Multiplicity and concentration results for magnetic relativistic Schrödinger equations

where ε > 0 is a parameter, m > 0, N ≥ 1, V : RN → R is a continuous scalar potential satis es V(x) ≥ −V0 > −m for any x ∈ RN and f : RN → R is a continuous function. Under a local condition imposed on the potential V, we discuss the number of nontrivial solutions with the topology of the set where the potential attains its minimum. We proof our results via variational methods, penalization techniques and LjusternikSchnirelmann theory.


Introduction and statement of main results
In this paper, we consider the mean eld limit of a quantum system with rest mass m > in the presence of a magnetic vector potential A(x) and an electric potential V(x). More precise, we focus our attention on the following time-depend pseudo-relativistic magnetic Schrödinger equation where ε > is a small positive constant, i is the imaginary unit, m > , N ≥ , ψ : R N × R → C is a wave eld, A : R N → R N is a continuous vector potential, V : R N → R is an external continuous scalar potential and function f : R N → R. The magnetic relativistic Schrödinger operator relate to the classical relativistic Hamiltonian symbol in Fourier variables which is the sum of the kinetic energy term. This operator is known as a spinless particle in electromagnetic elds where we ignore quantum eld theoretic e ect like particles creation and annihilation but should take relativistic e ect into consideration, see [1,2]. We should remark that there are three type of relativistic Hamiltonian depending on how we quantize the kinetic energy symbol ξ − A(x) + m . The rst two quantized operators de ned by mean formulas, that is, for any function φ ∈ C ∞ (R N , C), We note that the Weyl pseudo-di erential operator H A is not covariant under gauge transformations, that is, H A+∇ϕ ≠ e iϕ H A e −iϕ . The operator H A is a modi cation of operator H A , which is gauge covariant, see [3].
The third quantized H A is the square of the nonnegative selfadjoint operator −i∇ − A(x) + m , that is, The operator H A is gauge covariant and is used in the description of the stability of the matter in relativistic quantum mechanics, see for example [4,5]. All three quantized operators are di erent from one another (see [1,6]). As we know that they coincide if A(x) is linear, that is, A(x) = A · x, with A is a real symmetric constant matrix, see [1]. Particularly, this holds for constant magnetic eld when N = , that is, B = ∇ × A is constant. A solution of the form ψ(x, t) = e iEt/ε u(x) is called a solitary wave. Then ψ(x, t) is a solution of (1.1) if and only if the function u satis es where we write V instead of V + (E − m) for simplicity.
Recently, Cingolani and Secchi in [7] studied the interwining solutions of magnetic relativistic Hartree type equations, that is, where ≤ p < N/(N − ) and (N − )p − N < α < N. Their proofs are based on the variational methods and Ca arelli and Silvestre's type extension (see [8]) for pseudo-relativistic magnetic Schrödinger operator −i∇ − A(x) + m + V(x) when A(x) is uniformly bounded or linear in x. If N = and α = p = , which corresponds to the Coulomb kernel, equation (1.3) is often referred to a boson star in astrophysics, see for example [9,10]. If also assume A ≡ and V(x) = −m, equation (1.3) is reduced to the classical pseudorelativistic Hartree equation which introduced by Lieb and Yau [11], see also [12][13][14] and references therein.
In the literature, the existence of standing waves solutions to nonlinear magnetic Schrödinger equation (1.4) has been rst studied by Lions and Esteban [15], for ε > xed and special classes of magnetic elds. They have found existence results by solving appropriate minimization problems and concentration-compactness method for the corresponding energy functional in the cases N = and 3. Lately, Kurata [16] studied the existence of a least energy solution of (1.3) under a condition relating V(x) and A(x); Cingolani [17] and Alves et al. [18] investigated the multiplicity results of (1.3) by applying the Ljusternik-Schnirelmann theory. We refer readers to [17,[19][20][21] and references therein for other results about nonlinear magnetic Schrödinger equation.
