical point theory, it is proved in [6] that, for any external perturbation U , non singular and T -periodic in time, equation (1.4) has infinitely many T -periodic solutions and, in particular, at least two T -periodic solutions of winding number k around the origin, for any integer k = 0. Of course, such a result deeply relies on the presence of the singularity x = 0 for the potential V , which produces a non-trivial topology for the domain of the action functional: the set of T -periodic paths winding k times around the origin is nothing but a connected component of the domain, and each of them (but the one with k = 0) carries at least two periodic solutions of (1.4).Let us emphasize the universal character of this result, meaning that no assumptions on U (besides its smoothness) are needed: this is ultimately a consequence of the fact that a periodic path x winding around the origin with bounded velocity (since | ẋ| < c) is a priori-bounded.
In this paper, we provide a sort of generalization of the result in [6] applying to the Lorentz force equation (1.1).More precisely, we consider an electrostatic potential V < 0 defined in a set Ω of the form where the functions r 1 , . . ., r N : R → R 3 are T -periodic (for some T > 0), of class C 1 with ṙi ∞ < c and such that r i (t) = r j (t) for every t ∈ [0, T ] and i = j.Moreover, we assume that V has a Keplerian blow-up at the boundary of Ω (cf.assumption (V) in Section 3) and that the magnetic potential A satisfies the global condition for some κ ′ ∈ (0, 1).Under these conditions, if both A, V and their derivatives tend to zero at infinity, we prove that (1.1) has infinitely many T -periodic solutions (cf.Theorem 3.1).
Let us point out that the structure of the singularities of V , described via the set Ω, is modeled on the relevant case of Liénard-Wiechert potentials (cf.[16] and Section 4), corresponding to the motion of a charged particle under the effect of N moving charged particles q 1 , . . ., q N .In this situation, the functions r 1 , . . .r N are the motions laws of the particles generating the potentials and V and A are given by where t i = t i (t, x) is the so-called retarded time (see (4.2) in Section 4).In particular, let us notice that condition (1.6) is satisfied since ṙi ∞ < c, for every i = 1, . . ., N .As a second application of our main result, going back to the relativistic celestial mechanics framework, we can prove the existence of infinitely many T -periodic solutions for the relativistic forced Kepler problem in the space when U > 0 and U → 0 for |x| → +∞ together with its gradient (cf.Theorem 4.2 in Section 4).In particular, this provides a partial generalization of the result given in [6] for the planar case mentioned above.
For the proof of Theorem 3.1 we use a variational approach, combining arguments from both [3,4] and [6].More precisely, we consider the functional I : W 1,∞ T → (−∞, +∞] defined as where Ω is as in (1.5)) and ẋ ∞ ≤ c, and extended to +∞ otherwise.This functional satisfies the structural assumption of Skulzin non-smooth critical point theory and its critical points give rise to T -periodic solutions of equation (1.1); moreover, it is well-behaved near collisions, in the sense that if x n approaches the boundary of Λ, then I(x n ) → +∞.These properties can be proved by using arguments already developed in [3,4,6] and are collected in Lemma 3.4.
On the other hand, however, due to the three-dimensional setting, the approach of [6] based on the winding number cannot be used and a different strategy to achieve both existence and multiplicity has to be developed.In particular, inspired by classical results available in the setting of classical mechanics [1], we detect periodic solutions via a min-max principle of Lusternik-Schnirelmann type.For this, two main issues have to be faced.On one hand, we prove that the functional I satisfies a weak form of the Palais-Smale condition at any level c > inf I = 0, cf.Lemma 3.5.On the other hand, we show that the proper domain of the action functional I contains compact subsets of arbitrarily large category, allowing us to define the min-max levels where F j is the family of compact subsets of the domain of I having category at least j, cf.Lemma 3.6; moreover, c j > 0 for any j ≥ 3, cf.Lemma 3.7.Then, taking advantage of the general min-max principle for non-smooth functionals proved in [6] (cf.Theorem 2.5 in Section 2), we can prove that for j ≥ 3 the number c j is a critical level for the action functional.This would ensure the existence of infinitely many periodic solutions to equation (1.1) provided a sequence of distinct critical levels c j exists, a fact which however seems hard to be established in general.Thus, adapting the arguments in the proof of [4, Th. 1] we prove that whenever two critical levels coincide, the corresponding critical level carries infinitely many critical points.From this, we deduce that the functional I has infinitely many critical points.
To the best of our knowledge, a technique of this type seems to be completely new in a non-smooth setting and we think that the general Lusternik-Schnirelmann min-max principle we introduce can be of independent interested.
