Relativistic equations with singular potentials

Z. Angew. Math. Phys. (2023) 74:91 c 2023 The Author(s) 0044-2275/23/030001-22 published online April 17, 2023 https://doi.org/10.1007/s00033-023-01977-z Zeitschrift f¨ur angewandte Mathematik und Physik ZAMP Relativistic equations with singular potentials David Arcoya and Caterina Sportelli Abstract. 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) in the relevant physical configuration where the electric field −→ Ehas a singularity in zero. By using Szulkin’s critical point theory, we prove the existence of T-periodic solutions provided that Tand 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), admits at least a periodic solution when h∈L1(0,T)andGis singular at zero. Mathematics Subject Classification. Primary 35Q60; Secondary 58E05, 35B10. Keywords. Lorentz force equation, Relativistic pendulum equation, Mountain pass theorem, Global continuation theorem. 1. Introduction The main scope of this paper is to investigate the existence of T-periodic solutions of the relativistic Lorentz force equation q 1−|q|2 =−→ E(t, q)+q×−→ B(t, q).(1.1) Here, −→ Eand −→ Bdenote, respectively, the electric and magnetic fields and are given by −→ E=−∇qV−∂W ∂t ,−→ B= curlqW, (1.2) where V:[0,T]×(R3\{0})→Rand W:[0,T]×R3→R3. By a solution of Eq. (1.1) we mean a function q=(q1,q 2,q 3)∈C2satisfying (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 Poincar´ein[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 Vand Ware assumed to be of class C1, 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 −→ Eis a sufficiently large L1-pertubation of a Coulomb electric potential, or more specifically, −→ E(t, q)=−∇V(q)−h(t) with h∈L1([0,T],R3). The potential Vis assumed to be singular at zero and, for some γ≥1, c>0, satisfies the inequality q·∇V(q)≤−c/|q|γif |q|is small enough. On the other hand, the magnetic field −→ Bis 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 Tperiodic solution is guaranteed only when the mean value h of his greater than the supremum of C(t) := lim sup|q|→∞ |−→ B(t, q)|. In particular, this result clearly fails if e.g., his 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 −→ Eis singular, widening the range of possible choices of Vand 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. Asafirststeptostudy(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(qn) blows up when {qn}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 Vis dominated by the function −c/|q|(cis a positive constant) when qis located in a neighborhood of the origin, while the sum of the magnitudes of V(t, q)and∇qV(t, q) tend to 0 uniformly in t∈[0,T] when qapproaches infinity. Also, we suppose that Wis bounded, its modulus and the sum of the magnitudes of the components of its gradient at qconverge to 0 uniformly in twhen qgoes to infinity. Thus, if also there exists c0>0 such that π2 2T2|q|2−|W(t, q)|−V(t, q)≥c0∀q∈R3\{0}, 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 Vand Wdepends only on the variable q,Vis of class C2 in R3\{0}and there exists ξ∈R3\{0}such that V(ξ)<0, ∇V(ξ) = 0. In this case, we also prove that if the matrix 2∂Wj ∂qi (ξ)−∂2V ∂qi∂qj (ξ)i,j=1,2,3 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. ZAMP