ON THE CONCENTRATION OF SEMICLASSICAL STATES FOR NONLINEAR DIRAC EQUATIONS

. In this paper, we study the following nonlinear Dirac equation − iεα · ∇ w + aβw + V ( x ) w = g ( | w | ) w, x ∈ R 3 , for w ∈ H 1 ( R 3 , C 4 ) , where a > 0 is a constant, α = ( α 1 ,α 2 ,α 3 ), α 1 ,α 2 ,α 3 and β are 4 × 4 Pauli– Dirac matrices. Under the assumptions that V and g are continuous but are not necessarily of class C 1 , when g is super-linear growth at inﬁnity we obtain the existence of semiclassical solutions, which converge to the least energy solutions of its limit problem as ε → 0.


Equation (D) or a more general one
arises when one seeks for the standing wave solutions of the nonlinear Dirac equation Assuming that G(x, e iθ ψ) = G(x, ψ) for all θ ∈ [0, 2π], a standing wave solution of (2) is a solution of the form ψ(t, x) = e iµt w(x). It is clear that ψ(t, x) solves (2) if and only if w(x) solves (1) with a = mc, V (x) = M (x)/c + µI 4 and F (x, w) = G(x, w)/c.
Recent, equation (1) has been widely studied and the existence and multiplicity results for such a equation have been discussed in many papers under different assumptions on the potential and nonlinearity (see [2,14,16,19,20,21,29] for example).
When is small, the standing waves are referred to as semiclassical states. The concentration phenomenon of semiclassical states as → 0 reflects the transformation process between quantum mechanics and classical mechanics. So it possesses an important physical interest. As we know, there have been so many results that relate to the existence and concentration phenomenon of semiclassical states for nonlinear Schrödinger equations, [1,3,4,6,7,8,9,24,25,26,30,32,34,35]. However, such results on Dirac equations are relatively few: In [11], Ding considered (1) with V (x) ≡ 0 and F w (x, w) = P (x)|w| p−2 w, p ∈ (2, 3) and showed the existence of a family of ground states of the equation for small , which concentrates around the maximum points of P (x) as → 0. This result was later generalized to the case with F w (x, w) = g(|w|)w in [12]; F w (x, w) = P (x)g(|w|)w in [13] and F w (x, w) = P (x)(g(|w|) + |w|)w in [15]. In [18], Ding and Xu weakened the assumptions on V (x) to the following local version: there is a bounded domain Λ ⊂ R 3 such that min x∈Λ V (x) < min x∈∂Λ V (x).
And they obtain the existence and concentration results when F w (x, w) = g(|w|)w possesses a supper linear or a asymptotically linear growth at infinity. However, in all previous results involving semiclassical states for Dirac equations, there holds the strong differentiability conditions: g is of class C 1 . And the standard method is a reduction method in two steps: Using the differentiability conditions and the monotonicity conditions on g, one reduce the problem first to E + and then to the Nehari manifold on E + . By corresponding the least energy to the infimum of the Nehari manifold on E + , one could prove the concentration result. While, in this paper, we mainly consider the case g ∈ C(R + , R) but is not necessarily of class C 1 (0, +∞) in which such a standard method does not work.
To state our main results, we make the following assumptions: |V (x)| < a and there is a bounded domain Λ ⊂ R 3 , such In this paper, we denote by V the set V := {x ∈ Λ : V (x) = v}, and according to (V ), we know dist(V, ∂Λ) > 0.
Using the scaling u(x) = w(εx), it is easy to see that w is a solution of (D) if and only if u is a solution of Therefore, we will mainly focus on this equivalent equation in the following.
Our strategy of proof is as follows: We first make a slight modification of the functional corresponding to (D ε ) such that it satisfies the Palais-Smale condition, and show the existence of ground states of the modified problem via a classical linking theorem. Then by relating the least energy to the infimum of the modified functional restrained on a generalized Nehari set, we prove that the least energy accumulating to the least energy of the limit problem. Lastly, using this fact, we show that there exists a uniformly decay at infinity for the least energy solution as ε → 0, which implies that these solutions corresponding to the modified problem are indeed the solutions of (D ε ). And then Theorem 1.1 follows naturally. In fact, the generalized Nehari set mentioned above has been firstly presented by Pankov in [31] and later thoroughly studied by Szulkin and Weth in [33]. We also refer the readers to [21,36,37] for the applications of this set in Dirac equations.