The plan of the paper is the following.In Section 2, we describe the abstract variational setting and we provide the non-smooth min-max principle of Lusternik-Schnirelmann type (Theorem 2.6).In Section 3 we state and prove our main result (Theorem 3.1).Finally, in Section 4 we provide the above mentioned applications: the motion of a charged particle under the influence of periodic Liénard-Wiechert potentials (Theorem 4.1) and the perturbed relativistic Kepler problem (Theorem 4.2).

An abstract result
In this section, we present a result on the existence of infinitely many critical points for non-smooth functionals with singularities.More precisely, as in [6] we are concerned with functionals of the form described in the following assumption.
Remark 2.3.The functional Φ in Section 3 is actually of class C 1 in its domain D Φ and, thus, We decided to present this more abstract section in the setting of nonsmooth calculus since, on one hand, the assumption Φ ∈ C 1 (D Φ ) doesn't really simplify the argument and, on the other, locally Lipschitz functionals immediately appear as soon as one considers some truncation of a C 1 functional.
We also need to consider the following weak form of the Palais-Smale condition, as introduced in [3].
Definition 2.4.Let I : X → (−∞, +∞] satisfy Assumption 2.1 and assume that there exists a Banach space Y such that X ⊂ Y with continuous embedding.The functional I is said to satisfy the weak Palais-Smale condition at level c if for every PS-sequence {x n } in X such that I(x n ) → c, there exist x ∈ X and a subsequence {x The existence of infinitely many critical points for a functional of the form I is obtained using a general non-smooth min-max principle, together with the Lusternik-Schnirelmann category.For the readers convenience, we recall here the definition and basic properties of the category (cf.[1]) and the min-max principle we use (established in [6] as a generalization of [17,Theorem 3.1]).
Given M ⊂ X, the category of A ⊂ M relative to M , denoted by cat X (A, M ) is the least integer k, if it exists, such that where A i ⊂ M is closed and contractible in M , for every i = 1, . . ., k.The category is infinite if such a least integer does not exist.The category satisfies the following properties, which will be used in many situations:

it is homotopic to the inclusion ι
Theorem 2.5.[6, Th. 2.4] Let I = ψ + Φ be a functional satisfying Assumption 2.1, let B be a closed set in X and F be a family of compact sets in X such that: (1) F is homotopy stable with extended boundary B, that is, for each A ∈ F and each continuous deformation (3) there exists a closed set F in X such that Then, for any sequence {A n } in F such that lim We are now in a position to state our result.For every integer j ∈ N, let us define Moreover, let for every j ∈ N such that F j is not-empty.Then, we are able to prove the following result.
Theorem 2.6.Let I = ψ + Φ be a functional satisfying Assumption 2.1 and the weak Palais-Smale condition at each level c > inf I.Moreover, let us assume that there exists j 0 ∈ N such that (i) F j = ∅, for every j ≥ j 0 (ii) c j < +∞ for every j ≥ j 0 (iii) c j0 > inf I.
Then, the functional I has infinitely many critical points.More precisely: (a1) c j is a critical level of I, for every j ≥ j 0 (a2) whenever c j1 = c j2 for some j 2 > j 1 ≥ j 0 , then the functional I has infinitely many critical points at level c j1 .
Proof.(a1) Let us fix j ≥ j 0 .We claim that the assumptions of Theorem 2.5 with B = ∅, F = X and F = F j are satisfied.Indeed, assumption (1) is a consequence of property (P4) of the category.Moreover, assumption (2) is guaranteed by (ii) and assumption (3) is trivially fulfilled since Hence, we can apply Theorem 2.5 to obtain the existence of a PS-sequence at the level c j ≥ c j0 > inf I.
Since I satisfies the weak Palais-Smale condition at levels greater than inf I, we deduce that there exists a critical point at level c j .
(a2) The argument here follows closely the one in [4, Theorem 1], with three main changes.The first one is that our functional Φ is singular and locally Lipschitz continuous, instead of being even and in C 1 (X).The second is that we use the Lusternik-Schnirelmann category instead of the Krasnoselskii genus, a fact which is, however, linked to the first difference.Finally, we use item 2 in Assumption 2.1, which is weaker than the continuity of ψ on its proper domain D ψ required in [4].