Relativistic equations with singular potentials Page 3 of 22 91 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)∈Rof the scalar relativistic pendulum-type equation q 1−(q)2 +q=G(q)+h(t),(1.3) where h∈L1(0,T), the singular function Gdominates the function 1/|q|as q∼0 and its first derivative is bounded when qis far away from the singularity at 0. It is worth noting that, unlike (1.1), Eq. (1.3) identifies a scalar problem. Nonetheless, as Eqs. (1.1) and (1.3) share the same relativistic part, we employ again Theorem 2.1 to derive the existence of a T-periodic solution of (1.3). To be more precise, we improve the results in [1] and derive new existence results both in the case h≡ 0 (see Theorem 4.6) and h≡0 (see Theorem 4.8). Finally, we use the global continuation theorem to address Eq. (1.3) but assuming that there exists c0>0 such that G(q)qis bounded from below by the function c0/|q|when qis in a neighborhood of the origin. In fact, establishing a suitable homotopic system that drives the original problem into an autonomous system, we compute the Brouwer degree of the vector field and use [7, Theorem 2] to infer the existence of a T-periodic solution for the Eq. (1.3). We point out that, in contrast with [10, Theorem 1], our existence result does not require any hypothesis on the mean value of the function h. Our paper is organized as follows. In Sect. 2we present the required version of Mountain Pass Theorem for non-smooth functionals involving singularities. In Sect. 3we introduce the variational setting and apply our abstract result to the Lorentz force equation with a singular electric field, providing the claimed existence theorems. Finally, in Sect. 4we study the relativistic pendulum type equation and derive our main results both using variational methods (Sect. 4.1) and topological techniques (Sect. 4.2). 2. 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. Theorem 2.1. Assume that Λis an open subset of a Banach space Eand that the functional Iis the sum of two functionals I=Ψ+Fwhere (i) Ψ:E→(−∞,+∞]is a convex and proper functional with a closed domain Dom Ψ:={v∈E: Ψ(v)<∞} in Eand Ψis continuous in Dom Ψ. (ii) F:Λ→Ris a C1-functional. Assume also that lim n→∞ I(qn)=+∞,(2.1) for every sequence {qn}⊂Λwhose distance dist (qn,E\Λ) is converging to zero. Let also Kbe a compact metric space, K0⊂Kaclosedsubsetandγ0:K0→Λa continuous map. Consider the set ΓΛ={γ:K→Λγis continuous and γ|K0=γ0}. If c1:= sup t∈K0I(γ0(t)) <c:= inf γ∈ΓΛ sup t∈KI(γ(t)) <∞,(2.2) then, for every ε>0and γ∈ΓΛsuch that c≤max t∈KI(γ(t)) ≤c+ε 2,(2.3) 91 Page 4 of 22 D. Arcoya and C. Sportelli ZAMP there exist γε∈ΓΛand qε∈γε(K)⊂Esatisfying c≤max t∈KI(γε(t)) ≤max t∈KI(γ(t)) ≤c+ε 2, max t∈Kγε(t)−γ(t)≤√ε, c−ε≤I(qε)≤c+ε 2, and Ψ(ϕ)−Ψ(qε)+F(qε)[ϕ−qε]≥− √εϕ−qεfor all ϕ∈E. 