For the nonlocal magnetic Schrödinger equations have been investigated recently. The fractional magnetic Laplacian is de ned by which is deduced from the magnetic operator H A for smooth functions u. In quantum mechanics, when ε → , the existence and concentration of solution is of particular importance. The existence and concentration results for fractional magnetic Schrödinger equations were studied by Ambrosio and d'Avenia [22], Fiscella, Pinamonti and Vecchi [23], Zhang, Squassina and Xia [24], Mao and Xia [25]. We also refer to d'Avenia and Squassina [26] for the existence of ground states and other useful estimates. Lastly, for the existence and multiplicity results of semilinear or quasilinear Schrödinger equations, we refer readers to [27][28][29] and references therein. Motivated by the about results, in this paper we deal with multiplicity and concentration results of the more general class of pseudo-relativistic magnetic Schrödinger equation (1.2) . In what follows, on potentials we assume that (A) A : R N → R is a continuous functions and uniformly bounded.
Also, we suppose continuous function f satisfying (f3) There exists a constant θ ∈ ( , ) such that We shall establish a relation between the number of solutions of (1.2) and the topology of the set M. In order to make a precise statement let us recall that, for any closed subset Y of a topological space X, the Ljusternik-Schnirelmann category of Y in X, cat X (Y), stands for the least number of closed and contractible sets in X which cover Y.
The main result of this article is there exists ε δ > such that problem (1.2) has at least cat M δ (M) solutions provides ε ∈ ( , ε δ ). Moreover, if uε denotes one of these solutions and ηε ∈ R N its global maximum, then It should be pointed out that we only assume the potential V(x) satis es local conditions (V ) − (V ) and no information on the behavior of the potential V(x) at in nity, so we will use the penalization method introduced by del Pino and Felmer [30] rather than minimax theorem to prove our main results. It is worthwhile to remark that in the arguments developed in [30], one of the key points is the existence of estimates involving the L ∞ -bounds of the modi ed problem. Here we obtain the desired L ∞ -bounds via Moser's iteration method (see [31]) instead of Kato's inequality. Moreover, we get the multiplicity results by Ljusternik-Schnirelmann theory (see [32]). As far as we known, this is the rst time that penalization scheme and topological arguments are combined to get multiple solutions for magnetic pseudo-relativistic equations.
We also remark that we assume the nonlinearity term f is only continuous, so we can not use the standard arguments on the Nehari manifold. To overcome the non-di erentiability for the Nehari manifold, we shall use some variants of critical point theorems from Szulkin and Weth [32]. This idea has been used extensively for nonlocal elliptic problems, see for example [33,34].
Our proof based on the Ca arelli and Silvestre's type extension (see [8]) for pseudo-relativistic magnetic is uniformly bounded, which is prove by Cingolani and Secchi in [7]. However, some di culties appear since the nonlinearity is on the boundary. In particular, in order to obtain the L ∞ -bounds in Section 4 we will establish an inverse Hölder inequality for γ(w) = u and we my iterate the inequality for γ(w).
This paper is organized as follows. In section 2, we present the variational setting of the original and the extended variables problems, and we modify the original problem. We also prove the Palais-Smale condition for the modi ed functional and obtain some tools which are useful to establish a multiplicity result. In section 3, we study the autonomous problem associated which allow us to prove the modi ed problem has multiple solutions. Finally, we prove Theorem 1.1 via Morse iteration method.

Extension and modi ed problem
In this paper, we will systematically consider spaces of complex-valued functions. Precisely, the L (R N , C) space will be endowed with the real scalar product In what follows, we will write | · |p for the norm in L p (R N ) and · p for the norm in L p (R N+ + ). Moreover, for any w ∈ H (R N+ + , C), we denote LetÃ(x, y) = (A(x), ) : R N+ + → R N+ be the trivial lifting of a vector eld A(x) : R N → R N for every (x, y) ∈ R N+ + . Then, we de ne the magnetic Sobolev spaces on the half-space H Ã (R N+ + , C) as which endowed with the norm For simplicity, we will write H Ã (R N+ + , C) and w H Ã as H A (R N+ + , C) and w H A respectively.
Next, we recall the following result about trace in magnetic Sobolev space operator which proved in [7].