Another difference is the procedure of showing the exponential decay of the solutions. Noting that the potential function V is required to be C 1 in [12,13,15] and is required to be locally Hölder continuous in [18]. Combining with the C 1 assumption on g, if u is a solution of (D ε ), one can apply the operator −iα · ∇ on both sides of (D ε ). Then the desired result follows from the fact (−iα · ∇) 2 = −∆, Kato's inequality and the classical comparison theorem. However, since we only assume V and g to be continuous, such a approach that we mentioned above could not be applied directly, and we will give the details of proof at the end of Section 4.
The remainder of this paper is organized as follows. In Section 2, we derive a variational setting for the problem, introduce the modified functional, give some preliminary lemmas and by using a linking theorem, we show the existence of ground state solutions for the modified problem with energy m ε . In Section 3, we fist show the limit problem possesses a ground state and then by introducing the generalized Nehari set we prove the upper limit of the least energy m ε is less than or equal to the least energy of the limit problem as ε → 0. In Section 4, we finish the proof of Theorem 1.1.

Preliminaries.
To prove the main results, some preliminaries are first in order. We denote by | · | q the usual L q norm, and by ·, · 2 the L 2 inner product. 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 3 , C 4 ) possesses the orthogonal decomposition: such that H 0 is positive definite in L + and negative definite in L − . So for u ∈ L 2 (R 3 , C 4 ), we denote by u = u + + u − the decomposition of u with u + ∈ L + and u − ∈ L − .
Let |H 0 | denote the absolute value of H 0 and |H 0 | It is clear that (see [10] Section 7) E possesses the following decomposition where E + = E ∩ L + and E − = E ∩ L − and the sum is orthogonal with respect to both ·, · 2 and ·, · . Additionally, this decomposition of E induces a natural decomposition of L q (R 3 , C 4 ) for q ∈ (1, +∞) and we refer the readers to [18] Proposition 2.1 for the proof of the following proposition.
Proposition 1. Let E + ⊕ E − be the decomposition of E according to the positive and negative part of σ(H 0 ). Then, if we set E ± q := E ± ∩L q (R 3 , C 4 ) for q ∈ (1, +∞), there holds L q (R 3 , C 4 ) = cl q E + q ⊕ cl q E − q , where cl q denotes the closure in L q (R 3 , C 4 ). Moreover, for any q ∈ (1, +∞), there exists positive constant d q such that From standard arguments, the functional J ε : E → R defined by with u · v denoting the usual inner product in C 4 , i.e., u · v = 4 j=1 u jvj . And hence the critical points of J ε are weak solutions of (D ε ) (one can find more details in [13]).
However, we will not deal with J ε directly. Instead, we use a convenient truncation of the nonlinearity such as that used in [7,8] to modify the function such that the energy functional becomes coercive far from origin and satisfies the Palais-Smale condition.
According to the assumptions in (V ): V is continuous and v < min and g ∈ C(R + , R): Then using (g 1 ) − (g 3 ), we can verify that f is a Caratheodory function and it satisfies: From (g 1 ) and (g 2 ), it follows that there exist t 0 > 0 and c 1 > 0 so that and This, together with (g 3 ): By this and (7), we get Now we are ready to define the modified functional Φ ε : E → R, Similarly, Φ ε is of class C 1 and the critical points correspond to weak solutions of

XU ZHANG
For the sake of simplicity, in the following of this paper, we denote by Lemma 2.1. For every ε > 0, the functional Φ ε satisfies the Palais-Smale condition.
Proof. Suppose that {u n } ⊂ E is a sequence such that Φ ε (u n ) is bounded and Φ ε (u n ) → 0. Then there exists a constant C > 0 satisfying By (f 3 ) and (6), we have On the other hand, (8) yields that Using the Hölder's inequality, by χ ε ∈ [0, 1], we deduce that Then from (4) and (9), it follows which means that {u n } is bounded in E. Therefore u n u weakly in E and u n → u strongly in L q loc (R 3 , C 4 ), q ∈ [1, 3).
Subtracting the left and right sides of the two equations above and using the fact that Then it follows from (4) that Noting that g ≤ a−|V |∞ 2 and χ ε ∈ [0, 1], we have Since the support of χ ε is bounded for every fixed ε > 0 and z n → 0 strongly in L q loc (R 3 , C 4 ), we know that z n → 0, n → ∞ strongly in E.