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 Γ={γ:K→Eγis continuous and γ|K0=γ0}, which is a complete metric space endowed with the uniform distance dΓ(γ1,γ 2) = max t∈Kγ1(t)−γ2(t),(γ1,γ 2∈Γ). Since Λ is open in Eand Kis compact, the set ΓΛ={γ∈Γ:γ(t)∈Λ}is open in Γ. Consider the functional Υ : ΓΛ→(−∞,+∞] given by Υ(γ)=sup t∈KI(γ(t)),(γ∈ΓΛ). Observe that every γin 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◦γis continuous in the compact Kand we have Υ(γ) = max t∈KI(γ(t)). By (2.2) the functional Υ is proper and bounded from below by c1.Fixε>0 that, without loss of generality, can be assumed less than c−c1.Letγ∈ΓΛsatisfying (2.3). For 0 <μ<μ 0:= dΓ(γ,Γ\ΓΛ), we consider the set Nμ:= {η∈Γ: d Γ(η,Γ\ΓΛ)≥μ}, which clearly contains γ.By(2.2) and (2.3)wehave c≤cδ0:= inf γ∈Nμ sup t∈KI(γ(t)) ≤max t∈KI(γ(t)) ≤c+ε 2, i.e., c≤cδ0:= inf γ∈Nμ Υ(γ)≤Υ(γ)≤c+ε 2, Taking into account that Nμis closed in the complete metric space Γ, we obtain that it is also a complete metric space. In addition, Υ is lower semicontinuous in Nμby [13, Lemma 3.1]. Applying the Ekeland variational principle [8], we deduce from the above inequality that there exists γε,μ ∈N μsatisfying c≤cδ0≤Υ(γε,μ)≤Υ(γ)≤c+ε 2, dΓ(γε,μ,γ) = max t∈Kγε,μ(t)−γ(t)≤√ε, (2.4) and Υ(γε,μ)<Υ(ϑ)+√εdΓ(γε,μ,ϑ) for all ϑ∈N μ.(2.5) ZAMP Relativistic equations with singular potentials Page 5 of 22 91 We claim that there exists μ∈(0,μ 0) such that γε,μ ∈◦ Nμ(i.e., dΓ(γε,μ,Γ\ΓΛ)<μ). Indeed, assume by contradiction that dΓ(γε,μ,Γ\ΓΛ)=μ, ∀μ∈(0,μ 0). In this case, it is possible to choose sequences {μn}⊂(0,μ 0)and{ηn}⊂Γ\ΓΛsatisfying dΓ(γε,μn,η n) = max t∈Kγε,μn(t)−ηn(t)=μn (n→∞) −→ 0. Since ηn∈Γ\ΓΛ, there exists tn∈Ksuch that ηn(tn)∈E\Λ and using that γε,μn(tn)−ηn(tn)≤ dΓ(γε,μn,η n), we infer from the above convergence to zero that lim n→∞ γε,μn(tn)−ηn(tn)=0, and by assumption (2.1) that lim n→∞ I(γε,μn(tn)) = +∞. By the inequality I(γε,μn(tn)) ≤max t∈KI(γε,μn(t)) = Υ(γε,μn),it follows that lim n→∞ Υ(γε,μn)=+∞, contradicting (2.4). The claim has been proved and thus there exists μ∈(0,μ 0) such that γε,μ ∈◦ Nμ. In the sequel we fix this constant μand we denote γε,μ =γε. We conclude the proof by showing the existence of tε∈T:= {t∈K:I(γε(t)) ≥c−ε}such that, if qε=γε(tε), then Ψ(ϕ)−Ψ(qε)+F(qε)[ϕ−qε]≥− √εϕ−qεfor all ϕ∈E. Indeed, assume by contradiction that for every t∈T there exists ϕt∈E\{γε(t)}such that Ψ(ϕt)−Ψ(γε(t)) + F(γε(t))[ϕt−γε(t)] <−√εϕt−γε(t). 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 dΓ(γ∗, γε)≤δ, and Υ(γ∗)<Υ(γε)−δ√ε≤Υ(γε)−dΓ(γ∗, γε)√ε. The first inequality allows to choose δ>0 such that γ∗∈N μ(remind that γε∈◦ Nμ) and then the second inequality contradicts (2.5) and completes the proof.  3. The relativistic Lorentz force equation Consider the relativistic Lorentz force equation (1.1) when the electric and magnetic fields, respectively −→ Eand −→ B, are given by (1.2), with V:[0,T]×(R3\{0})→Rand W:[0,T]×R3→R3two C1-functions. In order to study it, we denote by W1,∞(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 W1,∞=[W1,∞(0,T)]3 endowed with the norm ·defined from the usual norm ·∞of [L∞(0,T)]3as q=q∞+q∞(q∈W1,∞). 91 Page 6 of 22 D. Arcoya and C. Sportelli ZAMP We consider also the subspace Eof all T-periodic vector functions q∈W1,∞(i.e. q∈W1,∞such that q(0) = q(T)). Let also Kbe the convex and closed set given by K={q∈E:q∞≤1}, and Λ={q∈E:q(t)=0,∀t∈[0,T]},KΛ=K∩Λ. 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 I(q)=Ψ(q)+F(q),q∈Λ. where the functionals Ψ and Fare defined by Ψ(q)=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ T 0 [1 −1−|q|2]dt, if q∈K, +∞,if q∈E\K, F(q):= T 0 [q·W(t, q)−V(t, q)]dt, ∀q∈Λ. 