ON MULTIPLICITY OF SEMI-CLASSICAL SOLUTIONS TO NONLINEAR DIRAC EQUATIONS OF SPACE-DIMENSION n

. In this paper, we study multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension n : where n ≥ 2, (cid:126) > 0 is a small parameter, a > 0 is a constant, and f describes the self-interaction which is either subcritical: W ( x ) | u | p − 2 , or critical: W 1 ( x ) | u | p − 2 + W 2 ( x ) | u | 2 ∗ − 2 , with p ∈ (2 , 2 ∗ ) , 2 ∗ = 2 n n − 1 . The number of solutions obtained depending on the ratio of min V and liminf | x |→∞ V ( x ), as well as max W and limsup | x |→∞ W ( x ) for the subcritical case and max W j and limsup | x |→∞ W j ( x ) ,j = 1 , 2 , for the critical case.

1. Introduction and main results. In quantum theory in order to describe the translation from quantum to classical mechanics, existence of semi-classical solutions of stationary quantum systems possesses an important physical interest. There have been large amounts of works on existence, multiplicity and concentration phenomenon of semi-classical solutions of nonlinear Schrödinger equations for n-dimensional space arising from non-relativistic quantum mechanics. We refer the reader to ,eg., [2,10,13,18,12,31] and the references therein. In comparison, only a few similar results are known for nonlinear Dirac equations of space-dimension n arising from relativistic mechanics. Mathematically, the nonlinear Dirac equation is more difficult because, unlike the spectrum of the Laplacian which is bounded below, the spectrum of the Dirac operator is neither bounded below nor above.
In this paper, we are mainly interested in utilizing variational methods to obtain multiplicity results for the Dirac equation of space-dimension n, by introducing some conditions depending on the behaviors of the potentials near the infinity, which can be directly verified. There are two main ingredients. One is to give a representation of ground state of the associated linear autonomous problem (the so-called limit equation) which yields the comparison conditions and etc. The other is to construct subspaces on which the relative energy functional is bounded above, say by b, and satisfies the Palais-Smale condition below the level b, and thus we are able to apply an abstract critical point theorem. Moreover, we will deal with the case of critical nonlinearity.
We now recall the problems and state our results. Consider the nonlinear Dirac equation of space-dimension n, given by for the function ψ : R × R n → C N which stands for the wave. Here c is the speed of light, m > 0 is the mass of the electron, denotes Planck's constant, and ({α k } n k=1 , β) is an (n + 1)-tuple of Dirac matrices: (1) β * = β and α * k = α k for k = 1, · · · , n, i.e. β and α k are self-adjoint. (2) β 2 = 1, α k β + βα k = 0 and α i α j + α j α i = 2δ ij I N ×N for i, j = 1, · · · , n. where the a k are N 2 × N 2 matrices (which are Hermitian if n is odd). For the nonlinear Dirac equation of space-dimension n, P (x) denotes the potential, and G ψ (x, ψ) describes the self-interaction in Quantum electrodynamics. It is considered to be possible basic models for unified field theories (see [28,26] etc. and references therein). For more physical background about the nonlinear Dirac equation of space-dimension n, we refer the reader to [16,33] and the references therein. Here, we assuming that G(x, e iθ ψ) = G(x, ψ) for all θ ∈ [0, 2π], a standing wave solution of (1) is a solution of the form ψ(t, x) = e iµt w(x). It is clear that where a = mc, V (x) = P (x) c + µI 4 and F (x, w) = G(x,w) c . In the two space-dimensional case, dealing with the Cauchy problem of (1), N.Bournaveas in [7] proved a local well-posedness result for the Yukawa interaction model in which the nonlinear term has the form G ψ (x, ψ)ψ = φβψ with φ being Klein-Gordon field determined by ψ. This result is later improved by P. d'Ancona, D. Foschi and S. Selberg [11]. Their proof relied on the null structure of the nonlinear system. And in [27] A. Grünrock and H. Pecher obtained the first global well-posedness result for large data in two space dimensions of Yukawa model is established by using Bourgain type function spaces.
