Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 66, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.66 EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO EIGENVALUE PROBLEMS FOR SCHRÖDINGER-BOPP-PODOLSKY EQUATIONS LORENA SORIANO HERNANDEZ, GAETANO SICILIANO Abstract. We study the existence and multiplicity of solutions for the Schrödinger-Bopp-Podolsky system −∆u+ φu = ωu in Ω a2∆2φ−∆φ = u2 in Ω u = φ = ∆φ = 0 on ∂Ω∫ Ω u2 dx = 1 where Ω is an open bounded and smooth domain in R3, a > 0 is the Bopp- Podolsky parameter. The unknowns are u, φ : Ω → R and ω ∈ R. By using variational methods we show that for any a > 0 there are infinitely many solutions with diverging energy and divergent in norm. We show that ground states solutions converge to a ground state solution of the related classical Schrödinger-Poisson system, as a→ 0. 1. Introduction In this article we prove the existence of solutions for the Schrödinger-Bopp- Podolsky system −∆u+ φu = ωu in Ω a2∆2φ−∆φ = u2 in Ω u = φ = ∆φ = 0 on ∂Ω∫ Ω u2 dx = 1 (1.1) on a bounded and smooth domain Ω ⊂ R3. Also we study the behavior of this system as the parameter a tends to zero. In (1.1), the first equation is a Schrödinger equation which relates the modulus of the charged wave function ψ(x, t) = u(x)e−iωt of a non relativistic particle, its fre- quency ω ∈ R and the electrostatic potential φ generated by its motion. The value of the charge has been settled to one for simplicity. In particular the electrostatic potential obeys to the electromagnetic field theory of generalized electrodynamics 2020 Mathematics Subject Classification. 35A15, 58E05. Key words and phrases. Schrödinger type systems; existence of solutions; variational methods; critical point theory. ©2023. This work is licensed under a CC BY 4.0 license. Submitted March 8, 2023. Published October 13, 2023. 1 2 L. SORIANO H., G. SICILIANO EJDE-2023/66 developed by Bopp and Podolsky. It is evident by the second equation that the source of the electrostatic field is the same wave function. In the above system the unknowns are the real functions u, φ : Ω → R and the real number ω. To these equations we associate suitable Dirichlet boundary conditions, that roughly speaking, indicate that the particle is constrained to “live” in the bounded region Ω, and the potential vanishes on its boundary. The normalizing condition is quite reasonable from a physical point of view since it is just the L2 norm of the wave function, that in the physical applications represents the probability of finding the particle in the region Ω. More details on the physical background as well as the deduction of the equa- tions can be found in [4, Section 2], where for the first time such a system was introduced. After the work [4] this kind of system has been extensively studied. In particular the problem has been addressed in the whole space and in bounded domains, where existence and multiplicity of solutions have been proved by using variational methods and critical point theory. We refer the reader to the recent papers [2, 5, 7, 9, 10, 11, 13]. In a natural way we can associate with (1.1) its “limit” problem, namely when a = 0. In particular, the difference is formally in the second equation which now is −∆φ = u2, namely the classical Poisson equation, highlighting the fact that in this case the Maxwell theory of the electromagnetic field has been used. Of course this affects also the first equation, since φ is different. This “limit” problem, called Schrödinger-Maxwell (or Schrödinger-Poisson system) is studied in [3], where the authors showed a general reduction method to study similar systems involving the interaction between matter and electromagnetic field. However there are some reasons for which the electromagnetic theory of Bopp-Podolsky is preferable to the Maxwell one, and this is discussed in [4]. In this paper we want to show once more in which sense the Bopp-Podolsky theory is an approximation of the Maxwell theory. Going back to problem (1.1), our approach is variational. Indeed as usual in these cases, we will see that a suitable energy functional on certain Sobolev spaces can be defined and its critical points are exactly the weak solutions we want, according to the definition given below. We work in the Sobolev spaces H1 0 (Ω) with the usual norm ‖u‖ := (∫ Ω |∇u|2 dx )1/2 and, for a fixed a > 0 we consider the space H = H1 0 (Ω)∩H2(Ω) endowed with the norm ‖φ‖a := ( a2 ∫ Ω |∆φ|2 dx+ ∫ Ω |∇φ|2 dx )1/2 , and the associated scalar product (φ, ψ)a = a2 ∫ Ω ∆φ∆ψ dx+ ∫ Ω ∇φ∇ψ dx. Throughout this work we denote by |u|p := (∫ Ω |u|p dx )1/p the norm in Lp(Ω). We define the L2−sphere in H1 0 (Ω) by