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 Fis a function of class C1in KΛ, Szulkin’s critical point theory from [13] is applicable for I. Recall what is it understood by a critical point in this theory. Definition 3.1. A function q∈K Λis a critical point of Iif Ψ(ϕ)−Ψ(q)+F(q)[ϕ−q]≥0 for all ϕ∈K Λ, or, equivalently, for E:[0,T]×R3×R3→R3given by E(t, q, p)=(p·Dq1W(t, q),p·Dq2W(t, q),p·Dq3W(t, q)),∀(t, q, p)∈[0,T]×R3×R3, if T 0 [1−|q|2−1−|ϕ|2]dt+ T 0 [E(t, q, q)−∇qV(t, q)] ·(ϕ−q)dt + T 0 W(t, q)·(ϕ−q)dt≥0,for all ϕ∈K Λ. By a similar argument to this one in Theorem 2 of Section 3 in [3], we have that the critical points q∈K Λof Iare just the T-periodic solutions of (1.1). The following lemma will be essential to control the singularity of Vat q=0. Lemma 3.2. Assume that Wis bounded and Vsatisfies the following hypothesis: (V0)There exists c0>0 such that lim sup q→0 V(t, q)|q|≤−c0<0. If a sequence {qn}in KΛconverges uniformly to some qwhich vanishes at some point of [0,T], then lim n→∞ T 0 V(t, qn)dt=−∞ and lim n→∞ I(qn)=+∞. ZAMP Relativistic equations with singular potentials Page 7 of 22 91 Remark 3.3. A sufficient condition for (V0) is that there exist α>1andc1>0 such that lim sup q→0 V(t, q)|q|α≤ −c1<0. Proof. Since {qn}⊂K Λis bounded in C([0,T],R)andWis bounded, the first and second integrals of I(qn)= T 0 [1 −1−|q n|2]dt+ T 0 W(t, qn)·q ndt− T 0 V(t, qn)dt are bounded. Hence to prove the lemma, it suffices to show that T 0V(t, qn)dtconverges to −∞. By the hypothesis (V0), there exists ε0>0 such that V(t, q)≤−c0 |q|for 0 <|q|≤ε0. Let t0∈[0,T] be such that q(t0) = 0. Two cases can occur: either q≡0, or (up to a change of the zero t0by other zero) we can assume that there exists t1∈(t0,T] such that q(t0)=0<|q(t)|≤ε0, for every t∈(t0,t 1]. In the first case, q≡0, we have |qn(t)|≤ε0for every t∈[0,T] provided that nis large enough. Thus, the above hypothesis implies T 0 V(t, qn)dt≤− T 0 c0 |qn|dt, for nlarge enough. By Fatou lemma, we deduce lim sup n→∞ T 0 V(t, qn)dt≤−lim inf n→∞ T 0 c0 |qn|dt≤− T 0 lim inf n→∞ c0 |qn|dt=−∞, which shows that lim n→∞ T 0 V(t, qn)dt=−∞ and the lemma is proved in this case. In the second case, observe that T 0 V(t, qn)dt=⎧ ⎪ ⎨ ⎪ ⎩ |qn|≤ε0 + |qn|>ε0 ⎫ ⎪ ⎬ ⎪ ⎭V(t, qn)dt≤−  |qn|≤ε0 c0 |qn|dt+ |qn|>ε0 V(t, qn)dt. Using V(t, ξ) is bounded from above for t∈[0,T]andε0<|ξ|≤supnqn∞<∞,wehave  |qn|>ε0 V(t, qn)dt is also bounded from above for all n. Therefore, to conclude the proof it suffices to show that lim n→∞  |qn|≤ε0 1 |qn|dt=∞. 91 Page 8 of 22 D. Arcoya and C. Sportelli ZAMP In order to prove it, since q n∞≤1, we note that  |qn|≤ε0 1 |qn|dt≥ t1  t0 1 |qn|dt≥ t1  t0 qnq n |qn|2dt= log |qn(t1)|−log |qn(t0)|. Consequently, using that qn(t1) converges to q(t1)=0andqn(t0)toq(t0) = 0, we deduce that lim n→∞  |qn|≤ε0 1 |qn|dt≥lim n→∞ log |qn(t1)|−log |qn(t0)|=∞, and the proof is concluded.  