Next we shall verify that Φ ε possesses the linking structure. Before that we give the following notations.
where d µ is the constant appeared in Proposition 1. Then setting we have the following property.
Proof. We only need to show Otherwise, there exist ε n → 0 and w n ∈ E(e) ∩ S R , such that Φ εn (w n ) ≥ − 1 n . Thus, we set w n = s n e + v n with s n ≥ 0 and v n ∈ E − . Since Φ εn (w n ) ≥ − 1 n , we have which implies that According to (3) and 0 ∈ V, we can assume that there exists δ > 0 such that Here we have used the fact that x ∈ Λ, t ≥ 0 and Proposition 1. Taking the above inequality into (12), we obtain this is a contradiction to (10).
Then, for the fixed e ∈ E + ∩ S 1 , we set with R the positive constant we defined in (11). Obviously, for any v ∈ E − and ε > 0, there holds Combining this with Lemma 2.2, we know Additionally, a standard argument shows that there exist r ∈ (0, R) and ρ > 0, such Then, using the fact that E embeds into L q (R 3 , C 4 ) continuously for q ∈ [2, 3] and embeds into L q loc (R 3 , C 4 ) compactly for q ∈ [1, 3), it is easy to see that: Lemma 2.3. Ψ ε is weakly sequentially lower semi-continuous and Ψ ε is weakly sequentially continuous.
According to (4), ·, · V is an inner product on E, the induced norm · V is equivalent to · . Moreover, computing directly, we have Let S be a countable dense subset of (E − ) * (the dual space of E − ) and D = {d s : s ∈ S, d s (u, v) = |s(u − v)| for u, v ∈ E − } be the associated family of semi-metrics on E − . Let P be the family of semi-norms on E consisting of all semi-norms: Thus P is countable and it induces the product topology on E given by the Dtopology on E − and the norm topology on E + . We denote this topology by (E, T P ) and denote the weak * topology on E * by (E * , T w * ). Then using (15), Lemma 2.3 and the arguments in Theorem 4.1 of [10], we can prove that: (Φ) For any ε > 0, c ∈ R and Φ ε,c = {u ∈ E : Φ ε (u) ≥ c}, there hold Φ ε : (E, T P ) → R is upper semi-continuous and Φ ε : (Φ ε,c , T P ) → (E * , T w * ) is continuous. According to this, (13) and (14), we may apply the linking theorem (see Theorem 4.4 in [10] for example), and obtain a Palais-Smale sequence at level
We end this section by two technical results, which play an important role in the following proof.
It is easy to check that for any δ > 0, h δ satisfies (h 1 ) − (h 3 ) of Lemma 2.4. And the desired result follows by applying Lemma 2.4 to h δ and then letting δ → 0.
3. Limit problems. For λ ∈ (−a, a), we consider such a problem Note that for any e ∈ E + ∩ S 1 and u ∈ E(e) with u = se + v, s ≥ 0 and v ∈ E − , there holds According to (g 1 ) and (g 3 ), G(t) ≥ C g t µ − a−λ 4 t 2 for all t ≥ 0, with C g a positive constant depending only on g and λ. Using this and Proposition 1, we deduce This implies that there exists R λ = R λ (e) > 0 such that where Q λ (e) = {u ∈ E(e) : u = se + v, s > 0, v ∈ E − and u < R λ }. Obviously, there exist r λ ∈ (0, R λ ) and ρ λ > 0, such that Then by the same procedure as that in Section 2, we obtain a Palais-Smale sequence for J λ at level Moreover, we have the following proposition. Otherwise, Lions' Lemma yields that u n → 0 in L q (R 3 , C 4 ) for q ∈ (2, 3). Noting that g(t)t ≤ a−λ 4 t + Ct p−1 for any t ≥ 0 with C a positive constant depending only on g and λ, by (4) and o n (1) = J λ (u n )(u + n − u − n ), we get that So u n → 0 as n → ∞. However, this is impossible since c λ ≥ ρ λ > 0.