Concerning the weak PS-condition, we denote • Y the norm in Y (see Definition 2.4) and we set which is open in X also w.r.t. the stronger topology induced by • .By contradiction, let us assume that I has only n ∈ N critical points at level c := c j1 = c j2 , which we label x 1 , . . ., x n , and let r > 0 be such that the sets B Y (x m , 2r) are pairwise disjoint and contained in D Φ (the closure is taken w.r.t. the norm • of X, if not otherwise specified).We define and observe that, arguing by contradiction and using the weak PS-condition, there exists ǫ ∈ (0, r 2 ) such that, for each Let A ∈ F j2 be chosen in such a way that sup In particular, A ⊂ D φ and sup A I = max A I since ψ is bounded on A and, thus, continuous in A by Assumption 2.1.The set B = A \ N 2r is compact in X and B ∈ F j1 , since by (P3).As a consequence we have We apply Ekeland variational principle (see also [4, Lemma 1(iii)]), to the map Π : and Π is lower semi-continuous w.r.t. the same matric.Then, we obtain C ∈ F j1 such that max where is the Hausdorff distance for compact sets of a metric space.In particular, C ∩ N r = ∅ and the set Since the mapping is upper semi-continuous in X × D Φ (by [9, Proposition 2.1.1])and negative for x 1 = x 2 = x ∈ S by (2.4), for each x ∈ S there is a positive δ Since S is compact, there exist y 1 , . . ., y ℓ ∈ S such that We observe that ξ y k ∈ B k , by construction, and, thus, we can fix some positive δ which is continuous and satisfies β t (x) − x ≤ δ < δ C for all (t, x) ∈ [0, 1] × C by construction.As a consequence β 1 is a deformation of C in D Φ (observe that β 0 is the identity on C) and D := β 1 (C) belongs to F j1 by property (P4).
From the estimate tδη(x) we deduce that β t (x) is a convex combination of x, ξ y1 , . . ., ξ y ℓ and, hence, On the side of Φ, by Lebourg's theorem [9, Theorem 2.3.7] for each x ∈ C there exists τ = τ (x) ∈ (0, 1) and . As a consequence we can estimate Now, by construction, β τ (x) − x ≤ δ ≤ δ y k for all k = 1, . . ., ℓ, which implies that Therefore, we have On the other hand, if x ∈ C \ S, we have η(x) = 0 and I(β As consequences we have that x 0 ∈ S and c ≤ max and, thus, η(x 0 ) = 1 and max which is a contradiction with (2.5).

The main result
In this section we state and prove our main result for the Lorentz force equation where, as usual, Notice that, without loss of generality, we have normalized the charge-to-mass ratio to 1 (while, on the other hand, we prefer to keep track of the value c of the speed of light).As already mentioned in the Introduction, our main interest is in covering the case when the potential V is singular.More precisely, we assume that the singularities of V are described by N functions r 1 , . . ., r N : R → R 3 which are T -periodic (for some T > 0), of class C 1 , with ṙi ∞ < c, and such that r i (t) = r j (t) for every t ∈ [0, T ] and i = j.Accordingly, we settle equation (3.1) on the open domain The following result holds true.endowed with its usual norm x = x ∞ + ẋ ∞ .We define the functional ψ : X → (−∞, +∞] as According to the notation of Section 2, we thus have Moreover, we consider the open subset of X and we define Φ : X → (−∞, +∞] as +∞ otherwise, so that D Φ = Λ.Finally, we define the action functional I : X → (−∞, +∞] as and we recall the notation For further convenience, we observe that assumption (AV1) implies that and then, by assumption (V), Taking into account that ψ ≥ 0 and that I = +∞ outside D I , we deduce that In the next Lemma, we show that this functional satisfies the structural Assumption 2.1 of Section 2 and that, moreover, its critical points correspond to classical T -periodic solutions of the Lorentz force equation (3.1).Proof.Most of the above statement has been already proved in [6, Proposition 3.2] (and, in turn, in corresponding results in [3]); notice indeed that the functional ψ is the same as the one considered therein, while Φ, despite the presence of the magnetic term, is still of class C 1 on the open set D Φ = Λ, with cf. [3, Lemma 1] (in the above formula, the term (D x A) T ẋ is meant as the product of the transpose of the Jacobian matrix D x A with the (column) vector ẋ).Notice that, by integrating by parts, From this, together with the identity we can rewrite dΦ in the equivalent form Using this formula, and recalling (3.2), the fact that critical points of I gives rise to classical T -periodic solutions of equation (3.1) can be proved with the very same arguments of [6, Proposition 3.3] (see also [3,Theorem 2]).