This result allows us to generalized the well-known Dirichlet-to-Neumann extension for fractional Laplacian to the magnetic pseudo-relativistic operator. Letting ∆Ã = −∇ Ã = ∆ A + ∂ ∂y where then Cingonali and Secchi in [7] showed that Proposition 2.2. For u ∈ H / (R N , C), then there exists one and only one function w ∈ H A (R N+ + , C) such that We remark that the key point of the proof of Proposition 2.2 is to show that magnetic Sobolev spaces H A (R N+ + , C) and H / A (R N , C) are equivalent to H (R N+ + , C) and H / (R N , C) respectively when A(x) is bounded. Therefore, the existence of trace operator follows immediately from the standard theory of Sobolev traces in non-magnetic spaces. Hence, by Proposition 2.1, we deduce that the embeddings are continuous when A is uniformly bounded. By Proposition 2.2, we know that every function w ∈ H A (R N+ + , C) possesses a trace γ(w) ∈ H / (R N , C). Moreover, the following inequality holds provides ≤ p ≤ . For the proofs of (2.5), one can nd in [7]. It is easy to see that problem (1.2) is equivalent, after a change of variable, to the following one where Aε(x) = A(εx) and Vε(x) = V(εx). Once we obtain a solution of (2.6), then the functionũε(x) = uε(x/ε) is a solution of (1.2). Moreover, the maximum ζε ofũε is related to the maximum point zε of uε by ζε = εzε. By applying Proposition 2.2, we are interested to the study of the relativistic magnetic nonlocal equation whereÃε = (Aε , ). We also observe that, for every m > , (2.5) implies that Since there is no information about the in nity of V(x), we adapt the penalization method introduced by del Pino and Felmer [30] to establish the multiplicity results. and where χ O (x) is the characteristic of set O. By the assumptions (f ) − (f ), it is easy to check that g is a Carathéodory function and satis es Therefore, we study the auxiliary problem where gε(x, w) = g(εx, w). Note that solution of (2.10) with w(x) ≤ a for each x ∈ Oε are also the solution of Consider the Euler-Lagrange functional associated to (2.10) given by where ∇Ã ε is de ned as (2.2). Next, we de ne the Nehari manifold Nε (see [35]) related to Iε. We say w ∈ Nε means w ∈ H Aε (R N+ + , C) and satis es We denote by H Aε (R N+ where Sε is the unit sphere of H Aε (R N+ + , C). Note that Sε is a non-complete C , -manifold of codimension 1, modeled on H Aε (R N+ + , C) and contained in the open H Aε (R N+ + , C) (see for example [32]). Then, H Aε (R N+ We can check the functional Iε satis es the Mountain pass geometry [35].
(ii) For every w ∈ H Aε (R N+ + , C) and t > , we can obtain that where we have used (g3) and the standard ODE computations. This and (2.4) imply the conclusion (ii) since θ ∈ ( , ).
Since we only assume f is continuous, in order to overcome the non-di erential of Nε and the in completeness of Sε, we need the following two results. (a) For each w ∈ H Aε (R N+ + , C), let hw : R + → R be de ned as hw(s) = Iε(sw). Then there exists a unique sw > such that h w (s) > in ( , sw) and h w (s) < in (sw , ∞). (b) There is a t > independent of w such that sw > t for all w ∈ Sε. Moreover, for each compact set W ⊂ Sε, there is C W > such that sw ≤ C W for all w ∈ W. (c) The map mε : Sε → Nε given by mε(w) = sw w is continuous and mε := mε| Sε is a homeomorphism between Proof. (a) Observe from the proof of Lemma 2.3 that hw( ) = , hw(s) > for s small and hw(s) < for s large. Thus max s≥ hw(s) is achieved at a s = sw > satisfying h w (sw) = and sw w ∈ Nε. On the other hand, we know By the de nition of g, the right hand side is nondecreasing in s for s > . Therefore, max s≥ hw(s) is achieved at a unique s = sw > such that h w (sw) = and sw w ∈ Nε.