Remark 3.4. Observe that, as a consequence of the above lemma, the functional Isatisfies the condition (2.1) of Theorem 2.1. Indeed, let {qn}⊂Ebe a sequence satisfying limn→∞ dist (qn,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(qn)}is bounded from above. In particular, qn∈KΛ. By using this and choosing pn∈E\Λ such that qn−pn=dist(qn,E\Λ) we get the uniform boundedness of p nin [0,T], which together to the fact that each pnvanishes at some point in [0,T], implies that the sequence {pn}is bounded in E.Bythe compact embedding of Einto the continuous functions in [0,T] we can extract a subsequence {pnk}which uniformly converges in [0,T] to some function qwhich vanishes at some point of [0,T]. The convergence to zero of qn−pngives also the uniform convergence of {qnk}to qin [0,T]. By the previous lemma we obtain the convergence of {I(qnk)}to infinity contradicting the boundedness from above of the sequence {I(qn)}. In the incoming results we will use this direct sum decomposition E=E⊕ E, q =q+q, q=1 T T 0 qdt, q=q−q. Lemma 3.5. (Palais–Smale condition) Assume in addition to (V0)that Vand Wsatisfy also [(V∞)] lim |q|→∞ |V(t, q)|+|∇qV(t, q)|=0uniformly in t∈[0,T],(W∞) lim |q|→∞ |W(t, q)|+|Dq1W(t, q)|+|Dq2W(t, q)|+ |Dq3W(t, q)|=0uniformly in t∈[0,T].If{εn}is a sequence of positive numbers converging to zero and {qn}is a sequence in KΛsatisfying that lim n→∞ I(qn)=c∈R\{0}(3.1) and T 0 [1−|q n|2−1−|ϕ|2]dt+ T 0 [E(t, qn,q n)−∇qV(t, qn)] ·(ϕ−qn)dt + T 0 W(t, qn)·(ϕ−q n)dt≥−εnϕ−qn,∀ϕ∈K,\{0},(3.2) then there exists a subsequence {qnk}of {qn}converging in C([0,T],R3)to a critical point q∈K Λof I with level I(q)=c. Proof. Let {qn}be a sequence satisfying (3.1) and (3.2). Since qn=qn+qnwith qn=qn∞+q n∞= qn∞+q n∞≤T+1, to deduce the boundedness of {qn}in Eit suffices to show that {qn}is bounded. ZAMP Relativistic equations with singular potentials Page 9 of 22 91 Suppose by contradiction that, up to a subsequence, |qn|converges to infinity. Choosing ϕ=qnin (3.2) we obtain T 0 [1−|q n|2−1]dt− T 0 [E(t, qn,q n)−∇qV(t, qn)] ·qndt − T 0 W(t, qn)·q ndt≥−εnqn≥−εn(T+1). Since |qn(t)|=|qn+qn(t)|converges to infinity, |[E(t, qn,q n)−∇qV(t, qn)] ·qn|≤T|E(t, qn,q n)|+|∇qV(t, qn)| ≤T[|Dq1W(t, qn)|+|Dq2W(t, qn)|+|Dq3W(t, qn)|]+|∇qV(t, qn)|, and |W(t, qn)·q n|≤|W(t, qn)|, by hypotheses (V∞) and (W∞), we have lim n→∞ T 0 [E(t, qn,q n)−∇qV(t, qn)] ·qndt+ T 0 W(t, qn)·q ndt=0. Thus, lim sup n→∞ T 0 [1 −1−|q n|2]dt≤lim sup n→∞ εn(T+1)=0; which means (by the positiveness of the function 1 −1−|s|2) that lim n→∞ T 0 [1 −1−|q n|2]dt=0. Therefore, again by (V∞) and (W∞) we would obtain from (3.1) that c= lim n→∞ I(qn) = lim n→∞ T 0 [1 −1−|q n|2]dt+ T 0 W(t, qn)·q ndt− T 0 V(t, qn)dt=0, a contradiction proving that the sequence {qn}is necessarily bounded. By the compact embedding of Einto C([0,T],R) we can assume, up to subsequences, that qn(t)→q(t) uniformly in [0,T]. Since each qnis Lipschitz with Lipschitz constant equal qn∞≤1, we deduce that qis also Lipschitz with Lipschitz constant smaller or equal to one; i.e., q∈K. By Lemma 3.2 and (3.1), we have q(t)=0,∀t∈[0,T], concluding the proof.  Theorem 3.6. Assume the hypotheses (V0),(V∞)and (W∞). If there exists c0>0such that π2 2T2|q|2−|W(t, q)|−V(t, q)≥c0,∀q∈R3,/{0}(3.3) then there exists a T-periodic solution of the Lorentz force equation (1.1). 