The only point which requires a bit of care is the proof of the property of blow-up on the boundary (that is, item 4 of Assumption 2.1), for which we give the complete details.At first, we notice that So, let us consider a sequence {x n } in D I such that d n := dist(x n , ∂D Φ ) → 0 and, accordingly, let y n ∈ ∂D Φ be such that x n − y n ≤ 2d n .Since ẋn ∞ ≤ c for any n, we find that for n large enough.Moreover, since y n (t n ) = r in (t n ) for some t n ∈ [0, T ] and i n ∈ {1, . . ., N }, we have and thus the sequence {y n } is bounded in X.Since x n − y n ≤ 2d n , the sequence {x n } is bounded in X as well.Therefore, the Ascoli-Arzelà theorem yields the existence of a continuous function z such that, up to subsequence, x n → z and y n → z uniformly on [0, T ].Hence, z(0) = z(T ) and z(t 0 ) = r i0 (t 0 ) for some t 0 ∈ [0, T ], limit point of the sequence t n , and i 0 ∈ {1, . . ., N }, limit point of the sequence i n .Moreover, passing to the limit in the Lipschitz-continuity condition we easily see that z ∈ D ψ ⊂ X.Hence, the function z − r i0 is Lipschitz continuous and so Therefore, by (V) and (AV1) and using Fatou's lemma we obtain Since 0 ≤ ψ(x n ) ≤ mc 2 T , we finally conclude that I(x n ) → +∞ as desired.
Let us now notice that inf Choosing z = xn in (2.1), we obtain that is, using (3.9), (3.12) Therefore, recalling the boundedness of xn ∞ , (3.10) and assumption (AV2) we obtain ψ(x n ) → 0. On the other hand, for the same reasons Φ(x n ) → 0 and so As a second step, we show that the boundedness of {x n } implies the existence of a subsequence {x n k } converging in L ∞ (0, T ) to a critical point x of the functional I at level c (that is, the condition required in the definition of weak Palais-Smale condition at level c).For this, we combine the arguments used in the proof of [6, Proposition 3.5] with the ones in the proof of [3,Lemma 5].
We now consider the sets F j and the min-max levels c j defined respectively in (2.2) and (2.3) and we turn to the proof of the validity of assumptions (i)-(ii)-(iii) of Theorem 2.6.
At first, we deal with (i)-(ii).
Lemma 3.6.For every integer j ≥ 1, it holds that: Proof.We first prove that (i) holds when there is only one curve r 1 of singularities.To this aim, we make use of the following auxiliary open sets in C T := {x ∈ C([0, T ], R 3 ) : x(0) = x(T )}, endowed with the topology of uniform convergence: and of the the continuous and dense immersion ι : X → C T .We have that X ∩ Λ 1 = ι −1 (Λ 1 ) and that ι| X∩Λ1 : X ∩ Λ 1 → Λ 1 is a homotopy equivalence by [20,Theorem 16].Since the affine isometry x → x − r 1 maps Λ 1 onto Λ 0 , we have that its composition with ι| X∩Λ1 provides a homotopy equivalence between X ∩ Λ 1 and Λ 0 .Using [12, Corollary 2.8], we deduce that the cup length in Z 2 of Λ 0 is infinite.Since the cup length is a homotopy invariant, we infer that X ∩Λ 1 contains compact sets with arbitrarily large category by [10, Lemma 2.9].Now, we are going to show that, for each j ≥ 1, there exists a compact A ⊂ Λ ∩ D ψ such that cat X (A, Λ) ≥ j; this will imply both (i) and (ii), since our functional ψ is bounded in D ψ .We just showed that there exists a compact A 1 ⊂ Λ 1 such that cat X (A 1 , Λ 1 ∩ X) ≥ j.For each λ > 0 and x ∈ X we define x λ = r 1 + λ(x − r 1 ) and observe that x λ ∈ Λ 1 ∩ X if and only if x ∈ Λ 1 ∩ X.We set A λ = {x λ : x ∈ A 1 } which is compact and omeomorphic to A 1 so that cat by property (P2).On the other hand, we have that Hence, if λ > 0 is small enough we have that A λ ⊂ D ψ ∩ Λ and A λ ∈ F j .