(b) Assume that w ∈ Sε, then by (2.15), (2.8), (2.12) and a similar argument as (2.14), we can get that sw ≥ t for some t > . Suppose by contradiction that there is On the other hand, denote vn = sn wn ∈ Nε and use (2.8) and property (g ), we have that Therefore, we can prove Iε(vn) > since K > V m−V and µ > , which yields a contradiction. (c) We rst show that mε, mε and m − ε are well de ned. In fact, by (a), we know that for each w ∈ H Aε (R N+ On the other hand, we claim that if w ∈ Nε, then w ∈ H Aε (R N+ + , C). Otherwise, we have |supp|γ(w)| ∩ Oε| = and by (2.8) and (g3)-(ii), we have ∈ Sε is well de ned and contin- we conclude that mε is a bijection. Next, we prove mε : Then, by passing to the limit as n → +∞, we have which means that s w ∈ Nε and sw = s . This implies mε(wn) → mε(w) in H Aε (R N+ + , C). So mε and mε are continuous functions.
Then by the Sobolev embedding, there exits constant C t > such that for all m ∈ N and t ∈ [ , ]. By (g1)-(g2), (g3)-(ii) and (2.19), for s > , we have On the other hand, from the de nition of mε, for all s > , we have and denote by Φε := Φε| Sε . A direct conclusion of Lemma 2.4 is the following. Proof. The details of the proof can be found in relevant material from Corollary 2.3 in [32], and we omit it here.
As in [32], we have the following variational characterization of the in mum of Iε under Nε: The main feature of the modi ed functional is that it satis es the local compactness condition, we will show it as follows.
, it follows that I ε (wn), η R wn = on( ). Therefore, Then, xing R > such that Oε ⊂ B R/ and using (g3)-(ii) and Hölder inequality, we can get We get the conclusion by choosing R large, using he boundedness of {wn} proved in Lemma 2.6 and passing to the limit in the last inequality. Proof. By Lemma 2.6, we know {wn} is bounded in H Aε (R N+ + , C) and thus we can suppose that wn w weakly in H Aε (R N+ + , C). In view of I ε (wn) → , the local compactness of H Aε (R N+ + , C) and the subcritical growth of g, one has I ε (w) = , that is (2.21) On the other hand, using Lemma 2.7, we can prove that as n → ∞. Combining (2.21)-(2.24) and I ε (wn) → , we have wn → w strongly in H Aε (R N+ + , C).

Multiplicity result of the modi ed problem
In this section, we prove a multiplicity result for problem (2.10). In what the follows we shall assume that δ > small such that M δ ⊂ O, where O is given in (V2). We start by considering the limit problem related to (2.10), that is, the following problem −∆w + m w = in R N+ + , − ∂w ∂y = V w + f (|w|)w in R N = ∂R N+ + . (3.1) The solutions of equation (3.1) are critical points of the functional given as Next, we de ne the Nehari manifold N related to I . We say w ∈ N means w ∈ H (R N+ + ) and satis es In the sequel, we state without proof of the following Lemma 3.1 and Proposition 3.2. The proofs are similar to those of Lemma 2.4 and Proposition 2.5.

Lemma 3.1. Assume that (V ) and (f ) − (f ) are satis ed. Then the following properties hold: (a) For each w ∈ H (R N+ + ), let hw : R + → R be de ned as hw(s) = I (sw). Then there exists a unique sw > such that h w (s) > in ( , sw) and h w (s) < in (sw , ∞). (b) There is a t > independent of w such that sw > t for all w ∈ S . Moreover, for each compact set W ⊂ S ,
there is C W > such that sw ≤ C W for all w ∈ W. The next lemma allows us to assume the weak limit of a (PS)c sequence is nontrivial. Due to Lions' Lemma (see for example [36]), we have Therefore, by (f1)-(f2), we have Since I (wn), wn → as n → ∞, that is, Proof. By Lemma 2.6, we know that {wn} is bounded in H (R N+ + ). Then, up to a subsequence, wn w weakly in H (R N+ + ) and γ(wn) → γ(w) in L p loc (R N ) and γ(wn) → γ(w) a.e. in R N . By Lemma 3.3, we know that problem (3.1) has a nontrivial ground state.
The next result is a compactness result of problem (3.1) which will be used later. In the following, we will relate the number of nontrivial solution of (2.10). So we consider δ > such that M δ ⊂ O and choose η ∈ C ∞ (R + , [ , ]) satisfying η ≡ in [ , δ ] and η ≡ in [δ, ∞). For any z ∈ M, we de ne and sε > such that max where w is a solution of (3.1) from Theorem 3.4 satisfying I (w) = c . Let Θε : M → Nε be as Θε(z) = sε Ψε,z .