B := {u ∈ H1 0 (Ω) : |u|2 = 1}. EJDE-2023/66 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 3 We recall that for a fixed a > 0, the triple (ωa, ua, φa) ∈ R × B × H is a weak solution of (1.1) if∫ Ω ∇ua∇v dx+ ∫ Ω φauav dx = ωa ∫ Ω uav dx for all v ∈ H1 0 (Ω) (1.2) and a2 ∫ Ω ∆φa∆v dx+ ∫ Ω ∇φa∇v dx = ∫ Ω u2 av dx for all v ∈ H. (1.3) However it worth saying that the weak solutions, if any, are sldo classical, as stated in the the first result. Theorem 1.1. If (ωa, ua, φa) ∈ R × B × H is a weak solution of (1.1) then ua ∈ C2,λ(Ω) and φa ∈ C4,λ(Ω) for some λ ∈ (0, 1). Here we are using the classical notation for Hölder spaces Cj,λ, 0 < λ ≤ 1. The proof of this result involves classical boot-strap arguments. Our main result concerning with the existence of solutions is stated as follows. Theorem 1.2. Let a > 0. Then there is a sequence {(ωa,n, ua,n, φa,n)}n ⊂ R × B ×H of solutions of (1.1) with ωa,n →∞, ‖ua,n‖ → ∞, as n→∞ We will see in the proof, that more precise information can be deduced. For example the energy levels are divergent, and we can assume that ua,1 is positive. See the proof of Theorem 1.2. In our approach the frequencies ω will appear as Lagrange multipliers associated to critical points of the energy functional on the constraint B. Our last result concerns with the asymptotic behavior of the ground state solu- tions, obtained for n = 1 in Theorem 1.2, whenever a tends to zero. By ground state we mean a solution of the system with minimal energy, in the sense specified later. To this aim we consider the “limit” problem −∆u+ φu = ωu in Ω −∆φ = u2 in Ω u = φ = 0 on ∂Ω∫ Ω u2 dx = 1 (1.4) studied by Benci and Fortunato in [3], where existence result of a ground state and even of multiple solutions {(ω0,n, u0,n, φ0,n)}n ⊂ R × B ×H1 0 (Ω) is obtained. This system has also been extensively studied in the last decades under different boundary conditions and/or the nonlinearity. For example the case of a Berestycki- Lions type nonlinearity has been studied in [8]. Theorem 1.3. Let {(ωa,1, ua,1, φa,1)}a>0 ∈ R×B×H be ground state solutions of (1.1) found in Theorem 1.2. Then as a→ 0, up to subsequences, we have ua,1 → u0 and φa → φ0 in H1 0 (Ω), ωa,1 → ω0 in R (1.5) where (ω0, u0, φ0) ∈ R×B ×H1 0 (Ω) is a ground state solution of (3.5). 4 L. SORIANO H., G. SICILIANO EJDE-2023/66 We will see in the proof that there is convergence also of the ground state levels. This last result corroborates the fact that Schrödinger-Poisson (also called Schrö- dinger-Maxwell) systems can be seen as limit of Schrödinger-Bopp-Podolsky sys- tems as already seen in [4, 12, 11]. This is essentially due to the fact that the Maxwell theory of electromagnetism is the limit of the generalized Bopp-Podolsky theory of electromagnetism. We spend few words on our methods. We use critical point theory to show how the solutions can be associated with a critical point of a functional on a suitable manifold in an Hilbert space. In view of the applications of variational methods and to use topological invariants of the Ljusternick-Schnirelmann Theory, some facts like compactness and geometry of the functional have to be shown. We remind that in many problems of this type, the frequency ω of the wave function is fixed. Then the approach in finding solutions is different, in particular the L2 norm of the solutions u is not given a priori. In our case, the wave function is completely unknown, so both u and ω are unknowns, and we are looking for solutions with a priori fixed L2 norm. Let us recall that the L2 norm is constant in time on the solutions of the evolution problem, so it is constantly equal to the L2 norm of the initial datum. As a consequence, the unknowns ω related to the solutions will be found as the Lagrange multipliers associated to the critical points on the manifold made by the unit sphere in L2. For these reasons, we think that it is natural to consider the frequencies of the wave function, ω, as an unknown and the L2 norm of u fixed, since it is more interesting also from a physical point of view. The paper is organized as follows. In the subsequent Subsection 1.1 we show once for all that the weak solutions are classical. This is a classical fact which is independent of the variational framework or the way we use to find weak solutions. Then we focus in proving the existence of solutions. In Section 2 the variational setting is implemented. This will be fundamental in order to define the energy functional and then look for its critical points, characterized as weak solutions of (1.1). In the final Section 3 the proofs of Theorem 1.2 and Theorem 1.3 are given. We use C,C ′, . . . to denote suitable positive constants whose value may change from line to line and which do not depend on the functions involved in the inequalities. 