In the literature, there are many results concerning existence of solutions of (2) with n = 3 under various hypotheses on the potential and the nonlinearity (see ON MULTIPLICITY OF SEMI-CLASSICAL SOLUTIONS TO DIRAC EQUATIONS   4107 [23] for a survey). When V (x) ≡ ω, F. Merle in [29] studied the following model nonlinearity: In [3,4,8,25] the so-called Soler model was investigated by using shooting methods. Such nonlinearities were later studied by using for the first time a variational method in [24], where more general superlinear subcritical F (w) independent of x were considered. Existence and multiplicity results for (2) with V (x) and F (x, w) depending periodically on x were obtained in [6] by using a critical point theory. For non-periodic potentials (the Coulomb-type potential is a typical example), existence and multiplicity of solutions were studied in [19] for asymptotically quadratic nonlinearities and in [21] for super-quadratic subcritical nonlinearities, where V (x) and F (x, w) were assumed to have limits as |x| → ∞.
For small , the standing waves are referred to as semi-classical states. To describe the translation from quantum to classical mechanics, existence of solutions w , small, is of great physical importance. Existence and concentration phenomena of semi-classical ground states of the Dirac equation (2) with n = 3, N = 4 where the nonlinearity F w (x, w) = W (x)h(w) have been studied, in [14] for V (x) ≡ 0 and h(w) super-linear and subcritical, in [17] for V : R 3 → R and h(w) super-linear and subcritical and in [20] for V : R 3 → R and h(w) = g(|w|) + |w|, g(|w|)w critical. In [16], Y. Ding, Q. Guo and T. Xu obtained existence and concentration phenomena of semi-classical ground states of the Dirac equation (2) with n ≥ 2 for subcritical or critical nonlinearities. For the multiplicity of semi-classical solutions of nonlinear Dirac equations, in [34] Z. Wang and X. Zhang obtained an interesting result for autonomous nonlinear Dirac equations with n = 3, N = 4 where the nonlinear term is subcritical. They constructed an infinite sequence of bound state solutions for small values of ε, particularly, these solutions are of higher topological type. Very recently, Y. Chen, Y. Ding and T. Xu in [9] treated subcritical and critical nonlinearities for autonomous nonlinear Dirac equations with n = 3, N = 4 and assumed that the nonlinearities f satisfied lim s→∞ f (s) s = κ. They got the number of solutions with the topology of the set where the potential attains its minimum for small κ. The main objective of this paper is to study non-autonomous nonlinear Dirac equations of space-dimension n ≥ 2 with subcritical or critical nonlinearities. The number of solutions obtained is described by the maximum and behavior at infinity of linear and nonlinear potentials.
Firstly, we deal with the subcritical case, writing ε ≡ and α · ∇ = n k=1 α k ∂ k , where p ∈ (2, 2 * ) with 2 * = 2n n−1 . Notations: In order to describe our results some notations are in order: Assuming that the potentials satisfies: If (P 1 ) holds, we can set If (P 2 ) holds, we can set Now we state our main results as follows.
Theorem 1.1. Assume that (P 0 ) and (P 1 ) hold, let (4) Then there is ε m > 0 such that equation (3) possesses at least m pairs of solutions Theorem 1.2. Assume that (P 0 ) and (P 2 ) hold. Let Then all the conclusions of Theorem 1.1 remain true.
Remark 1.1. If additionally ∇V and ∇W are bounded, then among the solutions, the ground state (the least energy solution) denoted by w ε satisfies (see [16]): (i) There exists a maximum point x ε of |w ε | such that, up to a subsequence, In (ii) There are positive constants C 1 , C 2 independent of ε such that Next we consider the equation with critical nonlinearity In the following let S denote the best constant of the Sobolev inequality

ON MULTIPLICITY OF SEMI-CLASSICAL SOLUTIONS TO DIRAC EQUATIONS 4109
We use the notationḢ where u →û is the Fourier transform of u ∈ L 2 . Let γ p denote the least energy of the equation Set We also need the following notations: for j = 1, 2 Assuming that the potentials satisfies: for j = 1, 2 If (Q 1 ) holds, we can set If (Q 2 ) holds, we can set Now we state our main results as follows.
If additionally ∇V and ∇W j , j = 1, 2, are bounded, then among the solutions, the ground state (the least energy solution), denoted by w ε satisfies (see [16]): (i) There exists a maximum point x ε of |w ε | such that, up to a subsequence, For the subcritical case, observe that in (4), m can be sufficiently large if τ closes sufficiently to −a. One may make similar comments on (5). For the critical case, we can obtain in (11) that if κ 1∞ , κ 2∞ are small, m can be sufficiently large. In this condition, κ 1 , κ 2 can be sufficiently large, which is an open problem in [9].
Notation. Throughout this paper, we make use of the following notations.
• For any R > 0 and for any x ∈ R n , B R (x) denotes the ball of radius R centered at x; • | · | q denotes the usual norm of the space L q (R n , • C or C i (i = 1, 2, · · · ) are some positive constants may change from line to line.
2. The functional-analytic setting. For convenience, let where σ(H 0 ) and σ c (H 0 ) denote the spectrum and the continuous spectrum of H 0 , respectively. Thus the space L 2 (R n , C N ) possesses the orthogonal decomposition: Note that this norm is equivalent to the usual H 1 2 -norm, hence E embeds continuously into L q for all q ∈ [2, 2 * ] and compactly into L q loc for all q ∈ [1, 2 * ). That is, there exists constant d q > 0 such that Moreover, it is clear that E possesses the following decomposition which is decomposition orthogonal with respect to inner products (·, ·) L 2 and (·, ·). Furthermore, from [22, Proposition 2.1], this decomposition induces of E also a natural decomposition of L q , hence there is c q > 0 such that In the following, let Making the change of variable x → εx, we can rewrite the equation (3) or (8) as the following equivalent equation On E, we define the energy functional Φ ε corresponding to equation (15) by for u = u + + u − ∈ E. It follows by standard arguments that Φ ε ∈ C 1 (E, R). Also, for any u, v ∈ E, one has Moreover, in [17, Lemma 2.1] and [20,Lemma 3.3] it is proved that critical points of Φ ε are weak solutions of equation (15). We will study the multiplicity of critical points of Φ ε via linking arguments. The following results can be found in [16].
Let c ε denote the minimax level of Φ ε deduced by the linking structure Denote Using the iterative argument of [24, Proposition 3.2] one checks easily the following lemma(see [16]).
In addition, we have Lemma 2.4. Let {u n = u + n + u − n } be a (P S) c sequence for Φ ε and set v n = u + n + h ε (u + n ) , z n = u − n − h ε (u + n ) . Then z n → 0 and {v n } is also a (P S) c sequence for Φ ε , that is, u + n is a (P S) c sequence for I ε . Consequently, either c = 0 or c ≥ c ε .
Proof. It suffices to show that z n → 0. Observe that Thus, Thus, z n → 0. Finally, it follows from (16) that if c = 0 then c ≥ c ε .
Below, for notational convenience, we denote by Φ 0 the energy functional of the equation We define correspondingly c 0 , the critical set K 0 , and the "Mountain-pass" reduction {h 0 , I 0 , N 0 } for (18). The proofs of the following conclusions can be found in [17] for the subcritical case and [20] for the critical case.

Lemma 2.5. We have
(1) For any u ∈ E + \{0}, there is a unique t ε = t ε (u) > 0 such that t ε u ∈ N ε . Moreover, lim In order to establish our multiplicity results, we recall an abstract critical point theorem, see [5,14]. Let X, Y be Banach spaces with X being separable and reflexive, and set E = X ⊕ Y . Let S ⊂ X * be a countable dense subset. Let P be the family of semi-norms on E: Denote by T P the topology on E induced by P. Let T w * be the weak * topology of E * .
For a functional Φ : E → R and numbers a, b ∈ R we write Φ a := {u ∈ E : . We consider the set M (Φ c ) of maps g : Φ c → E with the properties (i) g is P-continuous and odd; The pseudo-index of Φ c is defined by where gen(·) denotes the usual symmetric index. Additionally, set for d > 0 fixed Then we define for c ∈ [0, d] Theorem 2.1. (See [5,14]) Let (Φ 1 ) − (Φ 3 ) be satisfied, and assume that Φ is even and satisfies the (P S) c condition for c ∈ [ρ, b]. Then Φ has at least n := dim Y 0 pairs of critical points with critical values given by If Φ has only finitely many critical points in Φ b ρ , then ρ < c 1 < c 2 < · · · < c n ≤ b.
Remark 2.1. Setting X = E − and Y = E + , it follows from the definition and Lemma 2.1 that the functional Φ = Φ ε is even and satisfies (Φ 1 ) and (Φ 2 ).
3. Preliminary results. Firstly, we recall a result on the representation of the energy to certain constant coefficient systems.
3.1. The limit equation: subcritical case. Consider, for any τ ≤ µ ≤ τ ∞ and Denote the critical set, the least energy, and the set of least energy solutions of Γ µν as follows The following conclusions are from [17,20]: (i) L µν = ∅, γ µν > 0, and L µν ⊂ q≥2 W 1,q , (ii) γ µν is attained, and R µν is compact in H 1 R n , C N , (iii) there exist C, c > 0 such that |u(x)| ≤ C exp(−c|x|) for all x ∈ R n and u ∈ R µν . Using γ p we have the following representation (see [16,Lemma 3.4

]).
Lemma 3.2. The corresponding least energy of (19) denoted by γ µν , then The following Lemma describes a comparison between the ground state energy values for different parameters µ ∈ (−a, a) and ν > 0, which will play an important role in proving the existence result in Section 4. This conclusion follows directly from the representation of γ µν .
Below, let u be a least energy solution of with the energy denoted by γ ∞ which is attained.

YANHENG DING, XIAOJING DONG AND QI GUO
3.3. The cut-off functions. Finally, we consider where where W νj j = min{ν j , W j (x)}, W νj jε = W νj j (εx). The solutions of (27) are critical points of denote the Minimax level of Φ µν ε deduced by the linking structure (see (16)). Write h µν e , I µν e , M µν e , and so on, for the notations associated to the Mountain-Pass reduction. Recall that, for any u ∈ E + \{0}, there is a unique t = t(u) > 0 such that t(u)u ∈ N µν ε . It is easy to check that As a consequence of Lemma 2.5 we have Remark 3.1. Similarly, one obtains easily that lim ε→0 c µν ε = γ µν .
As a consequence one has Lemma 3.10. Φ µν ε satisfies the (P S) c condition for c < γ ∞ if ε small.
It is not difficult to check the following where o (1 r ) means arbitrary small as r → ∞, and o (1 rε ) means arbitrary small as r sufficiently large and ε sufficiently small. Now, by assumptions and Lemma 3.5 for any 0 < δ < γ ∞ − mγ τ κν , one may choose r > 0 large and then ε m > 0 small such that, for all ε ≤ ε m , max w∈Em Φ ε (w) ≤ γ ∞ − δ.
By Lemma 2.2 we see that the solutions are in s≥2 W 1,s . Now, as (30) one can choose r > 0 and ε m > 0 such that, if ε ≤ ε m Φ ε (w) < γ ∞ for all w ∈ E m .
It follows from Lemma 3.10 that Φ ε satisfies the (P S) c condition for all c < γ ∞ , that is, the general condition (Φ 3 ) is satisfied. Now by applying Theorem 2.1 one sees that either Φ ε has infinitely many critical points, or has at least m pairs of critical points with different critical values 0 < c 0 ε < · · · < c m−1 The proof is hereby complete.