Then, there exists {y n } ⊂ R 3 such that B1(yn) |u n | 2 ≥ η > 0. Consequently, {u n (· − y n )} is a bounded Palais-Smale sequence for J λ , which converges weakly to a nontrivial solution u of (D λ ). Moreover, by Fatou's Lemma, We denote the critical set of J λ by K λ := {u ∈ E \ {0} : J λ (u) = 0} and set m λ = inf u∈K λ J λ (u). Since for any u ∈ K λ , we know m λ ≥ 0. Then we claim m λ > 0. In fact, if m λ = 0, there exists {w n } ⊂ K λ such that J λ (w n ) → 0. According to this, similarly as the proof in Lemma 2.1, we can prove w n → 0. while, from (4) and the fact that g(t)t ≤ a−λ 4 t + Ct p−1 for t ≥ 0, it follows that Hence u ≥ η > 0 for any u ∈ K λ , which contradicts to w n → 0. Thus for {w n } ⊂ K λ such that J λ (w n ) → m λ > 0, using Lions' Lemma, we can obtain a sequence {z n } ⊂ R 3 such that w n (· − z n ) u λ weakly in E. Moreover, u λ is a nontrivial solution of (D λ ) with J λ (u λ ) ≤ m λ (see (17)), and hence is a ground state with J λ (u λ ) = m λ .
In Section 2, for ε ∈ (0, ε 0 ), we have obtained a least energy solution u ε for Φ ε with Φ ε (u ε ) = m ε . And if we can show that there exists a uniformly decay for u ε : |u ε (x)| → 0, as |x| → +∞, uniformly for small ε, then u ε is actually a solution of (D ε ). In order to show this, as well as the convergence property and the exponential decay results in Theorem 1.1, the following property is the key: where v = min x∈Λ V (x) and m v is the least energy of (D λ ) with λ = v.
As far as we know, in all the results involving semiclassical states for Dirac equations, there exists the differentiability conditions: g is of class C 1 (0, +∞). And the standard method is a reduction method in two steps, first to E + and then to a Nehari manifold on E + . By corresponding the least energy to the infimun of the Nehari manifold on E + , one could obtain (18). However, such a method does not work here, and since it is difficult to obtain the relation (18) directly, we need more characterizations on m ε and m λ .
Motivated by [21], we consider the generalized Nehari set for Φ ε and J λ : Observing that for ε ∈ (0, ε 0 ), Φ ε possesses positive least energy, so for any nontrivial solution u of Φ ε there holds Combining this with (4) and |V | ∞ < a, we know u + = 0 and hence u ∈ M ε . Similarly, we can prove nontrivial solutions of J λ for λ ∈ (−a, a) belong to M λ . In summary, the nontrivial critical set of Φ ε and J λ satisfy Then, M ε and M λ are nonempty and for each u ∈ M λ , J λ | E(u + ) attains its unique maximum at u; for u ∈ M ε , Φ ε | E(u + ) attains its maximum at u.
Since u ∈ M λ and w ∈ E − , we have J λ (u)( t 2 2 u − 1 2 u + tw) = 0, which means that Taking this into the above equation, using Lemma 2.4 and (4), we deduce that To show the second property, we set u = u + + u − ∈ M ε , t ≥ 0 and v ∈ E − . By an argument similar to that above, for w = v − tu − ∈ E − , we have From (f 1 ), we know that for any x ∈ R 3 , f ε (x, ·) satisfies the assumption (h 1 ) and (h 2 ) in Proposition 2. So applying Proposition 2 and (4), we get the conclusion.
Proposition 5. For any u ∈ E with u + = 0, the following statements hold true: , a).

From Lemma 2.3 and
To prove (iii), let u λ be the ground state solution we obtained in Proposition 3. Then (19) implies that u λ ∈ M λ and consequently m λ = J λ (u λ ) ≥ inf u∈M λ J λ (u). If we assume that m λ > inf u∈M λ J λ (u), then there exists w ∈ M λ satisfying m λ > J λ (w). From w ∈ M λ we know w + = 0. Hence we may apply the linking theorem in Q λ (w + ) to obtain a critical value c λ of (D λ ) such that c λ ≤ sup u∈Q λ (w + ) J λ (u) (see the proof at the beginning of this section and (17)). Then by Proposition 4 and w ∈ M λ , we deduce c λ ≤ sup u∈Q λ (w + ) J λ (u) ≤ J λ (w) < m λ , which is a contradiction.
We claim that Writing Taking (22) and the following fact (21), we obtain that Similarly, if we define G(u) = R 3 G(|u|), we will deduce that Computing directly, we have Therefore,

XU ZHANG
Taking (23) and (24) into the right hand side of the above equality, we deduce that Clearly, there hold and G (w)z ε = o ε (1), which lead us to On one hand, On the other hand, since f ε (x, |w ε |)|w ε | 2 → g(|w|)|w| 2 a.e. x ∈ R 3 , using Fatou's Lemma, we have Consequently, z − ε ≤ o ε (1) and z ε → 0 in E, which imply that the claim is true.

Proof of main result.
Lemma 4.1. Let u ε be the ground state solutions of Φ ε which we obtained in Section 2 with ε ∈ (0, ε 0 ), then for any q ≥ 2, u ε ∈ W 1,q (R 3 , C 4 ) and u ε W 1,q ≤ C q , where C q depends only on q. In particular, for q > 3 and y ∈ R 3 , there exists positive constant C depending only on q such that, |u ε (y)| ≤ C u ε W 1,q (B1(y)) .
Proof. According to Proposition 6, we have Φ ε (u ε ) = m ε is uniformly bounded for ε ∈ (0, ε 0 ). By a proof similar to that of Lemma 2.1, we deduce that {u ε } is bounded in E. Then using the same iterative argument as that in [20] Proposition 3.2 and [17] Lemma 3.19, we obtain that u ε ∈ W 1,q (R 3 , C 4 ) with u ε W 1,q ≤ C q , where C q depends only on q for any q ≥ 2. In particular, u ε is uniformly bounded in L ∞ (R 3 , C 4 ) and |u ε (y)| ≤ C u ε W 1,q (B1(y)) for ε ∈ (0, ε 0 ), where C is independent of ε and the choice of y.
Proposition 7. Let u ε be ground state solutions which we obtained in (16). Then |u ε | attains its maximum at x ε . Moreover, if we set v ε (x) = u ε (x + x ε ), up to a subsequence, as ε → 0, there hold Proof.
Step 1. We claim that there exists δ > 0, such that Arguing indirectly, we assume From Lemma 4.1, we know that u ε is bounded in E. Then according to Lions' Lemma [27], up to a subsequence if necessary, u ε → 0 in L q (R 3 , C 4 ) for q ∈ (2, 3). Note that for any small > 0, there exist C > 0 and p ∈ (2, 3) such that, f ε (x, t) ≤ + C t p−2 uniformly holds for ε > 0 (see (f 2 ), (g 1 ) and (g 2 )). Then a standard argument shows that While according to (19), u ε ∈ M ε . This, together with Proposition 4 and (14), leads us to which is impossible.

XU ZHANG
Taking the scalar product of this equation with u + − u − , since g(t) ≤ a−|V |∞ 2 , we know This is a contradiction and hence as ε → 0, εy ε → y 0 ∈ Λ δ1 . Let f ∞ (t) = χ(y 0 )g(t)+(1−χ(y 0 )) g(t), where χ is the cut-off function we defined in Section 2. Then similarly, u satisfies Testing this equation by u, we have combining which with the fact u = 0, we have u + = 0. Denoting by Φ ∞ the associate energy functional corresponding to the above equation with F ∞ (t) = t 0 f ∞ (s)sds and setting Obviously, f ∞ satisfies the assumptions of Proposition 2 and hence by the same procedure as the proof in Proposition 4, we know that if u ∈ M ∞ , then Φ ∞ | E(u + ) attains its maximum at u. Thus by u + = 0 and Φ ∞ ( u) = 0, we know u + ∈ M ∞ and hence Φ ∞ ( u) ≥ Φ ∞ (w), for any w ∈ E( u + ).
It sufficient to prove that there is a subsequence { u εj } such that the above convergence holds.
Since y 0 ∈ V, we know f ∞ (t) = g(t). So Φ ∞ ( u) = J v ( u), and hence u is a nontrivial solution of (D λ ) with λ = v. Moreover, from (26), (27) and Proposition 6, it follows that J v ( u) = m v and u is a ground state.
Therefore, if we denote by Φ ε the functional corresponding to (25), it follows from (29), (31) and Proposition 6 that Similarly, using (30), we deduce that uniformly holds for ϕ ∈ E. Then we have .