The energy of the above function has the following behavior: Proof. We prove the lemma by contradiction arguments and assume that there is some δ > , {zn} ⊂ M and εn → such that |Iε n (Θε n (zn)) − c | ≥ δ . (3.5) Observe that for each n ∈ N and for all x ∈ B δ/εn ( ), we have εn x ∈ B δ ( ). Then, we have By using the change of variable x := (εn x − zn)/εn, we can write  (w(x , ))|. This yields a contradiction. Therefore, sε n → s ≥ . By (3.7), (2.8) and (f1), we can get s > . Next, we claim that s = . Indeed, by applying the Dominated Convergence Theorem and taking a similar argument as Lemma 3.2 in [17], we have R N+ + |∇x Ψε n ,zn | + ∂Ψε n ,zn ∂y + m Ψ εn ,zn dxdy (3.10) Therefore, by passing the limit in (3.7), we can obtain that On the other hand, since w is a solution of (3.1), we have Iε n (Θε n (zn)) = I (w) = c , which contradicts to (3.5). This completes the proof.
For the δ > given before Lemma 3.6, choose ρ = ρ(δ) > such that M δ ⊂ Bρ( ). De ne χ : Then let us consider the barycenter map βε : Nε → R N given by Since O ⊂ Bρ( ), by the de nition of χ and Lebesgue's Theorem, we conclude that lim ε→ βε(Θε(z)) = z uniformly in z ∈ O. (3.14) The next compactness result is fundamental for proving that the solutions of the modi ed problem are solution of the original problem. Proof. By Lemma 2.6, we know that wn H Aε n ≤ C for n ∈ N. Note that c > , and since wn H Aε n → would imply Iε n (wn) → , we can argue as in Lemma 3.3 to get a subsequence {zn} ⊂ R N and constants R, β > such that lim inf Let vn(x) = wn(x +zn), then {vn} is also bounded and therefore, along a subsequence, we have vn v ≢ weakly in H (R N+ + ). Take tn > such thatṽn := tn vn ∈ N , and set zn = εnzn. Since wn ∈ Nε n , we have where implies limn→∞ I (ṽn) = c . Moreover, {ṽn} is bounded in H (R N+ + ) andṽn ṽ. We may assume that tn → t * > . By the uniqueness of the weak limit, we have thatṽ = t * v ≢ . By Lemma 3.5,ṽn →ṽ in H (R N+ + ), and thus vn → v in H (R N+ + ). Moreover, I (ṽ) = c and I (ṽ),ṽ = .
Next, we prove that {zn} has a bounded subsequence. In fact, suppose by contradiction that |zn| → ∞. Choose R > such that O ⊂ B R ( ). Then for n large enough, we have |zn| > R and for each x ∈ B R/εn ( ) we have Therefore, by vn → v in H (R N+ + ), the above expression, the de nition of g and Lebesgue's theorem, we can get Then, we have vn → in H (R N+ + ), which contradicts with v ≢ . So {zn} is bounded and we can assume that zn → z ∈ R N . If z ∉ O, we can proceed as above to conclude that vn → . Then, we have that z ∈ O.
Then, we can obtain that Iε n (wn) → c . So we can invoke Proposition 3.7 to obtain a sequence {zn} ⊂ R N such that zn = εnzn ∈ M δ and zn → z ∈ M. Therefore, Since εn x + zn → z ∈ M δ , we see that βε n (wn) = zn + on( ) and thus the sequence {zn} satis es (3.18) and the lemma is proved.