Finally, we prove that (iii) of Theorem 2.6 is satisfied with j 0 = 3 (while it can be shown that c 1 = c 2 = 0).In particular, noticing that and so, from Sobolev inequality (see, for instance, [18, Proposition 1.3]), where we have written as usual x(t) = x + x(t), with x = 1 T T 0 x(t) dt.Now, we claim that, fixed an arbitrary constant R > 0 with the property that there exists n * ∈ N such that, for every n ≥ n * , Indeed, let ι * be defined by and observe that ι * > 0 by assumption (V); moreover, let n * ∈ N be such that which contradicts (3.18).At this point, we notice that from (3.15) and (3.18) it follows that Hence, taking into account (3.17) we deduce that, for every n ≥ n * ,  From Lemmas 3.4, 3.5, 3.6 and 3.7 we deduce that all the assumptions of Theorem 2.6 are satisfied and then Theorem 3.1 is proved.

Applications
In this section, we give some applications of our main result.The first one deals with the motion of a charge under the effect of the electric and magnetic field generated by N moving charges.
For the second one, we move to the interpretation of equation (3.1) in relativistic celestial mechanics, dealing with the motion of a particle in a perturbed Kepler potential.

The Liénard-Wiechert potentials
Les us consider the motion of a charged particle with m/q = 1 under the effect of N moving electric point charges.
We denote by q 1 , . . ., q N the moving charges and by r 1 , . . ., r N their trajectories, which we assume to be C 2 functions r j : R → R 3 , T -periodic and such that | ṙj (t)| < c for every t ∈ [0, T ] and r i (t) = r j (t) for every t ∈ [0, T ] and i = j (cf.Section 3).Let us now set, for i = 1, . . ., N , and observe that Moreover, we define η i : Ω → R 3 by and t i : Ω → R by the implicit relation It is well-known that, for every i = 1, . . ., N , the number t i is the retarted time.The existence and uniqueness of a solution of (4.2) for a fixed (t, x) ∈ Ω is a standard fact in special relativity and it can be proved by means of a plain implicit function argument, which also implies that t i is a function of class C 1 .Moreover, the periodicity of r i implies that t i is T -periodic as a function of the time variable t.
The Liénard-Wiechert scalar and vector potentials generated by the point charge source q i , i = 1, . . ., N , acting on a charge at the point (t, x), are given, respectively, by and where t i = t i (t, x) and ε 0 is the vacuum permittivity.For future reference, let us recall that the corresponding electric and magnetic fields are given by , is the Lorentz factor, and respectively (cf.[16]).Let us notice that V i and A i (and then E i and B i ) are well-defined in Ω: indeed, from (4.2) we first deduce that x − r i (t i ) = 0 ⇐⇒ t = t i , thus implying that (t, x) = (t i , r i (t i )), which is impossible if (t, x) ∈ Ω.On the other hand, if (t, x) ∈ Ω we have We are now in a position to state our result on periodic motions under Liénard-Wiechert potentials.
Proof.The result follows from Theorem (3.1).We need to show that V and A satisfy assumptions (V), (AV1) and (AV2).

Lemma 3 . 4 .
The functional I satisfies Assumption 2.1.Moreover, the functional ψ is lower semicontinuous with respect to uniform convergence, namely: if x ∈ X and {x n } is a sequence in D ψ such that x n → x uniformly on [0, T ], then x ∈ D ψ andψ(x) ≤ lim inf n→+∞ ψ(x n ).Moreover, each critical point x ∈ D I of I satisfies | ẋ(t)| < c for every t ∈ [0, T ] and correspond to a classical T -periodic solution of equation (3.1).

XILemma 3 . 5 .
= 0. Indeed, we have already observed that I > 0, cf.(3.7).Moreover, for a sequence x n (t) ≡ ξ n with |ξ n | → +∞ we readily see, by assumption (AV2), that I(x n ) = Φ(x n ) → 0. With this in mind, the next result ensures that the functional I satisfies, at each level c > inf I = 0, the weak Palais-Smale condition, according to Definition 2.4 with Y = L ∞ (0, T ).The functional I satisfies the weak Palais-Smale condition at each level c > 0.Proof.Let {x n } ⊂ X be a Palais-Smale sequence at level c > 0; incidentally, let us notice that {x n } ⊂ D I , since otherwise I(x n ) = +∞.The proof will be divided in two steps.At first, we show that the sequence {x n } is bounded in L ∞ (and, thus, in X).To see this, let us write x n = xn + xn , where xn = 1 T T 0 x n and T 0 xn dt = 0. Since ẋn ∞ = ẋn ∞ ≤ c, we have that xn ∞ is bounded.So, assuming by contradiction that x n ∞ is not bounded yields, up to subsequences, |x n | → +∞.Then |x n (t)| ≥ |x n | − ˙xn ∞ and so min t |x n (t)| → +∞. (