91 Page 16 of 22 D. Arcoya and C. Sportelli ZAMP Theorem 4.8. Assume that hypotheses (G0),(G∞)and (G ∞)hold true with the function Gof class C2 in R\{0}.Ifξ∈R\{0}verifies ξ=G(ξ)and G(ξ)>1, then the Eq. (4.8)has at least one T-periodic solution different from the trivial one ξ. Proof. Consider the functional Fdefined for q∈Eby F(q)=−1 2 T 0 q2dt+ T 0 G(q)dt. Observe that it is of class C2(because G∈C2) with the first and second derivatives given for every q,w1,w 2∈Eby F(q)[w1]=− T 0 qw1dt+ T 0 G(q)w1dt and F(q)[w1,w 2]=− T 0 w1w2dt+ T 0 G(q)w1w2dt. In particular, the condition ξ=G(ξ) implies q=ξis a critical point of Fand the hypothesis G(ξ)>1 means that the second derivative of Fat q=ξis positive definite. Thus, Fhas a strict local minimum at q=ξ. Taking into account that I(q)=Ψ(q)+F(q),∀q∈K Λ with Ψ(q):= T 0 [1 −1−(q)2]dt≥0(=Ψ(ξ)),∀q∈K, we deduce that q=ξis 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∈Ewe have lim q→∞ I(q) = lim q→∞ −T 2q2+TG(q)=−∞ and we can choose a constant η∈Esuch that I(η)<I(ξ). In conclusion, the functional Isatisfies the geometry of the Mountain Pass Theorem [2]and applying Theorem 2.1 with K=[0,1], K0={0,1},γ0(0) = ξ,γ(1) = ηand Γ = {γ:[0,1] →E: γis continuous and γ(0) = ξ, γ(1) = η}we deduce the existence of a sequence (qn)⊂Esuch that lim n→∞ I(qn)=c:= inf γ∈Γsup t∈[0,1] I(γ(t)) >I(ξ)>I(η), and a sequence 0 <ε n→0 such that Ψ(ϕ)−Ψ(qn)+F(qn)[ϕ−qn]≥−εnϕ−qn for all positive integer nand for all ϕ∈Dom Ψ. By Lemma 4.5, there exists a subsequence {qnk}of {qn}converging in C([0,T],R) to a critical point q∈K Λof Iwith critical level I(q)=c. Since I(q)=c≥inf q−ξ=rI(q)>I(ξ), we obtain q=ξand the proof is finished.  ZAMP Relativistic equations with singular potentials Page 17 of 22 91 4.2. Existence by global continuation theorem In contrast with the variational techniques used so far, here we will derive the existence of at least one T-periodic solution of (1.3) by using the global continuation theorem to the homotopic system q 1−(q)2 +q=gλ(q)+hλ(t),λ∈[0,1],(4.9) with gλ(q)=λG(q)+(1−λ)c0 q|q|and hλ(t)=λh(t). Observe that (4.9) turns into (1.3) when λ=1. By considering the position qand the momentum p=q/1−(q)2as new coordinates (and taking into account that the inverse of φ(s)=s/√1−s2is given by φ−1(p)=p/1+p2), Eq. (4.9) turns into the first order system of differential equations ⎧ ⎨ ⎩ q=p 1+p2 p=−q+gλ(q)+hλ(t), that is, (q,p)=fλ(t, (q,p)) := p 1+p2,−q+gλ(q)+hλ(t). Setting x:= (q,p) and denoting by Nfλthe Nemitskii operator associated to the function fλ(t, x), the previous problem can be written as the first order ordinary differential equation x=Nfλ(x) in the Banach space X={x∈C([0,T],R2):x(0) = x(T)}. If P:X→Xis the projection given by Px=x:= 1 T T 0 x(t)dt∀x∈X, then x=x+x, x=Px, x:= (I−P)(x),∀x∈X. Note that, if  X:= Ker P=x∈X:1 TT 0x(t)dt=0 , we can consider the operator K:L1([0,T],R2)→ Xdefining for each g∈L1([0,T],R2), the function Kg as the unique solution x∈ Xof the equation x(t)=g(t). Thus, using the previous notations, (4.9) turns into x=Px+KNfλ(x)=: Tλx, (4.10) where •Phas a finite range, •N fλis continuous with Nfλ(Ω) bounded in X, •and K|X:X→C1([0,T],R2) is linear and continuous. Thus, by the compact embedding of C1([0,T]) into C([0,T],R2) (due to the Ascoli-Arzel`a theorem) we have that Tλ:X→Xis 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. Theorem 4.9. Let Ω⊂X={x∈C([0,T],R2):x(0) = x(T)}be an open bounded subset and assume that the operators