1.1. Proof of Theorem 1.1. This subsection is devoted to show that every weak solution is necessary a classical solution. For the sake of simplicity we omit here the parameter a in the solutions. Let (ω, u, φ) ∈ R× B ×H be a weak solution of (1.1), then ψ := −a2∆φ+ φ is a weak solution of the Dirichlet problem ∆ψ = u2 in Ω, ψ = 0 on ∂Ω. Now, if u ∈ H1 0 (Ω), then u ∈ L6(Ω) and u2 belongs to L3(Ω). Thus, by [6, Theorem 9.9] we have − a2∆φ+ φ = ψ ∈W 2,3(Ω). (1.6) Recall that Ω is a bounded set. If φ ∈ H is a solution of (1.6) with ψ ∈ W 2,2(Ω), the interior regularity increases because [6, Theorem 8.10] implies that φ ∈W 4,2(Ω) which leads us to the fact that φ ∈ C2,λ(Ω) with λ ∈ (0, 1 2 ] by the Sobolev embed- ding [1, Theorem 5.4]. EJDE-2023/66 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 5 Now, considering the first equation of (1.1), −∆u+ φu = ωu in Ω, we have that u ∈ H1 0 (Ω) is the unique solution of ∆u = (φ− ω)u ∈ L2(Ω) because φ ∈ C2,λ(Ω). Then, by [6, Theorem 9.9], it holds ∆u = (φ− ω)u ∈ H2 0 (Ω). Therefore [6, Theorem 8.10] implies that φ ∈ H4 0 (Ω) which leads us to the fact that u ∈ C2,λ(Ω) with λ ∈ (0, 1/2] by [1, Theorem 5.4, part II]. Since u ∈ H1 0 (Ω) and u ∈ C2,λ(Ω), λ ∈ (0, 1/2], we obtain −∆ψ = u2 ∈ H2(Ω). By [6, Theorem 8.10] it follows that −a2∆φ+ φ = ψ ∈ H4(Ω), and then the interior regularity of φ increases by the same Theorem, i.e. φ ∈ H6(Ω). Finally, by Part II of the Sobolev embedding [1, Theorem 5.4], φ ∈ H6(Ω) ↪→ C4,λ(Ω), where λ ∈ (0, 1/2]. 2. Variational setting To prove the existence of solutions we set the right variational framework. Since the system has a Lagrangian derivation (see [4]), it is natural to look at solutions as critical point of a suitable energy functional. We define the functional on H1 0 (Ω)×H by Fa(u, φ) = 1 2 ∫ Ω |∇u|2 dx+ 1 2 ∫ Ω φu2 dx− a2 4 ∫ Ω |∆φ|2 dx− 1 4 ∫ Ω |∇φ|2 dx. (2.1) Straightforward computations show that Fa is C1 with derivatives given by ∂uFa(u, φ)[v] = ∫ Ω ∇u∇v dx+ ∫ Ω uvφ dx, ∀v ∈ H1 0 (Ω) (2.2) ∂φFa(u, φ)[v] = 1 2 ∫ Ω u2v dx− a2 2 ∫ Ω ∆φ∆v dx− 1 2 ∫ Ω ∇φ∇v dx, ∀v ∈ H. (2.3) Then we have a first variational principle. Theorem 2.1. Let a > 0. The triple (ωa, ua, φa) ∈ R × H1 0 (Ω) × H is a weak solution of (1.1) if, and only if, (ua, φa) is a critical point of Fa restricted to B×H having ωa as a Lagrange multiplier. Proof. An ordered pair (ua, φa) ∈ H1 0 (Ω) × H is a critical point of Fa constrained to B ×H if and only if there exists a Lagrange multiplier ωa ∈ R such that ∂uFa(ua, φa) = ωaua and ∂φFa(ua, φa) = 0 Taking into account the expressions of the partial derivatives in (2.2) and (2.3) this is equivalent to (1.2) and (1.3), namely to say that (ωa, ua, φa) ∈ R×H1 0 (Ω)×H is a weak solution of system (1.1). � 6 L. SORIANO H., G. SICILIANO EJDE-2023/66 2.1. Reduced functional. The functional Fa in (2.1) is unbounded both from above and below. Then the usual methods of critical point theory cannot be directly applied. To deal with this issue, we shall reduce the functional in (2.1) to the study of another functional depending on the single variable u, following a procedure introduced by Benci and Fortunato in [3] for these kind of problems. Proposition 2.2. Given a > 0 and u ∈ B, the problem a2∆2φ−∆φ = u2 in Ω ∆φ = φ = 0 on ∂Ω (2.4) has a unique (and non trivial) weak solution Φa(u) ∈ H. Moreover it minimizes the functional Ea(φ) = 1 2 ∫ Ω |∇φ|2 dx+ a2 2 ∫ Ω |∆φ|2 dx− ∫ Ω u2φdx. Proof. For every u ∈ B, we define the linear functional Lu : v ∈ H 7→ ∫ Ω u2v dx ∈ R The Hölder inequality and the Sobolev embedding imply, for v ∈ H, and suitable constants C,C ′ > 0 | ∫ Ω u2v dx| ≤ |u|24|v|2 ≤ C ′|u|24|∇v|2 ≤ C‖u‖2‖v‖a. (2.5) Then, the functional Lu is continuous, and by Riesz’s Theorem, there exists a unique vector, that we denote with Φa(u) ∈ H such that Lu[v] = (Φa(u), v)a = ∫ Ω ∇Φa(u)∇v dx+ a2 ∫ Ω ∆Φa(u)∆v dx, ∀v ∈ H. In other words Φa(u) ∈ H is the unique weak solution of (2.4) and satisfies∫ Ω u2v dx = a2 ∫ Ω ∆Φa(u)∆v dx+ ∫ Ω ∇Φa(u)∇v dx, ∀v ∈ H. (2.6) Finally it is standard to see that Φa(u) is the unique minimizer of Ea. � In particular from (2.6), by taking v = Φa(u), it follows that∫ Ω u2Φa(u) dx = a2 ∫ Ω |∆Φa(u)|2 dx+ ∫ Ω |∇Φa(u)|2 dx = ‖Φa(u)‖2a. (2.7) Since by (2.5) it holds ∫ Ω u2Φa(u) dx ≤ C‖u‖2‖Φa(u)‖a, (2.8) from (2.7) we have the estimate ‖Φa(u)‖a ≤ C‖u‖2. (2.9) Set now Γa := {(u, φ) ∈ H1 0 (Ω)×H : ∂φFa(u, φ) = 0}. Take the level set B = {u ∈ H1 0 (Ω) : |u|2 = 1} and define the map Φa : u ∈ B 7→ Φa(u) ∈ H (2.10) EJDE-2023/66 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 7 where Φa(u) is the unique solution given in Proposition 2.2. Actually Φa(u) = (a2∆2 −∆)−1u2 where (a2∆2 −∆)−1 : H′ → H is the Riesz isomorphism. Proposition 2.3. The map Φa is C1 and Γa is its graph. Proof. By the Sobolev embedding, H1 0 (Ω) ↪→ L6(Ω) is continuous and it is easy to see that the map u 7→ u2 is C1 from H1 0 (Ω) into L3(Ω) which is continuously embedded into H′. Since the operator (a2∆2 −∆)−1 is the Riesz isomorphism it is C1 and then the map Φa, as composition of C1 maps, is C1 too. Finally, the graph of Φa is Gr(Φa) := {(u, φ) ∈M : (a2∆2 −∆)−1u2 = φ}. Note that (u, φ) ∈ Gr(φa) means that (a2∆2−∆)φ = u2 which is equivalent to say that ∂φFa(u, φ) = 0, which in turn is also equivalent to having (u, φ) ∈ Γa. � We are in a position to define the reduced functional Ja(u) := Fa(u,Φa(u)). (2.11) From (2.7) we have a2 4 ∫ Ω |∆Φa(u)|2 dx+ 1 2 ∫ Ω |∇Φa(u)|2 dx = 1 2 ∫ Ω u2Φa(u) dx− a2 4 ∫ Ω |∆Φa(u)|2 dx and hence the functional Ja takes the form Ja(u) = 1 2 ∫ Ω |∇u|2 dx+ a2 4 ∫ Ω |∆Φa(u)|2 dx+ 1 2 ∫ Ω |∇Φa(u)|2 dx − 1 4 ∫ Ω |∇Φa(u)|2 dx = 1 2 ∫ Ω |∇u|2 dx+ a2 4 ∫ Ω |∆Φa(u)|2 dx+ 1 4 ∫ Ω |∇Φa(u)|2 dx . (2.12) Note that the dependence of Ja on a, is “explicit” because of the presence of a2, but also “implicit” via the map Φa. The functional Ja is then bounded from below, by Proposition 2.3, C1. Then, the Fréchet derivative of Ja at u is given by J ′a(u) = ∂uFa(u,Φa(u)) + ∂φFa(u,Φa(u))Φ′a(u) = ∂uFa(u,Φa(u)) (2.13) as linear and continuous operators on H1 0 (Ω). Taking into account (2.2) we obtain J ′a(u)[v] = ∫ Ω ∇u∇v dx+ ∫ Ω uvΦa(u) dx, ∀v ∈ H1 0 (Ω). (2.14) Recall by Theorem 2 that we are reduced to find critical points (ua, φa) of Fa on B × H with the associated Lagrange multiplier ωa. The following is a second variational principle and describes the relation between critical points of Fa on B ×H and critical points of Ja restricted to B. Proposition 2.4. Let (ua, φa) ∈ B×H and ωa ∈ R. The following statements are equivalent. (i) The pair (ua, φa) is a critical point of Fa constrained to B × H having ωa as Lagrange multiplier. 8 L. SORIANO H., G. SICILIANO EJDE-2023/66 (ii) The function ua is a critical point of Ja constrained to B having ωa as Lagrange multiplier and φa = Φa(ua). Proof. Condition (i) means that ∂uFa(ua, φa) = ωaua and ∂φFa(ua, φa) = 0. But then by Proposition 2.3 it has to be φa = Φa(ua) and by (2.13), J ′a(ua) = ωaua. This is exactly (ii). On the other hand, (ii) implies J ′a(ua) = ωaua and (ua,Φa(ua)) ∈ Gr(Φa) and then ∂φFa(ua, φa) = 0. Consequently, again by (2.13), we infer ωaua = J ′a(ua) = ∂uFa(ua,Φa(ua)) so (i) is proved. � In particular the above result says that all the solutions are of type (ωa, ua,Φa(u)). In view of the previous result, for brevity we may refer just to the unknown u as a solution of the system (ω and φ are then univocally determined), and Ja to its energy. 2.2. Properties of the functional Ja. A useful tool in critical point theory to obtain the compactness is the well known Palais-Smale condition that we recall now. We say that Ja satisfies the Palais-Smale condition on the manifold B ⊂ H1 0 (Ω) if any sequence {wn}n ⊂ B such that {Ja(wn)}n is bounded and J ′a(wn)→ 0 in Twn B, called also a Palais-Smale sequence, has a convergent subsequence in the H1 0 (Ω) norm to some element w (which is then necessarily in B). Lemma 2.5. The functional Ja constrained to B satisfies the Palais-Smale condi- tion. Proof. Let {wn}n ⊂ B be a Palais-Smale sequence for Ja. Then, there exist two sequences {λn}n ⊂ R and {εn}n ⊂ H−1(Ω), where H−1(Ω) is the dual space of H1 0 (Ω), such that εn → 0 and, see (2.12), J ′a(wn) = λnwn + εn, (2.15) Ja(wn) = 1 2 ‖wn‖2 + 1 4 ‖Φa(wn)‖2a → c. (2.16) In particular {wn}n and {Φa(wn)}n are bounded in H1 0 (Ω) and H, respectively. By (2.14) and (2.15) we obtain∫ Ω |∇wn|2 dx+ ∫ Ω w2 nΦa(wn) dx = λn + εn[wn]. Using the boundedness of {wn}n, (2.8) and the fact that εn → 0, we see that also {λn}n has to be bounded. Equation (2.15) is rewritten as −∆wn + wnΦa(wn) − λnwn = εn and applying the inverse Riesz isomorphism ∆−1 : H−1(Ω)→ H1 0 (Ω), we obtain that wn = ∆−1(wnΦa(wn))− λn∆−1wn −∆−1εn, (2.17) EJDE-2023/66 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 9 and {∆−1εn}n is a convergent sequence. Now {wnΦa(wn)}n is bounded in L2(Ω) because of the estimates∫ Ω |wnΦa(wn)|2 dx ≤ |Φa(wn)|24|wn|24 ≤ C‖Φa(wn)‖2a|wn|24. Then {wnΦa(wn)}n is also bounded in H−1(Ω). Actually since ∆−1 is compact, we deduce that (up to subsequences){ ∆−1(wnΦa(wn)) } n , { λn∆−1wn } n are convergent. Going back to (2.17), we infer that {wn}n is convergent (up to subsequences) in H1 0 (Ω), and the limit is of course in B. � Let us recall also some basic facts