Abstract
This paper is concerned with the following Hamiltonian elliptic system
where z = (u, v) : ℝN → ℝ2, V(x) and W(x, z) are 1-periodic in x. By making use of variational approach for strongly indefinite problems, we obtain a new existence result of nontrivial solution under new conditions that the nonlinearity
1 Introduction and main result
Consider the coupled nonlinear Schrödinger system
where ϕ = (ϕ1, ϕ2), i is the imaginary unit, m is the mass of a particle,
which propagate without changing their shape and thus have a soliton-like behavior. In general, the above coupled nonlinear Schrödinger system leads to the following elliptic system
where N ≥ 3, z := (u, v) : ℝN → ℝ × ℝ, V : ℝN → ℝ and W : ℝN × ℝ2 → ℝ. Moreover, according to [5], this type of system is called Hamiltonian elliptic system, which has strongly indefinite variational structure from a viewpoint of variational methods. In the present paper, our purpose is to establish new existence results of nontrivial solutions of system (1.2) under new nonlinear conditions.
In the past few decades, by using variational techniques, a number of important results of the existence and multiplicity of solutions for system (1.2) defined on the bounded domain Ω have been established with W satisfying various conditions, see for example [7, 9, 10]. Recently, many authors began to focus on system (1.2) defined on the whole space ℝN. Their most interesting studies were to establish the existence of multiple solutions, ground states and semiclassical states, see [1, 2, 4, 6, 12, 18, 21, 25, 26, 27, 28, 29, 31, 32] and their references therein. In these works, a huge machinery is needed to obtain existence and multiplicity of solutions, such as fractional Sobolev spaces, generalized mountain pass theorem, generalized linking theorem, reduction Nehari method and many others. Besides, based on variational arguments, some related problems involving the nonlocal elliptic equations have been received increasingly more attention on mathematical studies . Tang and Chen [20] studied the ground state solution of Nehari-Pohozaev type for the nonlocal Schrödinger-Kirchhoff problems by developing some new analytic techniques. More relevant results and recent developments, we refer the readers to [16] for elliptic problems and the monograph [17] for nonlocal fractional problems.
One of the main difficulties in dealing with system (1.2) relies on the lack of embedding compactness due to the unboundedness of the domain. In some of the above quoted papers this difficulty was overcame by imposing periodicity condition both on the potential V and the nonlinearity W. Along this direction, the papers [8, 13, 14, 32] studied system (1.2) with periodic and global super-quadratic growth, and the existence and multiplicity results are obtained. Subsequently, the authors in [30]weakened the global super-quadratic case to the local super-quadratic case and proved the existence of ground state solutions and infinitely many geometrically distinct solutions by using a new perturbation approach developed by Tang and his collaborators [22, 23].
Motivated by the researches about the Hamiltonian elliptic systems, we continue to study system (1.2) under general conditions, and assume the following basic assumptions.
(V) V ∈ C(ℝN , ℝ), V(x) is 1-periodic in each of x1, x2, . . . , xN, and maxℝN V(x) := β ≥ V(x) ≥ minℝN V(x) := β0 > 0;
(W1) W ∈ C1(ℝN × ℝ2), W(x, z) is 1-periodic in each of x1, x2, . . . , xN, W(x, z) ≥ 0;
(W2) |Wz(x, z)| = o(|z|), as |z| → 0 uniformly in x ∈ ℝN.
In the aforementioned references [12, 25, 26, 27, 28], the following asymptotically quadratic condition and other technique conditions for the nonlinearity W are commonly assumed:
(W3)
(W4)
(W5)
Observe that conditions (W4) and (W5) play an important role for showing that any Palais-Smale sequence or Cerami sequence is bounded in the aforementioned works. However, there are many functions do not satisfy these conditions, for example,
In a recent paper [14], making use of some special techniques, Liao, Tang, Zhang and Qin studied the existence of solutions for system (1.2) under more general super-quadratic conditions, that is,
(SQ) there exist a, b > 0 such that
Clearly, this condition is weaker than the usual super-quadratic condition
Very recently, the singularly perturbed problem with super-quadratic condition (SQ)
has been investigated in [21], where the authors proved the existence of semiclassical ground state solutions and generalized the results in [4].
Inspired by super-quadratic case [14] and [21], we further consider the general periodic asymptotically quadratic case and establish the existence result of solutions. In addition to (V), (W1) and (W2), we introduce the following new asymptotically quadratic conditions for the nonlinearity W:
(W3′) there exists symmetric non-negative definite matrix
such that
(W4′)
(W5′)
It is worth pointing out that conditions (W3′), (W4′) and (W5′) are different from usual conditions (W3), (W4) and (W5) and weaken these conditions. To the best of our knowledge, it seems that there is no work considered this problem in the literature before. So this result obtained in this paper is new, moreover, it can be viewed as a complement and an extension of [14] and [21]. However, it is difficult for us to prove the linking geometry and boundedness of Cerami sequences under the conditions (W3′), (W4′) and (W5′) since the arguments as [26, 27] (depending on the behavior of W(x, z) as
Based on the conditions given above on V and W, we can get the following theorem.
Theorem 1.1
Assume that V and W satisfy (V), (W1), (W2), (W3′), (W4′) and (W5′). Then system (1.2) has a nontrivial solution.
Before proceeding to the proof of Theorem 1.1, we give a nonlinear example to illustrate the assumptions.
Example 1.2
For example, let
where V∞(x) ∈ (ℝN , ℝ) is 1-periodic in each of x1, x2, . . . , xN, and infℝN V∞(x) > β0. By a straightforward computation, we can see that all conditions (W1), (W2), (W3′), (W4′) and (W5′) are satisfied with
The remainder of this paper is organized as follows. In Sect. 2, we introduce the variational setting of system (1.1). In Sect. 3, we analyze the geometry structure of the functional and property of Cerami sequence, and give the proof of Theorem 1.1.
2 Variational setting
Throughout this paper, we make use of the following notations.
In the following, we establish the variational setting of system(1.2). According to condition (V), we define the following Hilbert space
with the inner product
and the reduced norm
Let the working space E = H × H. Then E is a Hilbert space with the standard inner product
for zi = (ui , vi) ∈ E, i = 1, 2, and the corresponding norm
Observe that, the natural functional associated with system (1.2) is given by
Moreover, according to conditions (V), (W1) and (W2), it is easy to prove that Φ ∈ C1(E, ℝ), and for any z = (u, v) and η = (φ, ψ) ∈ E, there holds
Following the idea of De Figueiredo and Felmer [7] (see also Hulshof and Van Der Vorst [9]), for any z = (u, v) and w = (w1, w2) ∈ E, we introduce a bilinear form on E × E as
It is clear that
By a direct computation, we can deduce that
Moreover, it is easy to see that 1 and −1 are two eigenvalues of the operator
Hence, based on the above fact, the working space E has the following decomposition
Clearly, E = E− ⊕ E+. Furthermore, for z = (u, v) ∈ E, set
Then we have
Now we define the functional
Computing directly, we get
Therefore, the functional Φ defined by (2.1) can be rewritten the following form
Obviously, Φ is strongly indefinite and the critical points of Φ are solutions of system (1.2) (see [3]), and for z, φ ∈ E we have
On the other hand, according to the embedding theorem and condition (V), H embeds continuously into Lp(ℝN) for all p ∈ [2, 2*] and compactly into
3 Proof of main result
In this section, we will in the sequel focus on the proof of Theorem 1.1. Firstly, we need verify the linking geometry structure of the functional Φ.
Lemma 2.1
Assume that (V), (W1) and (W2) hold, then there exists a constant ρ > 0 such that
Applying the embedding theorem and some standard arguments (see [3] and [21]), one can check easily the Lemma 2.1, and omit the details of the proof.
Lemma 2.2
Assume that (V), (W1), (W2), (W3′) and (W4′) hold. Let e = (e0, e0) ∈ E+ with
Proof
Obviously, it follows from (W1) that Φ(z) ≤ 0 for z ∈ E− . Next, it is sufficient to show that Φ(z) → −∞ as z ∈ E− ⊕ ℝe and
By (3.2) we can deduce that τ > 0. Observe that, since e ∈ E+, there is a bounded domain Ω ⊂ ℝN such that
Moreover, by (W3′) and (3.2) we obtain
Clearly, according to (W3′) and (W4′), we know that
for some c0 > 0 and any z = (u, v) ∈ E, and
Since
First, according to τ > 0, we need to show (a11 + a12)(τe0 + v) + (a21 + a22)(τe0 − v) ≠ 0. Arguing indirectly, we assume that
which yields a contradiction. Therefore, from the above fact we can deduce that
Let ϕn = (a11 + a12)(τne0 + vn) + (a21 + a22)(τne0 − vn). By Lebesgue dominated convergence theorem and (W4′) we have
Moreover, by the embedding theorem, (3.4) and (3.5) we deduce that
which implies a contradiction to (3.3).
In order to end the proof of Theorem1.1, we will use the following generalized linking theorem developed by Li and Szulkin [11].
Lemma 2.3
Assume that Z is a Hilbert space with the decomposition Z = Z− ⊕ Z+, and let Φ ∈ C1(Z, ℝ) be of the form
Assume that the following assumptions are satisfied:
(Φ1)Ψ ∈ C1(Z, ℝ) is bounded from below and weakly sequentially lower semi-continuous;
(Φ2)
(Φ3)there exist R > ρ > 0 and e ∈ Z+ with
where
Then there exist a constant c ∈ [κ, supΦ(Q)] and a sequence {un} ⊂ X satisfying
For the sake of convenience, let
Using some standard arguments, we can verify Ψ is nonnegative, weakly sequentially lower semi-continuous, and Ψ′ is weakly sequentially continuous. Moreover, applying Lemma 2.1, Lemma 2.2 and Lemma 2.3, we can demonstrate the following result.
Lemma 2.4
Assume that (V), (W1), (W2), (W3′), (W4′) hold. Then there exists a constant c* ∈ [κ0, supΦ(Q)] and a sequence {zn} = {(un , vn)} ⊂ E satisfying
Lemma 2.5
Assume that (V), (W1), (W2), (W3′), (W4′) and (W5′) are satisfied. