Concentration behavior of semiclassical solutions for Hamiltonian elliptic system

1 Introduction and main results

In this paper, we will consider the following Hamiltonian elliptic system with gradient term

where η = (ψ, φ) : ℝ^N → ℝ², ϵ is small positive parameter, b⃗ 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 (𝓟_ϵ) when ϵ → 0.

This type of systems arises when one is looking for the standing wave solutions to system of diffusion equations

which comes from the time-space diffusion processes and is related to the Schrödinger equations. It appears in various fields, 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 b⃗ = 0. More specifically, 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 b⃗ ≠ 0 and ϵ = 1, as we all know, there are a few works devoted to the existence and multiplicity of solutions of the following system under different assumption

see [24, 35, 39, 40, 43]. For this case, since the appearance of the gradient term in system, system (1.1) has some differences and difficulties compared with the case b⃗(x) = 0. For example, the variational framework for the case b⃗(x) = 0 cannot work any longer in this case, then the first problem is how to establish a suitable variational framework. To solve this problem, Zhao and Ding [39] handled (1.1) as a generalized Hamiltonian system, and established a strongly indefinite 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 indefinite 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

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 0.

When ϵ is small, the standing waves of system (𝓟_ϵ) are referred to as semiclassical states. The concentration phenomenon of semiclassical states, when ϵ goes to zero, reflects 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 briefly 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 system

with p ∈ (2, 2^*), where 2^* 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 → 0. 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,

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 infinity 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 find solutions which concentrate around local minima of the potential. As we will see, the answer is affirmative. Hence, based on the above facts, in this paper we will investigate the existence and localized concentration phenomenon of semiclassical states of system (𝓟_ϵ) with potential satisfying local condition. More precisely, for the potential V, we assumed that the following local condition first introduced by del Pino and Felmer in [8]:

1. V ∈ C(ℝ^N, ℝ), max|V| < 1, and there is a bounded domain Ω in ℝ^N such that

   ν=minx∈ΩV(x)<minx∈∂ΩV(x). (1.3)

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 infinity and its properties are all unknown in this paper. So, from a variational point of view, one of the major differences 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 𝓢_ϵ = –ϵ²Δ + 1. We denote

Then system (𝓟_ϵ) can be rewritten as

Before stating our results, we make the following assumptions on the nonlinearity f:

1. f ∈ C(ℝ⁺, ℝ), f(s) → 0 as s → 0 and f is increasing on ℝ⁺ = [0, ∞);

2. there exist c₀ > 0 and p ∈ (2, 2^*) such that f(s) ≤ c₀(1 + s^p–2) for s ≥ 0;

3. there exists μ > 2 such that 0 < μ F(s) ≤ f(s)s² for all s > 0, where F(s) = ∫0s f(t)tdt.

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 0 ∈ 𝓥 throughout the paper. Moreover, according to (V), we know that

Now we are ready to state the main results of this paper as follows.

2 Variational setting and preliminaries

Below by |⋅|_q we denote the usual L^q- norm, (⋅, ⋅)₂ denotes the usual L² inner product, c, c_i or C_i stand for different positive constants. Denote by σ(A) and σ_e(A) the spectrum and the essential spectrum of the operator A, respectively. In order to establish a suitable variational framework for system ( Pϵ′ ), 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 2.1 and Lemma 2.2 that the space L² possesses the following orthogonal decomposition

such that A is negative definite (resp. positive definite) in L^–(resp. L⁺). Let |A| denote the absolute value of A and |A|12 be the square root of |A|. Let E = 𝓓( |A|12 ) be the Hilbert space with the inner product

and norm ∥z∥=〈z,z〉12. There is an induced decomposition

which is orthogonal with respect to the inner products (⋅, ⋅)₂ and 〈⋅, ⋅〉. According to [39], ∥⋅∥ and ∥⋅∥_H¹ are equivalent norms, and thus E embeds continuously into L^p := L^p(ℝ^N, ℝ²) for any p ∈ [2, 2^*] and compactly into Llocp := Llocp (ℝ^N, ℝ²) for any p ∈ [1, 2^*), and there exists positive constant π_p such that

Additionally, the decomposition of E induces also a natural decomposition of L^q, hence there exists a positive constant d_q such that

Now we define the following functional on E as follows

where ⋅ denotes the usual inner product in ℝ². Lemma 2.2 implies that I_ϵ is strongly indefinite. Moreover, our hypotheses imply that I_ϵ ∈ C¹(E, ℝ) and a standard argument shows that critical points of I_ϵ are solutions of problem ( Pϵ′ ) (see [10]), and for z, φ ∈ E, there holds