about Genus Theory. Let A be a closed and symmetric subset A of a Banach space. The set A has genus n ∈ N, denoted by γ(A) = n, if there exists an odd map h ∈ C(A,Rn \ {0}) and n is the smallest integer having this property. If A = ∅, we say that γ(A) = 0 and if there is not any integer satisfying the property, we set γ(A) =∞. Lemma 2.6. For any integer m there exists a compact and symmetric subset K of B such that γ(K) = m. Proof. Let Hm := span{u1, . . . , um} be a m-dimensional subspace of H1 0 (Ω). Define K := B ∩Hm = {u ∈ Hm : |u|2 = 1}. We consider the odd homeomorphism h : K→ Sm−1 defined by h(u) = x ‖x‖Rm , where x = (x1, . . . , xm) ∈ Rm. By the genus invariance via odd homeomorphism (see e.g. [14, Proposition 5.4]), we obtain γ(K) = γ(Sm−1) = m. The proof is complete. � The next result is well known in critical point theory, however we revise the argument. Lemma 2.7. For any c ∈ R the sublevel set Jca := {u ∈ B : Ja(u) ≤ c} has finite genus. Proof. Suppose by contradiction that there exists a real number c such that γ(Jca) = ∞. This means that D := {b ∈ R : γ(Jba) =∞} 6= ∅. We know that Ja is bounded from below on B, hence −∞ < b∗ := inf D <∞. We claim that b∗ /∈ D. Indeed, since Ja satisfies the Palais-Smale condition on B (Lemma 2.5), the set Kb∗ := {u ∈ B : Ja(u) = b∗, J |′B(u) = 0} is compact. By properties of the genus (see [14, Proposition 5.4]), there exists a closed symmetric neighborhood Z of Kb∗ such that γ(Z) <∞, then b∗ /∈ D. 10 L. SORIANO H., G. SICILIANO EJDE-2023/66 By the deformation lemma (see [14, Theorem 3.11]), there exist ε > 0 and an odd homeomorphism η such that η(1, Jb ∗+ε a \Z) ⊂ Jb∗−εa . Using properties (2), (3) and (5) of [14, Proposition 5.4], we obtain γ(Jb ∗+ε a ) ≤ γ(Jb ∗+ε a \ Z) + γ(Z) ≤ γ(Jb ∗−ε a ) + γ(Z) <∞, which goes against the fact that b∗ is equals to inf D. Therefore for all c ∈ R it has to be γ(Jca) <∞. � 3. Proof of the main results Proof of Theorem 1.2. We show that for any a > 0, the functional Ja restrict to B has infinitely many critical points. Let n be a positive integer. By Lemma 2.7, there exists a positive integer k = k(a, n) such that γ(Jna ) = k. Now, consider the collection Ak+1 := {A ⊂ B : A is symmetric and closed with γ(A) ≥ k + 1}. (3.1) By Lemma 2.6, there exists a compact set K ⊂ B such that K ∈ Ak+1, then Ak+1 6= ∅. Since by the definition, γ(A) > γ(Jna ), for all A ∈ Ak+1, by the monotonicity property of genus A 6⊂ Jna , it follows that sup Ja(A) > n, for all A ∈ Ak+1. Consequently ba,n := inf{sup Ja(A) : A ∈ Ak+1} ≥ n. We know by Lemma 2.5 that Ja satisfies the Palais-Smale condition on B and it is an even functional. Then it follows from [14, Theorem 5.7] that ba,n is a critical value of Ja on B, achieved on some ua,n ∈ B. By the Lagrange multipliers theorem, for any n ∈ N there exist ωa,n ∈ R such that J ′a(ua,n) = ωa,nua,n with Ja(ua,n) = ba,n ≥ n. Now evaluating J ′a(ua,n) = ωa,nua,n on the same ua,n we find that 1 2 ∫ Ω |∇ua,n|2 dx+ 1 2 ∫ Ω Φa(ua,n)u2 a,n dx = 1 2 ωa,n. (3.2) In particular ωa,n > 0. Replacing the above equation in the functional given by Ja(ua,n) = 1 2 ∫ Ω |∇ua,n|2 dx+ a2 4 ∫ Ω |∆Φa(ua,n)|2 dx+ 1 4 ∫ Ω |∇Φa(ua,n)|2 dx, (3.3) we have ba,n = Ja(ua,n) = 1 2 ωa,n − 1 4 ∫ Ω |∆Φa(ua,n)|2 dx− a2 4 ∫ Ω |∇Φa(ua,n)|2 dx or ωa,n = 2ba,n + a2 2 ∫ Ω |∆Φa(ua,n)|2 dx+ 1 2 ∫ Ω |∇Φa(ua,n)|2 dx > 2n (3.4) EJDE-2023/66 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 11 which shows that ωa,n →∞ as n→∞. We note that (3.4) implies also that ωa,n ≥ 1 2 ‖Φa(ua,n)‖2a. Recalling (2.9), we rewrite (3.2) as ωa,n = ∫ Ω |∇ua,n|2 dx+ a2 ∫ Ω |∆Φa(ua,n)|2 dx+ ∫ Ω |∇Φa(ua,n)|2 dx = ‖ua,n‖2 + ‖Φa(ua,n)‖2a ≤ ‖ua,n‖2 + C‖ua,n‖4 and then ‖ua,n‖ → ∞ as n→∞. Summing up, for any a > 0 fixed, we have found, for any n ∈ N: ua,n ∈ B ⊂ H1 0 (Ω), φa,n := Φa(ua,n) ∈ H, ωa,n ∈ R solutions of (1.1), proving Theorem 1.2. Furthermore the above computations provide the additional information and estimates on the norm of the solutions and the energy levels of the functional: (1) Ja(ua,n) = 1 2‖ua,n‖ 2 + 1 4‖φa,n‖ 2 a ≥ n, (2) ωa,n = ‖ua,n‖2 + ‖φa,n‖2a > 2n, (3) ‖φa,n‖2a ≤ 2ωa,n, (4) ‖φa,n‖a ≤ C‖ua,n‖2. It is well known that ua,1 is the minimum of Ja, for this reason we say that (ωa,1, ua,1, φa,1) is a ground state solution of (1.1). Correspondingly, ba,1 is the ground state level. Observe that since Ja(|u|) = Ja(u), the ground state ua,1 can be assumed positive. The proof complete. � Remark 3.1. Besides ba,n, the functional Ja may have other critical levels. Hence system (1.1) may have solutions other than the ones we found above. Whenever we need, we use the generic notation (ωa, ua,Φa(ua)) for a solution of (1.1), which is not necessarily at