Concentration behavior of semiclassical solutions for Hamiltonian elliptic system

where η = (ψ, φ) : RN → R2, ε is a small positive parameter and ~b is a constant vector. We require that the potential V only satis es certain local condition. Combining this with other suitable assumptions on f , we construct a family of semiclassical solutions. Moreover, the concentration phenomena around local minimumof V, convergence and exponential decay of semiclassical solutions are also explored. In the proofs we apply penalization method, linking argument and some analytical techniques since the local property of the potential and the strongly inde nite character of the energy functional.


Introduction and main results
In this paper, we will consider the following Hamiltonian elliptic system with gradient term where η = (ψ, φ) : R N → R , ϵ is small positive parameter, b is constant vector, V is linear potential and f is continuous, superlinear and subcritical nonlinearity. We are interested in the existence, convergence and concentration phenomenon of semiclassical solutions of system (Pϵ) when ϵ → .
This type of systems arises when one is looking for the standing wave solutions to system of di usion equations which comes from the time-space di usion processes and is related to the Schrödinger equations. It appears in various elds, such as physics and chemistry, quantum mechanics, control theory and Brownian motions. For more details in the application backgrounds, we refer the readers to see the monographs [19] and [21]. In recent years, there has been increasing attention to Hamiltonian elliptic system on obtaining existence of solutions, ground state solutions, multiple solutions and semiclassical solutions by using variational methods. But most of them focused on the case b = . More speci cally, based on various hypotheses on the potential and nonlinearity, the existence and multiplicity of solutions have been established by many authors. For example, see [11,12,15] for the case of a bounded domain, and [2,4,7,29,37,38] for the case of the whole space R N .
When b ≠ and ϵ = , as we all know, there are a few works devoted to the existence and multiplicity of solutions of the following system under di erent assumption see [24,35,39,40,43]. For this case, since the appearance of the gradient term in system, system (1.1) has some di erences and di culties compared with the case b(x) = . For example, the variational framework for the case b(x) = cannot work any longer in this case, then the rst problem is how to establish a suitable variational framework. To solve this problem, Zhao and Ding [39] handled ( . ) as a generalized Hamiltonian system, and established a strongly inde nite variational framework by studying the structure of essential spectrum of Hamiltonian operator. At the same time, the existence and multiplicity of solutions were obtained by using critical point theorems of strongly inde nite functional [10] and reduction method [1] for system (1.1) with periodic and non-periodic asymptotically quadratic growth condition. After that, Zhang et al. [40] studied the periodic super-quadratic case and proved the existence of ground state solutions by means of the linking and concentration compactness arguments. Later, this result has been extended to more general nonlinearity model by Liao et al. [24]. An asymptotically periodic case was considered in [43], and some properties of ground state solutions were obtained, by constructing linking levels and analyzing behavior of Cerami sequence. Besides, the existence of least energy solution for the non-periodic super-quadratic case was studied in [35]. Recently, the paper [45] studied the Hamiltonian elliptic system with inverse square potential of the form −∆u + b(x) · ∇u + V(x)u − µ |x| v = Hv (x, u, v) (x, u, v) in R N , and the ground state solutions was obtained by using non-Nehari manifold developed by Tang [32]. Moreover, some asymptotic behaviors of ground state solutions, such as the monotonicity and convergence property of ground state energy, were also explored as parameter µ tends to .
When ϵ is small, the standing waves of system (Pϵ) are referred to as semiclassical states. The concentration phenomenon of semiclassical states, when ϵ goes to zero, re ects the transformation process between quantum mechanics and classical mechanics. So it possesses an important physical interest. For such case, the asymptotic behaviors of semiclassical states, such as concentration, convergence and exponential decay, etc., are very interesting problem in mathematics and physics. To put our result in perspective, we review brie y here the background and relate results. There have been intensive interests in studying the existence and qualitative properties of semiclassical states. In [41] the authors considered the singularly perturbed sys- with p ∈ ( , * ), where * is the usual critical exponent. They proved that, by a global variational technique and reduction Nehari method, the semiclassical ground state solution concentrates around the maxima point of the nonlinear potential K as ϵ → . This method and result were later generalized in [42] to the critical nonlinearities case. Further investigations to system with competing potentials have also appeared in [44,46]. For such a problem, the solutions depend not only on the linear potential but also on the nonlinear potential. As was shown in [44,46], the semiclassical ground state solution concentrates around the global minimum points of linear potential V and the global maxima points of nonlinear potential K. Observe that, the method and results mentioned above basically depend on the global condition of the potential V, that is, inf It is worth pointing out that the global condition used in [44,46] plays an important role in proving the existence and concentration of semiclassical solutions. Indeed, the key point is that the property of the potential V at in nity can help us to restore the necessary compactness by comparing energy levels of original problem and limit problem. So, an interesting question, which motivates the present work, is whether one can nd solutions which concentrate around local minima of the potential. As we will see, the answer is a rmative. Hence, based on the above facts, in this paper we will investigate the existence and localized concentration phenomenon of semiclassical states of system (Pϵ) with potential satisfying local condition. More precisely, for the potential V, we assumed that the following local condition rst introduced by del Pino and Felmer in [8]: Compared with [44] and [46], the condition (V) is rather weak, without restriction on the global behavior of V is required, and the behavior of V outside Ω is irrelevant. This fact shows that the limit problem at in nity and its properties are all unknown in this paper. So, from a variational point of view, one of the major di erences between the global condition (1.2) and the local condition (1.3) is that the energy functional, under the local condition (1.3), does not satisfy the so-called compactness condition (such as (PS) or Cerami condition) in general.
Let us now describe the results of the present paper. For notational convenience, let and Sϵ = −ϵ ∆ + . We denote Then system (Pϵ) can be rewritten as Before stating our results, we make the following assumptions on the nonlinearity f : For showing the concentration phenomenon, we denote by V the set V := {x ∈ Ω : V(x) = ν}. Without loss of generality, below we may assume that ∈ V throughout the paper. Moreover, according to (V), we know that dist(V, ∂Ω) > . (1.5) Now we are ready to state the main results of this paper as follows.
as ϵ → , and η is a ground state solution of the following system Due to the above observations, we have an immediate consequence of our main results.

Corollary 1.2.
Assume that | b| < and (F )-(F ) are satis ed. If there exist mutually disjoint bounded domains Ω j , j = , · · · , k and constants ν < ν < · · · < ν k such that Then for all su ciently small ϵ > , (a) system (Pϵ) at least has k nontrivial solution η j ϵ ∈ H ,q for any q ≥ , j = , · · · , k; (b) |η j ϵ (x)| attains its maximum at p j ϵ in Ω j , moreover, up to a subsequence, there holds (c) η j ϵ (ϵx + pϵ) → η j (x) in H (R N , R ) as ϵ → , and η j is a ground state solution of the following system We remark here that in Corollary . , the solutions can be separated provided ϵ > is small since Ω j are mutually disjoint. Furthermore, if ν is a global minimum of V, then Corollary . describes a multiple concentrating phenomenon.
For the proof of our results, we do not handle the system (Pϵ) directly, but instead we handle an equivalent system to (Pϵ). For this purpose, set z(x) = (u(x), v(x)) = (ψ(ϵx), φ(ϵx)) = η(ϵx). Then the system (Pϵ) is equivalent to the following: (P ϵ ) Moreover, system (P ϵ ) can be expressed as where Clearly, ( . ) is equivalent to ( . ). We will, in the sequel, focus on this equivalent problem. As a motivation we recall that there are many enormous investigations concerning with the semiclassical states of Schrödinger equations (1.7) In particular, initiated by Rabinowitz [27], the positive ground state solution of (1.7) for ϵ > small under the global condition (1.2) was proved via mountain pass theorem. After that, Wang [34] showed that the positive solution obtained in [27] concentrates at global minimum points of V. It should be pointed out that, under the local condition (1.3), del Pino and Felmer [8] rst succeeded in proving a localized version of concentration of single-peak solution by using a new penalization approach, moreover, the multi-peak bound state solution was obtained in [9,18]. Based on a singular perturbation argument, the localized bound state solutions concentrating at an isolated component of the local minimum of V were also constructed in [5] and [6]. For further related topics including the Hamiltonian system and Dirac equation, we refer the reader to [3,13,14,17,26,28,33,36] and their references. From the commentaries above, it is quite natural to ask if the localized concentration results of semiclassical states can be obtained for the Hamiltonian elliptic system (Pϵ) as in Schrödinger equation (1.7)? In the present paper, we shall give some answers for this system. However, compared with the Schrödinger equation (1.7), system (Pϵ) becomes more complicated since system (Pϵ) is strongly inde nite in the sense that both the negative and positive parts of the spectrum are unbounded and consist of essential spectrum, and the energy functional has complex geometric structure. Hence our problem poses more challenges in the calculus of variation.
Our argument is based on variational method, which can be outlined as follows. The solutions are obtained as critical points of the energy functional associated to system (Pϵ). We emphasize here that, since the energy functional is strongly inde nite, the classical critical point theory, such as mountain pass lemma and Nehari manifold arguments, cannot be applied directly. On the other hand, the reduction method [1] used in [42,44,46], which reduces the strongly inde nite case to the mountain case, also do not seem to be applicable to our problem. Because such method depends on the convexity of the nonlinearities, speci cally, it requires that the second order derivative of the energy functional in negative de nite on negative space. And by the anti-coercion and concavity properties of the energy functional, one can de ne a reduction functional such that critical points of original functional and reduction functional are in one-to-one correspondence via reduction map. So, along this line, the nonlinearity f requires the strong di erentiability condition: f is of class C . However, we only assume that f satis es continuous condition, and such a reduction method does not work. In addition, the main di culty caused by the unboundedness of the domain is the lack of compactness of Sobolev embedding. Based on the above reasons, some new methods and techniques need to be introduced in the present paper.
More precisely, to prove our results, some arguments are in order. Firstly, since we have no global information on the potential V, we employ the truncation trick and make a slight modi cation of the energy functional corresponding to system (Pϵ). In such a way, the modi ed functional satis es the so-called Cerami compactness condition. Here the modi cation of the energy functional corresponds to a penalization technique "outside Ω"(see [8,9]). Secondly, to overcome the strongly inde niteness of the functional, we utilize the generalized linking theorem and the diagonal method to construct a minimizing Cerami sequence for the modi ed functional, moreover, together with the generalized Nehari manifold, we prove the existence and relation of ground state solution for the modi ed problem and the limit problem. Lastly, the sub-solution estimates of |z| seem not work well since the e ect of the gradient term, we establish the sub-solution estimate of |z| . Moreover, using this fact, we prove the uniformly exponential decay of ground state solution for the modi ed problem, which implies that the solution corresponding to the modi ed problem is indeed the solution of original problem (Pϵ) for ϵ su ciently small. And then Theorem . follows naturally.
The remainder of this paper is organized as follows. In Section , we present the variational setting of the problem, introduce the modi ed functional, and give some useful preliminaries. In Section , we prove the modi ed problem has a ground state solution with ground state energy mϵ. In Section , we show the limit problem possesses a ground state solution, and prove the upper limit of the ground state energy mϵ is less than or equal to the ground state energy of the limit problem as ϵ → . At last, we give the proof of Theorem . in Section .

Variational setting and preliminaries
Below by | · |q we denote the usual L q -norm, (·, ·) denotes the usual L inner product, c, c i or C i stand for di erent positive constants. Denote by σ(A) and σe(A) the spectrum and the essential spectrum of the operator A, respectively. In order to establish a suitable variational framework for system (P ϵ ), we need to analyze some properties of the spectrum of the associated Hamiltonian operator A. The proof can be seen [39], so we omit the details here. It follows from Lemma . and Lemma . that the space L possesses the following orthogonal decomposition such that A is negative de nite (resp. positive de nite) in L − (resp. L + ). Let |A| denote the absolute value of A and |A| be the square root of |A|. Let E = D(|A| ) be the Hilbert space with the inner product and norm z = z, z . There is an induced decomposition which is orthogonal with respect to the inner products (·, ·) and ·, · . According to [39], · and · H are equivalent norms, and thus E embeds continuously into L p := L p (R N , R ) for any p ∈ [ , * ] and compactly into L p loc := L p loc (R N , R ) for any p ∈ [ , * ), and there exists positive constant πp such that Additionally, the decomposition of E induces also a natural decomposition of L q , hence there exists a positive constant dq such that Now we de ne the following functional on E as follows where · denotes the usual inner product in R . Lemma . implies that Iϵ is strongly inde nite. Moreover, our hypotheses imply that Iϵ ∈ C (E, R) and a standard argument shows that critical points of Iϵ are solutions of problem (P ϵ ) (see [10]), and for z, φ ∈ E, there holds As we have mentioned in the introduction, the energy functional Iϵ does not satisfy compactness condition under local potential condition in general, we will not deal with Iϵ directly. Instead, we need make use of the penalization approach developed by del Pino and Felmer [8,9] to modify the energy functional such that the modi ed functional satis es the Cerami condition. After constructing solutions of the modi ed problems we will make these solutions localized, so they are solutions of the original problem for small ϵ.
In virtue of the assumption (V), we can x a small δ > such that where We de ne It is easy to check that (F )-(F ) implies that g is a Caratheodory function and it satis es the following assumptions: From (F ), (F ) and the de nition off , it follows that there exists c > such that According to ( . ) and ( . ), we have This, together with ( . ), we get By (F ) and the de nition off , we can see thatf ≤ −|V|∞ , and Now we are ready to de ne the modi ed functional Φϵ : E → R, Similarly, Φϵ is of class C and the critical points correspond to weak solutions of the following modi ed For the sake of simplicity, in what follows, we denote by Recall that for a functional Φ ∈ C (E, R), Φ is said to be weakly sequentially lower semi-continuous if for any un u in E one has Φ(u) ≤ lim inf n→∞ Φ(un), and Φ is said to be weakly sequentially continuous if We say that Φ satisfy (C)c-condition if any (C)c-sequence has a convergent subsequence in E.
To prove the main results, we need the generalized linking theorem due to [23].
Suppose that the following assumptions are satis ed: (Ψ )Ψ ∈ C (X, R) is bounded from below and weakly sequentially lower semi-continuous; (Ψ )Ψ is weakly sequentially continuous; (Ψ )there exist R > ρ > and e ∈ X + with e = such that Then there exist a constant c ∈ [κ, sup Φ(Q)] and a sequence {un} ⊂ X satisfying We introduce two technical results (see [22,31]), which play an important role in the following proof.
According to Lemma . , we can prove a weaker version result than Lemma . .
Then for t ≥ , x ∈ R N and z, w ∈ R such that z ≠ tz + w, there holds Proof. Applying the method in the proof of [22,Lemma 3.2], for ε > , we de ne hε : It is easy to check that hε(x, s) satis es the corresponding (h )-(h ) of Lemma . . The desired result follows by applying Lemma . to hε and then letting ε → .

The modi ed problem
In this section, we will in the sequel focus on the modi ed problem ( . ) and study the existence of ground state solution. In order to seek for the ground state solutions of the modi ed problem ( . ), we consider the following set which is introduced in Pankov [25] Mϵ := {z ∈ E\E − : Φ ϵ (z)z = and Φ ϵ (z)w = for any w ∈ E − }.
Following from Szulkin and Weth [31], we will call the set Mϵ the generalized Nehari manifold. Obviously, the set Mϵ is a natural constraint and it contains all nontrivial critical points of Φϵ. Let If mϵ is attained by zϵ ∈ Mϵ, then zϵ is a critical point of Φϵ. Since mϵ is the lowest level for Φϵ, then zϵ is called a ground state solution of the modi ed problem ( . ).

De ne
Then, using the fact that E embeds into L q continuously for q ∈ [ , * ] and embeds into L q loc compactly for q ∈ [ , * ), we can check easily the following lemma, and omit the details of proof.
Proof. Observe that On the one hand, by (V) and (2.1) (π = by Lemma . ) we deduce that On the other hand, from (g ), we know that for any x ∈ R N , gϵ(x, s) satis es the assumptions (h )-(h ) in Lemma . . So applying Lemma . , we get the rst conclusion. If z ∈ Mϵ, then Φ ϵ (z)z = Φ ϵ (z)w = , then the second conclusion holds.
For convenience of notation, we write E(z) := E − ⊕ R + z = E − ⊕ R + z + for z ∈ E\E − . Let z ∈ Mϵ, then Lemma 3.2 implies that z is the global maximum of Φϵ| E(z) . Next we shall verify that Φϵ possesses the linking structure.
Proof. (i) Observe that, π = by Lemma . . For z ∈ E + , by (F ), (g ), (g ) and (2.1), we obtain It is easy to see that there exist ρ > and α > both independent of ϵ such that κ := inf Sρ Φ ≥ α since |V|∞ < . So the second inequality holds. Note that for every z ∈ Mϵ there is s > such that sz + ∈ E(z) ∩ Sρ. Clearly, the rst inequality follows from Lemma 3.2.
Using Fatou's lemma, we obtain Since g(x, s)s ≥ , G(x, s) is nondecreasing in s, we deduce from (2.2), and (3.1) that Applying Lemmas . , . , . and . , we have In order to prove the existence of ground state solutions for the modi ed system (2.8), next we construct a (C)c ϵ -sequence for somecϵ ∈ [κ, mϵ] via a diagonal method (see [32]), which is very important in our arguments. Proof. Choose ξ k ∈ Mϵ such that mϵ ≤ Φϵ(ξ k ) < mϵ + k , k ∈ N.

Now, we can choose a sequence {n k } ⊂ N such that
Φϵ(z k,n k ) < mϵ + k and Φ ϵ (z k,n k ) ( + z k,n k ) < k , k ∈ N.
Let z k = z k,n k , k ∈ N. Then, going if necessary to a subsequence, we have , there exist tϵ > and wϵ ∈ E − such that tϵ z + wϵ ∈ Mϵ.

Lemma 3.8. For every ϵ > , let {zn} be a sequence such that Φϵ(zn) is bounded and ( + zn )Φ ϵ (zn) → . Then {zn} has a convergent subsequence.
Proof. We rst show that the sequence {zn} is bounded in E. In fact, suppose that {zn} is a sequence such that Φϵ(zn) is bounded and ( + zn )Φ ϵ (zn) → . Then there exists a positive constant C > , there holds This, together with (g ), we obtain On the other hand, by (V) and (2.7) we get By the Hölder inequality and the fact that χϵ ∈ [ , ], we deduce that Taking the above inequality in (3.10), we obtain − |V|∞ zn ≤ c zn + c , which implies that {zn} is bounded in E. Therefore, after passing to a subsequence, we may assume that zn z in E and zn → z in L q loc for q ∈ [ , * ). Let wn = zn − z, then wn in E, and wn → in L q loc for q ∈ [ , * ). It easy to see that Φ ϵ (zn)(w + n − w − n ) = on( ) and Φ ϵ (z)(w + n − w − n ) = . Then we have (3.11) By the exponential decay of z and the fact that wn → in L q loc for q ∈ [ , * ), we have (3.12) Subtracting the left and right sides of the two equations of (3.11) and using (3.12) we obtain It follows from (V) and (2.1) that Observe thatf ≤ −|V|∞ and χϵ ∈ [ , ], we get Since the support of χϵ is bounded for every xed ϵ > , and wn → in L q loc for q ∈ [ , * ), we know that wn → in E. So zn → z in E.
Next we show the existence of ground state solutions of the modi ed problem (2.8).
By Lemma 3.8, {zn} is bounded, then passing to a subsequence, zn → z in E, zn → z in L q for all q ∈ [ , * ) and zn(x) → z(x) a.e. on R N . Observe that, since zn → z in E, then z ≠ and Φ ϵ (z) = . Hence, z is a nontrivial critical point of Φϵ. Moreover, from (g ) and Fatou's lemma, it follows that which implies that Φϵ(z) ≤ mϵ. On the other hand, by the de nition of mϵ, we know Φϵ(z) = mϵ and z is a ground state solution of the modi ed problem ( . ).

The autonomous problem
In order to prove our main results, we need some results on related autonomous system. For the constant a ∈ (− , ), we consider the autonomous system It is well known that the solutions of system (Pa) are critical points of the functional de ned by |zn| > .
Let us de nezn(x) = zn(x + yn) so that Since system (Pa) is autonomous, we have z n = zn and Ia(zn) → ca ≤ ma and ( + z n )I a (zn) → . (4.4) Passing to a subsequence, we assume thatzn z in E,zn →z in L p loc for ≤ p < * , andzn(x) →z(x) a.e. on R N . Hence it follows from ( . ) and ( . ) thatz ≠ and I a (z) = . This shows thatz ∈ Ma and Ia(z) ≥ ma. On the other hand, from Fatou's lemma, it follows that which implies that Ia(z) ≤ ma. Hence Ia(z) = ma = inf z∈Ma I andz is a ground state solution of system (Pa).
According to (V), we know that ν ∈ (− , ). Taking a = ν, we replace ma, Ia and Ma by mν, Iν and Mν. Thus, as a special case of Lemma . , the system (Pν) has a ground state solution z such that Iν(z ) = mν = inf z∈Mν Iν(z) > . Here, system (Pν) is corresponding limit problem of system (P ϵ ). In the following, we give the relationship of the ground state energy between (P ϵ ) and (Pν).
Observe that zϵ ∈ Q(z + ), then zϵ ≤ R(z + ). Hence, up to a subsequence if necessary, tϵ → t and wϵ w with t ≥ and w ∈ E − . From (F ), (2.1) and Lemma . , we deduce that this shows that t > . Noting that zϵ ∈ Mϵ, we get (4.5) this, together with the fact zϵ tz + w in E, we have Thus by t > , we know tz + w ∈ Mν ∩ E(z), moreover, according to (4.1), we know tz + w = z and hence t = and w = .
Next, we show that zϵ → z in E. Setting ϕϵ = z − zϵ and ϵ(t) = Φϵ(zϵ + tϕϵ), there holds (4.7) Computing directly, we obtain Thus, from (4.6), (4.7) and (4.8) we get Similarly, we can also obtain By (4.9) and (4.10) we have On the other hand, since by a straightforward computation, we deduce that According to (4.11) and (4.12), we obtain (4.13) Since ϕϵ in E and the exponential decay of z, we know and This, jointly with (4.13), implies that (4.14) On the one hand, by (2.1) and the fact that tϵ → we have On the other hand, since gϵ(x, |zϵ|)|zϵ| → f (|z|)|z| a.e. x ∈ R N , by Fatou's lemma we get Consequently, from (4.14) we know that ϕ − ϵ → , and together with tϵ → , then ϕϵ → in E. So we have zϵ → z in E and lim The proof is completed.

Proof of the main result
In this section we give the proof of the main results. Let Kϵ := {z ∈ E\{ } : Φ ϵ (z) = } denote the set of all critical points of Φϵ. To describe some properties of ground state solutions, by using the standard bootstrap argument (see, e.g., [16,29] for the iterative steps) we can obtain the following regularity result.
Let Lϵ be the set of all ground state solutions of Φϵ. If z ∈ Lϵ, then by Lemma . , we know Φϵ(z) = mϵ is uniformly bounded for all small ϵ > . Moreover, by a similar argument as Lemma . , we can also show that Lϵ is bounded in E, hence, |z| ≤ C for all z ∈ Lϵ and some C > . Therefore, as a consequence of Lemma 5.1 we see that, for each q ∈ [ , +∞), there is Cq such that This, together with the Sobolev embedding theorem, implies that there is C∞ > with |z|∞ ≤ C∞ for all z ∈ Lϵ . (5.1)

Lemma 5.2.
Assume that (F ) and | b| < hold, then there is C > independent of x and z ∈ Lϵ such that
The proof of Lemma 5.2 can be found in [43,46], here we omit the details.
Proof. Let ϵ → , zϵ ∈ Lϵ, we rst claim that there exists σ > such that Arguing indirectly, we assume Similar to the proof of Lemma . , we can obtain {zϵ} is bounded in E. Then by Lions' vanishing lemma [20], zϵ → in L q for q ∈ ( , * ), and by (F ) and (g ) we have and for any φ ∈ E Now, the rest of the proof is divided into several steps.
Thus, by (5.1), the exponential decay ofẑ and the fact thatẑϵ j →ẑ in E, it is easy to show that which implies thatẑϵ j →ẑ in H (R N ).