Tλ:X→Xare compact on Ω. If the following conditions hold true 91 Page 18 of 22 D. Arcoya and C. Sportelli ZAMP (i) there is no solution x∈∂Ωof the homotopic problem (4.10)for every λ∈[0,1]; (ii) degB(f0,Ω∩R2,0) =0,whereΩ∩R2denotes the subset of the constant functions in Ω(observe that T0|Ω∩R2=f0)anddegBdenotes the Brouwer degree; then the problem (4.10)has at least a solution x∈Ω. Remark 4.10. Observe that if condition (i) in the previous theorem were not satisfied for λ= 1, then a (trivial) solution of (4.10) would exist on ∂Ω. In order to properly define the open bounded subset Ω required to apply the previous theorem, we derive some a priori bounds for the position qand the momentum p=q/1−(q)2of every T–periodic solution q(t)of(4.9). Lemma 4.11. If condition (G∞)and ( G0) lim inf s→0G(s)s|s|>0 hold true, then there exist positive constants C,cand Msuch that for every T-periodic solution q(t) of (4.9)we have (i) q∞≤C, (ii) |q(t)|≥c, for every t∈[0,T], (iii) q 1−(q)2≤M, for every t∈[0,T]. Remark 4.12. Similarly to Remark 4.1, assumption (  G0) implies that lim sup s→0 G(s)|s|<0. Proof. (i) By hypothesis (G∞), there exists a positive constant η(independent of λ∈[0,1]) such that gλ(s)≤G(s)+ c0 s|s|≤η, ∀|s|≥1. Fixing R≥max{η+hL1/T, 1}we claim that the above identity implies that for every solution qof (4.9) there exists at least one t0∈[0,T] such that q(t0)≤R. Indeed, supposing by contradiction that a solution qof (4.9) satisfies q(t)>R(≥1) for all t, we would have T 0 gλ(q(t))dt<Tη≤TR−hL1≤ T 0 q(t)dt− T 0 hλ(t)dt ≤ T 0 q(t)dt− T 0 hλ(t)dt. This inequality contradicts that choosing v= 1 as test function in (4.9), we have T 0 gλ(q(t))dt= T 0 q(t)dt− T 0 hλ(t)dt. Therefore, for every T-periodic solution qof (4.9) there is at least one t0∈[0,T] such that q(t0)≤R. Thus |q(t)|=q(t0)+ t  t0 q(s)ds<R+T, i.e. every solution qof the problem (4.9) satisfies q∞≤R+Tand it suffices to choose C=R+Tto conclude the proof of (i). ZAMP Relativistic equations with singular potentials Page 19 of 22 91 (ii) Choosing v=qas test function in (4.9) we deduce that − T 0 (q)2 1−(q)2dt+ T 0 q2(t)dt= T 0 gλ(q(t))q(t)dt+ T 0 hλ(t)q(t)dt, for every T-periodic solution of (4.9). Now, since T 0 (q(t))2 1−(q(t))2dt≥0, we infer that T 0 q2(t)dt≥ T 0 gλ(q(t))q(t)dt+ T 0 hλ(t)q(t)dt. Since assumption (  G0) implies that there exists c0,ε 0>0 such that G(s)s≥c0 |s|,∀0<|s|≤εo, we have gλ(s)s=λG(s)s+(1−λ)c0 |s|≥c0 |s|,∀0<|s|≤ε0,∀λ∈[0,1]. Then, it follows that  |q(t)|<ε0 c0 |q(s)|ds= |q(t)|<ε0 gλ(q(s))q(s)ds ≤ T 0 q2(s)ds− |q(t)|>ε0 gλ(q(s))q(s)ds− T 0 hλ(s)q(s)ds ≤ T 0 q2(s)ds+ |q(t)|>ε0 gλ(q(s))q(s)ds + T 0 hλ(s)q(s)ds. By (i) we obtain the inequality  |q(t)|<ε0 c0 |q(s)|ds≤C2T+Tmax ε0<|s|<C G(s)s+c0 |s|+ChL1. that allows us to prove the item (ii). Indeed, observe first that if the T-periodic solution q(t)of(4.9)is always out the ball BL∞(0,ε 0) the result is proved. On the other hand, if there exists a closed interval [t1,t 2]⊂[0,T] such that |q(t1)|=ε0and |q(t)|<ε 0∀t∈(t1,t 2], 91 Page 20 of 22 D. Arcoya and C. Sportelli ZAMP then, using that |q|≤1, we deduce for t∈(t1,t 2) that log |q(t)|=log |q(t1)|+ t  t1 q(s)q(s) q(s)2ds≤|log ε0|+ t2  t1 1 |q(s)|ds ≤|log ε0|+ |q(t)|<ε0 1 |q(s)|ds ≤|log ε0|+C2T c0 +T c0 max ε0<|s|<C G(s)s+c0 |s|+C c0hL1=: c, that is, |q(t)|>e −c, and (ii) is also proved in this case. (iii) Let q(t)beaT–periodic solution of (4.9). Choosing t0∈[0,T] such that q(t0) = 0 and using the mean value theorem, we have q(t) 1−(q(t))2= t  t0q(s) 1−(q(s))2ds Taking again v= 1 as test function in (4.9) (as in item (i)) we deduce that q(t) 1−(q(t))2= t  t0 (−q(s)+gλ(q(s)) + hλ(s)) ds, and item (iii) holds true by (i) and (ii) with M=Tmax c<|s|<C(|s|+|G(s)|+c0 |s|2)+hL1. Remark 4.13. Observe that, unlike in [10], in our setting we have no conditions on hto get the upper bound of case (i) in the previous lemma. We prove our main result. Theorem 4.14. If h∈L1(0,T)and the function Gdefined in R\{0}satisfies assumptions (G∞)and ( G0), then problem (1.3)admits at least one T-periodic solution. Proof. We define the open bounded subset Ω in Xas Ω:={x=(q,p)∈X:c<|q(t)|<C, |p(t)|<M, ∀t∈[0,T]}, which, by Lemma 4.11, contains each T–periodic solution of system (4.10). The proof will be concluded by applying Theorem 4.9 if we show that degB(f0,Ω∩R2,0) =0, where Ω ∩R2stands for the identification with constants functions and f0∈C∞(Ω ∩R2) is given by f0(x)=f0(t, x)=f0(t, (q,p)) = p 1+p2,−q+c0 q|q|. To compute this degree we observe that det Jacf0(x)=(1+p2)−3/2−1−2c0 |q|3. ZAMP Relativistic equations with singular potentials Page 21 of 22 91 and that the only zeroes of f0(x)are(±3 √c0,0). We apply the additivity property of the degree to infer that degB(f0,Ω∩R2,0) = det Jacf0(3 √c0,0) + det Jacf0(−3 √c0,0) = −6=0, and the proof is concluded.  Acknowledgements The first author is supported by FEDER-MINECO PGC2018-096422-B-I00 and PID2021-122122NB-I00 and Junta de Andaluc´ıa FQM-116. The second author is partially supported by MIUR–PRIN project “Qualitative and quantitative aspects of nonlinear PDEs” (2017JPCAPN 005) and is member of the Research Group INdAM-GNAMPA. This work was completed while the second author was visiting the Departamento de An´alisis Matem´atico at Universidad de Granada. She is very grateful for the kind hospitality of the host institution. The authors thank Cristian Bereanu for bringing to their attention a mistake in the proof of Theorem 2.1 found in the early version of this paper. The results outlined in this paper were presented by the first author during the day devoted to the 70th birthday of Prof. Patrizia Pucci at the Summer School in Nonlinear Analysis held in Viterbo, Italy (June 20–24, 2022) and also during the NLEPDE in Hauts-de-France held in Valenciennes, France (June 27–30, 2022). He thanks the organizers for the kind invitation. Funding Information Funding for open access publishing: Universidad de Granada/CBUA Open Access. 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Academic Press, New York (1978) [13] Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 3, 77–109 (1986) David Arcoya Departamento de An´alisis Matem´atico Universidad de Granada 18071 Granada Spain e-mail: darcoy[email protected] Caterina Sportelli Dipartimento di Matematica Universit`a degli Studi di Bari Aldo Moro Via E. Orabona 4 70125 Bari Italy e-mail: caterina.sp[email protected] (Received: November 13, 2022; revised: February 24, 2023; accepted: March 13, 2023)