As we have mentioned in the introduction, the energy functional I_ϵ does not satisfy compactness condition under local potential condition in general, we will not deal with I_ϵ 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 modified functional satisfies the Cerami condition. After constructing solutions of the modified 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 fix a small δ > 0 such that

where Ω^δ = {x ∈ ℝ³ : dist(x, Ω) := inf_y∈Ω |x – y| < δ} is the δ-neighborhood of Ω and Ω^δ is the closure of Ω^δ. Let ζ ∈ C(ℝ^N, ℝ) be a function such that 0 ≤ ζ(s) ≤ 1, ζ(s) = 1 if s ≤ 0 and ζ(s) = 0 if s ≥ δ. We choose a suitable constant a₀ > 0 such that f(a0)=1−|V|∞2, and set χ(x) = ζ(dist(x, Ω)) and f̃ ∈ C(ℝ⁺, ℝ):

We define

then

It is easy to check that (F₀)-(F₂) implies that g is a Caratheodory function and it satisfies the following assumptions:

1. g(x, s) → 0 uniformly for x ∈ ℝ^N, and g(x, s) is nondecreasing in s ∈ ℝ⁺ for x ∈ ℝ^N;

2. 0 ≤ g(x, s) ≤ f(s) for every (x, s) ∈ ℝ^N × ℝ⁺;

3. f(s)s² – 2F(s) ≥ 0 and f̃(s)s² – 2F̃(s) ≥ 0 for all s ≥ 0.

From (F₀), (F₁) and the definition of f̃, it follows that there exists c₁ > 0 such that

and f(s) ≤ c₁s^p–2 for s ≥ a₀. So for s ≥ a₀, there holds (f(s)s)1p−1≤c1s and

By (F₃), we obtain

According to (2.5) and (2.6), we have

This, together with (2.4), we get

By (F₀) and the definition of f̃, we can see that f~≤1−|V|∞2, and

Now we are ready to define the modified functional Φ_ϵ : E → ℝ,

Similarly, Φ_ϵ is of class C¹ and the critical points correspond to weak solutions of the following modified system

For the sake of simplicity, in what follows, we denote by

Recall that for a functional Φ ∈ C¹(E, ℝ), Φ is said to be weakly sequentially lower semi-continuous if for any u_n ⇀ u in E one has Φ(u)≤lim infn→∞Φ(un), and Φ^′ is said to be weakly sequentially continuous if limn→∞Φ′(un)v=Φ′(u)v for each v ∈ E. We recall that a sequence {u_n} ⊂ E is called Cerami sequence for Φ at the level c ((C)_c-sequence in short) 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].

We introduce two technical results (see [22, 31]), which play an important role in the following proof.

According to Lemma 2.4, we can prove a weaker version result than Lemma 2.4.

3 The modified problem

In this section, we will in the sequel focus on the modified problem (2.8) and study the existence of ground state solution. In order to seek for the ground state solutions of the modified problem (2.8), we consider the following set which is introduced in Pankov [25]

Following from Szulkin and Weth [31], we will call the set 𝓜_ϵ the generalized Nehari manifold. Obviously, the set 𝓜_ϵ is a natural constraint and it contains all nontrivial critical points of Φ_ϵ. Let

If m_ϵ is attained by z_ϵ ∈ 𝓜_ϵ, then z_ϵ is a critical point of Φ_ϵ. Since m_ϵ is the lowest level for Φ_ϵ, then z_ϵ is called a ground state solution of the modified problem (2.8).

Define

Then, using the fact that E embeds into L^q continuously for q ∈ [2, 2^*] and embeds into Llocq compactly for q ∈ [1, 2^*), we can check easily the following lemma, and omit the details of proof.

For convenience of notation, we write E(z) := E^–⊕ ℝ⁺z = E^– ⊕ ℝ⁺z⁺ for z ∈ E ∖ E^–. Let z ∈ 𝓜_ϵ, then Lemma 3.2 implies that z is the global maximum of Φ_ϵ|_E(z). Next we shall verify that Φ_ϵ possesses the linking structure.

It follows from (F₂) and the definition of g that there exists a positive constant c₄ such that

For any fixed e ∈ E⁺ with ∥e∥ = 1, there exists R > 0 such that

where d_μ is given in (2.2). Setting