a minimax level ba,n, reserving the notation (ωa,n, ua.n, φa,n) for the solutionsfound at the minimax energy level ba,n. In this case, it is still true that the “generic” solutions satisfy (1) Ja(ua) = 1 2‖ua‖ 2 + 1 4‖Φa(ua)‖2a > 0, (2) ωa = ‖ua‖2 + ‖Φa(ua)‖2a > 0, (3) ‖Φa(ua)‖2a ≤ 2ωa, (4) ‖Φa(ua)‖a ≤ C‖ua‖2. Being solutions, they of course satisfy J ′a(ua) − ωaua = 0 in H−1(Ω). These facts will be used later. Proof of Theorem 1.3. Let us consider the classical Schrödinger-Poisson system in Ω given by −∆u+ φu = ωu in Ω −∆φ = u2 in Ω u = φ = 0 on ∂Ω∫ Ω u2 dx = 1. (3.5) 12 L. SORIANO H., G. SICILIANO EJDE-2023/66 Note that, when a = 0 system (1.1) reduces formally to system (3.5). In this sense (3.5) can be seen as the “limit” problem of (1.1). Benci and Fortunato in [3, Theorem 1], obtained multiple solutions for the Schrödinger-Poisson system. They study the problem by variational methods by considering the functional on H1 0 (Ω)×H1 0 (Ω) F0(u, φ) = 1 2 ∫ Ω |∇u|2 dx+ 1 2 ∫ Ω φu2 dx− 1 4 ∫ Ω |∇φ|2 dx. (3.6) Denoted by Φ0 : u ∈ B 7→ Φ0(u) ∈ H1 0 (Ω) the map which assigns to u the unique solution of the second equation in (3.5) satisfying Φ0(u) = 0 on ∂Ω, they reduced to find critical points of J0(u) = 1 2 ∫ Ω |∇u|2 dx+ 1 4 ∫ Ω |∇Φ0(u)|2 dx on B = {u ∈ H1 0 (Ω) : |u|2 = 1}. In this case J ′0(u)[v] = ∫ Ω ∇u∇v dx+ ∫ Ω uvΦ0(u) dx, ∀v ∈ H1 0 (Ω) (3.7) and J0 satisfies the Palais-Smale condition. Then, by applying the genus index theory, they find infinitely many critical points, denoted hereafter coherently with {u0,n}n ⊂ B. To any u0,n are associated Lagrange multipliers on ω0,n ∈ R and φ0,n := Φ0(u0,n) in such a way that {(ω0,n, u0,n, φ0,n)}n are solution of (3.5), namely J ′0(u0,n)− ω0,nu0,n = 0 in H−1(Ω), or∫ Ω ∇u0,n∇v dx+ ∫ Ω u0,nvφ0,n dx− ω0,n ∫ Ω u0,nv dx = 0, ∀v ∈ H1 0 (Ω). Moreover b0,n := J0(u0,n)→ +∞, ‖u0,n‖ → +∞, ω0,n → +∞ as n→∞, and the critical values are characterized by b0,n = inf{sup J0(A) : A ∈ Ak+1}, Ak+1 as in (3.1). In particular u0,1 is the minimum of J0 on B and b0,1 the ground state level. Also in this case the solutions are classical and it follows that ∆φ0,n = 0 on the boundary ∂Ω and φ0,n ∈ H = H1 0 (Ω) ∩H2(Ω). For all these facts see [3]. We denoted by (ω0,n, u0,n, φ0,n) the solutions of (3.5) obtained with the genus index theory, then characterized by the levels b0,n above. Again, as in Remark 3.1, since J0 may have also other critical levels, we denote with (ω0, u0,Φ0(u0)) a generic solution of (3.5), then not necessarily at the minimax level b0,n for J0. It is obvious now that, if a > 0, systems (1.1) and (3.5) can not have the same solutions, then J ′0(ua)− ω0ua 6= 0 and J ′a(u0)− ωau0 6= 0 (as operators on H1 0 (Ω)). In particular this happens for the solutions obtained at the minimax levels: ua,n is not a critical point of J0, as well as u0,n is not a critical point of Ja. The following result is fundamental for the convergence of the solutions of the second equation of systems (3.5) and (1.1). EJDE-2023/66 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 13 Lemma 3.2. For a fixed v ∈ H1 0 (Ω) let Φ0(v) and Φa(v) be the unique solutions of −∆φ = v2 in Ω φ = 0 on ∂Ω and of −∆φ+ a2∆2φ = v2 in Ω ∆φ = φ = 0 on ∂Ω, respectively. Then, as a→ 0 we have (up to subsequences) Φa(v)→ Φ0(v) in H1 0 (Ω) and a∆Φa(v)→ 0 in L2(Ω). Proof. We already know that, since the solutions are classical, Φa(v),Φ0(v) ∈ H. From |∇Φa(v)|22 + a2|∆Φa(v)|22 = ∫ Ω v2Φa(v) dx ≤ C|v|24|∇Φa(v)|2 we see that {Φa(v)}a∈(0,1] is bounded in H1 0 (Ω). Then there exists φ ∈ H1 0 (Ω) such that Φa(v) ⇀ φ in H1 0 (Ω) and strongly in Lp(Ω), p ∈ [1, 6). Going back in the equality above we deduce that {|a∆Φa(v)|2}a∈(0,1] is bounded (in fact, convergent). In particular lim a→0 a ∫ Ω a∆Φa(v)ζ dx = 0 ∀ζ ∈ L2(Ω). (3.8) Then for every ξ ∈ C∞0 (Ω), passing to the limit in the equality∫ Ω ∇Φa(v)∇ξ dx+ a ∫ Ω a∆Φa(v)∆ξ dx = ∫ Ω v2ξ dx, we infer that ∫ Ω ∇φ∇ξ dx = ∫ Ω v2ξ dx and then, by unicity, that φ = Φ0(v). Finally, |∇Φ0(v)−∇Φa(v)|22 + |a∆Φa(v)|22 = |∇Φ0(v)|22 − 2 ∫ Ω ∇Φa(v)∇Φ0(v) dx+ |∇Φa(v)|22 + |a∆Φa(v)|22 = |∇Φ0(v)|22 − 2 ∫ Ω ∇Φa(v)∇φ0(v) dx+ ∫ Ω v2Φa(v) dx → −|∇Φ0(v)|22 + ∫ Ω v2Φ0(v) dx = 0 which shows that Φa(v)→ Φ0(v) in H1 0 (Ω) and a∆Φa(v)→ 0 in L2(Ω). � Now we can study the behavior of the generic solutions of (1.1) whenever a tends to zero. Roughly speaking it says that if we have a priori bound, then there is compactness for the solutions. Proposition 3.3. Let {(ωa, ua,Φa(ua))}a>0 ∈ R×B ×H be solutions of (1.1). If {ua}a∈(0,1] is bounded in H1 0 (Ω), then as a→ 0 (up to subsequence), ua → u0 and Φa(ua)→ Φ0(u0) in H1 0 (Ω), ωa → ω0 in R, where (ω0, u0,Φ0(u0)) ∈ R×B ×H1 0 (Ω) is a solution of (3.5). Moreover