We nish this section by presenting a relation between the topology of M and the number of solutions of the modi ed equation (2.10). Since Sε is not a complete metric space, we will invoke the abstract category result in [32]. Therefore, there is numberε > such that the set is nonempty for all ε ∈ ( ,ε) since πε(M) ⊂ Sε. Here h is given in the de nition of Nε. From the above considerations, together with Lemma 3.6, Lemma 2.4-(c), (3.14) and Lemma 3.8, we see that there exists aε =ε δ > such that the diagram of continuous mappings below is well-de ned for ε ∈ ( ,ε) From (3.14), we can choose a function τ(ε, z) with |τ(ε, z)| < δ/ uniformly in z ∈ M for all ε ∈ ( ,ε), such that βε(Θε(z)) = z + τ(ε, z) for all z ∈ M. De ne H(t, z) = z + ( − t)τ(ε, z). It follows from Corollary 2.9 and the category abstract theorem (see [32],

Proof of the main results
In this section, we will prove of main results. The idea is to show the solutions obtained in Theorem 3.9 satisfy the estimate wε(x) ≤ a for any x ∈ O c ε as ε is small. This fact implies that these solutions are indeed solutions of the original problem (2.7). The following lemma plays an important role in the study of behavior of the maximum points of the solutions, whose proof is related to the Morse iterative method [31] (see also [18,33,38]). with β > to be determined later. Since This inequality, by the de nition of φ L,n and I εn (vn), φ L,n = imply that here we use the fact V + V ≥ for all x ∈ R n in the last inequality. By (2.12), then from (4.1) we have for some ε > . On the other hand, let ω L,n = v β− L,n |vn| and then We deduce from (4.2) and (4.3) that for positive constant C. By the Sobolev embedding, we have where constant C > , see for example [39]. So combining (4.4) and (4.5), we have for constant C > . Next, we claim |γ(|vn|)| ∈ L ( ) (R N ). In fact, choosing β = in (4.6) and using Hölder inequality, we have Choosing proper ε > , we can obtain (4.7) Now we let it follows that t/(t − ) < . We estimate the right-and side of (4.6). By Hölder inequality On the other hand, set a = ( − ) (β− ) and b = β − a, we see that a, b ∈ ( , ).Then by Young's inequality, we have For i ≥ , we de ne β i+ inductively so that This implies that γ(|vn|) L ∞ (R N ) ≤ C A .
We complete the proof by using the fact By a standard arguments as Proposition 2.5 in [20] and Theorem 7.1 in [40], we can prove that γ(vn) is exponential decay and we omit the details here.
We are now ready to prove the main result of this paper.
Therefore, there holds γ(wn) < a, ∀ R N \ Oε n , (4.14) which contradicts to (4.13) and the claim holds true. Letε δ given by Theorem 3.9 and let ε δ := min{ε δ ,ε δ }. We will prove the theorem for this choice of ε δ . Let ε ∈ ( , ε δ ) be xed. By using Theorem 3.9 we can get cat M δ (M) nontrivial solutions of (2.10). If w ∈ H Aε (R N+ + , C) is one of these solutions, we have that w ∈ Nε and we can use (4.14) and the de nition of g to conclude that g(·, |γ(w)|) = f (|γ(w)|). Hence, u(x) = γ(w(x, y)) is also a solution of problem (2.6). By an easy calculation we see that v(x) := u(x/ε) is a solution of the original problem (1.2). Then problem (1.2) has at least cat M δ (M) nontrivial solutions. Now we consider εn → + and take a sequence wn ∈ H Aε n (R N+ + , C) of solutions of problem (2.10) as above. In order to study the behavior of the maximum points of un = γ(wn), we rst note that, by the de nition of g and (f ) − (f ), there exists < τ < a small such that g(εn x, s)s = f (s)s ≤ V K s (4.15) for all x ∈ R N and s ≤ τ.
Using a similar argument as above, we can take R > such that un L ∞ (B c R (zn)) < τ. (4.16) Up to subsequence, we may also assume that un L ∞ (B R (zn)) > τ. (4.17) Otherwise, if this is not the case, we have un L ∞ (B R (zn)) ≤ τ, and so it follows from I εn (wn) = , (4.15) and take a same calculation as (2.17)-(2.18), we can get a contradiction. Therefore, (4.17) holds. By observing (4.16) and (4.17), we see that the maximum points pn ∈ R N of un belongs to B R (zn)) . Hence pn =zn+qn for some qn ∈ B R ( ). Recalling that the associated solution of (1.2) is of the form vn(x) := un(x/εn), we conclude that the maximum point ηε n of vn is ηε n := εnzn + εn qn. Since {qn} ⊂ B R ( ) is bounded and εnzn → z ∈ M (according Proposition 3.7), we obtain lim n→∞ V(ηε n ) = V(z ) = −V .