the following convergences hold: 14 L. SORIANO H., G. SICILIANO EJDE-2023/66 (i) a∆Φa(ua)→ 0 in L2(Ω), (ii) Ja(ua), J0(ua), Ja(u0)→ J0(u0), (iii) J ′a(u0)− ωau0, J ′ a(ua)− ω0ua, J ′ a(u0)− ω0u0 → 0 in H−1(Ω), (iv) J ′0(ua)− ω0ua, J ′ 0(ua)− ωaua, J ′0(u0)− ωau0 → 0 in H−1(Ω). The limits in (iv) say that mixing the solutions of (1.1) and (3.5), we obtain almost solution of the limit problem: the triples (ω0, ua,Φ0(ua)), (ωa, ua,Φ0(ua)) and (ωa, u0,Φ0(u0)) are almost solution of (3.5). Proof. The boundedness of {ua}a∈(0,1] implies from (2.9) the boundedness of the sequence {‖Φa(ua)‖a}a∈(0,1], then of the sequences {|∇Φa(ua)|2}a∈(0,1] and of the sequence {|a∆Φa(ua)|2}a∈(0,1]. Therefore there exists u ∈ H1 0 (Ω) and φ ∈ H1 0 (Ω) such that, as a→ 0, ua ⇀ u, Φa(ua) ⇀ φ in H1 0 (Ω). (3.9) It follows that |∇φ|22 ≤ lim inf a→0 |∇Φa(ua)|22. (3.10) From (3.9), and using the compact Sobolev embeddings, for any ξ ∈ C∞0 (Ω), we have ∫ Ω u2 aξ dx→ ∫ Ω u2ξ dx , ∫ Ω ∇Φa(ua)∇ξ dx→ ∫ Ω ∇φ∇ξ dx and, for a suitable C > 0,∣∣ ∫ Ω a∆Φa(ua)∆ξ dx ∣∣ ≤ |∆Φa(ua)|2|∆ξ|2 ≤ C. We conclude, passing to the limit as a→ 0 in the equality∫ Ω ∇Φa(ua)∇ξ dx+ a2 ∫ Ω ∆Φa(ua)∆ξ dx = ∫ Ω u2 aξ dx, that ∫ Ω ∇φ∇ξ dx = ∫ Ω u2ξ dx. (3.11) Moreover for ua a solution, using (2.8), we infer 0 < ωa = |∇ua|22 + ∫ Ω Φa(ua)u2 a dx ≤ |∇ua|22 + C‖ua‖2‖Φa(ua)‖a and then {ωa}a∈(0,1] is bounded too, and we can assume ωa → ω. We know also that for any ξ ∈ C∞0 (Ω),∫ Ω ∇ua∇ξ dx+ ∫ Ω Φa(ua)uaξ dx = ωa ∫ Ω uaξ dx and passing to the limit as a → 0, using that ua → u,Φa(ua) → φ in L2(Ω), we obtain ∫ Ω ∇u∇ξ dx+ ∫ Ω φuξ dx = ω ∫ Ω uξ dx. (3.12) By density, (3.11), and (3.12) we deduce that (ω, u, φ) is a solution of the (3.5) system, then we can rename it (ω0, u0,Φ0(u0)) and we have proved that ua ⇀ u0, Φa(ua) ⇀ Φ0(u0) in H1 0 (Ω) and ωa → ω0. EJDE-2023/66 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 15 The strong convergence of {ua}a∈(0,1] is actually a consequence of the compact- ness, because of the boundedness of the domain. Since∫ Ω |Φa(ua)ua|2 dx ≤ |Φa(ua)|24|ua|24 ≤ C‖Φa(ua)‖2a|ua|24 ≤ C, from −ua + ∆−1(Φa(ua)ua) = ωa∆−1ua, using the compactness of ∆−1, we see that indeed {ua}a∈(0,1] has to be convergent in H1 0 (Ω), and the limit is necessarily u0. Let us pass to the strong convergence of {Φa(ua)}a∈(0,1] in H1 0 (Ω). We know that Φa(ua) minimizes the functional Ea(φ) = 1 2 |∇φ|22 + a2 2 |∆φ|22 − ∫ Ω u2 aφdx and then if {ξn}n ⊂ C∞0 (Ω) is such that ξn → Φ0(u0) in H1 0 (Ω) as n → ∞, we obtain Ea(Φa(ua)) ≤ Ea(ξn), namely 1 2 |∇Φa(ua)|22 ≤ 1 2 |∇Φa(ua)|22 + a2 2 |∆Φa(ua)|22 ≤ 1 2 |∇ξn|22 + a2 2 |∆ξn|22 − ∫ Ω u2 aξn dx+ ∫ Ω u2 aΦa(ua) dx. (3.13) Observe that lim a→0 ∫ Ω u2 aξn dx = ∫ Ω u2 0ξn dx and lim a→0 ∫ Ω u2 aΦa(ua) dx = ∫ Ω u2 0Φ0(u0) dx . Then from (3.13) we obtain lim sup a→0 1 2 |∇Φa(ua)|22 ≤ 1 2 |∇ξn|22 − ∫ Ω u2 0ξn dx+ ∫ Ω u2 0Φ0(u0) dx. Passing to the limit in n in the above inequality we deduce lim sup a→0 1 2 |∇Φa(ua)|22 ≤ 1 2 |∇Φ0(u0)|22 that joint with (3.10) gives |∇Φa(ua)|2 → |∇Φ0(u0)|2 and so Φa(ua) → Φ0(u0) in H1 0 (Ω). The strong convergence to a solution of the (3.5) system is proved. As a consequence, as a→ 0, |a∆Φa(ua)|22 = ∫ Ω u2 aΦa(ua) dx− |∇Φa(ua)|22 → ∫ Ω u2 0Φ0(u0) dx− |∇Φ0(u0)|22 = 0 proving (i). Clearly, by (i) and the above strong convergence, it is Ja(ua) = 1 2 |∇ua|22 + a2 2 |∆Φa(ua)|22 + 1 4 |∇Φa(ua)|22 → 1 2 |∇u0|22 + 1 4 |∇Φ0(u0)|22 = J0(u0). By using the continuity of the map Φ0 we obtain J0(ua) = 1 2 |∇ua|22 + 1 4 |∇Φ0(ua)|22 → 1 2 |∇u0|22 + 1 4 |∇Φ0(u0)|22 = J0(u0). 16 L. SORIANO H., G. SICILIANO EJDE-2023/66 Moreover, by Lemma 3.2 with v := u0 we have Ja(u0) = 1 2 |∇u0|22 + a2 2 |∆Φa(u0)|22 + 1 4 |∇Φa(u0)|22 → 1 2 |∇u0|22 + 1 4 |∇Φ0(u0)|22 = J0(u0) and these last three limits prove (ii). The proof of the limits in (iii) and (iv) follows the same lines we use until now: just use Lemma 3.2 with v := u0, the strong convergence of the solutions proved above and (i). As an example let us verify just the first limit in (iii). For any ξ ∈ C∞0 (Ω), using Lemma 3.2 with v = u0, we have J ′a(u0)[ξ] = ∫ Ω ∇u0∇ξ dx+ a2 ∫ Ω ∆Φa(u0)∆ξ dx+ ∫ Ω ∇Φa(u0)∇ξ dx → ∫ Ω ∇u0∇ξ dx+ ∫ Ω ∇Φ0(u0)∇ξ dx = J ′0(u0)[ξ]. By density the convergence is true for any v ∈ H1 0 (Ω). Since it is also easy to see that the limit is uniform in v and ωa → ω0, we have J ′a(u0)−ωau0 → J ′0(u0)−ω0u0 = 0, being u0 a critical point of J0 on B with Lagrange multiplier ω0. The proof is then complete. � Remark 3.4. In addition to the convergence Ja(u0) → J0(u0), we have further information. By (2.1) and (3.6), for any a > 0, u ∈ H1 0 (Ω), and φ ∈ H, we have Fa(u, φ) < F0(u, φ). Then if u0 is a critical point of J0, since Φ0(u0) ∈ H, we infer that Ja(u0) = Fa(u0,Φ0(u0)) < F0(u0,Φ0(u0)) = J0(u0) . We stress the fact that in Proposition 3.3 a fundamental assumption has been the a priori bound, namely the boundedness of {ua}a∈(0,1]. In particular Proposition 3.3 and Remark 3.4 hold for the solutions of Theorem 1.2. We state for convenience explicitly the result for n fixed. Corollary 3.5. Fixed n∗ ∈ N, let {(ωa,n∗ , ua,n∗ , φa,n∗)}a>0 ∈ R×B×H be solutions of (1.1) found in Theorem 1.2. If {ua,n∗}a∈(0,1] is bounded in H1 0 (Ω), then as a→ 0 (up to subsequence) ua,n∗ → u0 and φa,n∗ → Φ0(u0) in H1 0 (Ω), ωa,n∗ → ω0 in R. where (ω0, u0,Φ0(u0)) ∈ R×B ×H1 0 (Ω) is a solution of (3.5). Moreover the following convergences hold: (i) a∆Φa,n∗ → 0 in L2(Ω), (ii) Ja(ua,n∗), J0(ua,n∗), Ja(u0)→ J0(u0), and Ja(u0) < J0(u0), (iii) J ′a(u0)− ωa,n∗u0, J ′ a(ua,n∗)− ω0ua,n∗ , J ′a(u0)− ω0u0 → 0 in H−1(Ω), (iv) J ′0(ua,n∗)−ω0ua,n∗ , J ′0(ua,n∗)−ωa,n∗ua,n∗ , J ′0(u0)−ωa,n∗u0 → 0 in H−1(Ω). Remark 3.6. By (2.9), we see that the boundedness of {ua,n∗}a∈(0,1] in H1 0 (Ω) is equivalent (i) by (3.2), to require that {ωa,n∗}a∈(0,1] be bounded; or (ii) by (3.3), to require that {Ja(ua,n∗)}a∈(0,1] be bounded. This fact will be important in the proof of Theorem 1.3. An analogous observation can be made for the generic solutions (ωa, ua,Φa(ua)), however we will not use it. EJDE-2023/66 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 17 Two natural questions arise from Corollary 3.5: (1) in which case the solutions {ua,n∗}a∈(0,1] are bounded? (2) Even if they are bounded, then by the limit in (ii) in the Corollary, ba,n∗ → J0(u0), can we say that J0(u0) = b0,n∗? In other words, does the minimax levels converge to the respective minimax levels? In case n∗ = 1, namely in case of ground state solutions, we can give a positive answer to both questions: not only the solutions are automatically bounded as a goes to zero, but the limit is a ground state solution of (3.5), i.e. ba,1 → b0,1. In fact we can give the proof of Theorem 1.3. Let ua,1 be the ground state of Ja, and u0,1 the ground state of J0. We have Ja(ua,1) ≤ Ja(u0,1) < J0(u0,1) = b0,1, (3.14) where the strict inequality is due to Remark 3.4 replacing the generic solution u0 of (3.5) with the particular one u0,1. Then lim sup a→0 Ja(ua,1) ≤ b0,1 (3.15) and by (ii) of Remark 3.6 we have the boundedness of {ua,1}a∈(0,1] in H1 0 (Ω), the a priori bound we were looking for. Corollary 3.5 gives, as a→ 0, ua,1 → u0, φa,1 → Φ0(u0), ωa,1 → ω0 and (ω0, u0,Φ0(u0)) ∈ R × B ×H1 0 (Ω) solves (3.5). We do not know yet if u0 is a minimum of J0. However by the first limit in (ii) in Corollary 3.5 and (3.15), ba,1 = Ja(ua,1)→ J0(u0) ≤ b0,1 On the other hand it holds b0,1 ≤ J0(u0), so that, as a tends to zero, ba,1 → b0,1 and u0 is a minimum of J0 on B. The proof of Theorem 1.3 is complete. We conclude by saying that for the other solutions {ua,n∗}a∈(0,1] which are not at the ground state level, namely for n∗ 6= 1, although it is always true that (see Remark 3.4), Ja(u0,n∗) < J0(u0,n∗) = b0,n∗ , (3.16) we cannot guarantee the first inequality in (3.14), i.e. Ja(ua,n∗) ≤ Ja(u0,n∗), which would give, joint with (3.16), the boundedness Ja(ua,n∗) < b0,n∗ . We leave this as an interesting open problem. Acknowledgments. L. Soriano H. was supported by CNPq, proc. n. 160953/2022- 7, 465591/2014-0 INCT de Matemática (Brazil) G. Siciliano was supported by Capes, CNPq, FAPDF Edital 04/2021 - Demanda Espontânea, Fapesp grant no. 2022/16407-1 (Brazil) and Indam (Italy). References [1] R. A. Adams, J. Fournier; Sobolev Spaces, Elsevier, Academic Press, 2003. [2] D. G. Afonso, G. Siciliano; Normalized solutions to a Schrödinger-Bopp-Podolsky system under Neumann boundary conditions, Commun. Contemp. Math., 25 No. 2 (2023) 2150100. 20 pages. [3] V. Benci, D. Fortunato; An Eigenvalue Problem for the Schrödinger-Maxwell Equations, Topol. Methods Nonlinear Anal., (1998), vol 11, 283-293. [4] P. d’Avenia and G. Siciliano; Nonlinear Schrödinger equation in the Bopp-Podolsky elec- trodinamics: Solutions in the electrostatic case, J. Differential Equations, (2019), vol 269, 1025-1065. [5] G. M. Figueiredo, G. 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Lorena Soriano Hernandez Departamento de Matemática, Universidade de Braśılia - UnB, 70910-900, Campus Uni- versitário Darcy Ribeiro, Asa Norte, Braśılia, DF, Brazil Email address: loresohe199@gmail.com Gaetano Siciliano Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil Email address: sicilian@ime.usp.br 1. Introduction 1.1. Proof of Theorem 1.1 2. Variational setting 2.1. Reduced functional 2.2. Properties of the functional Ja 3. Proof of the main results Proof of Theorem 1.3 Acknowledgments References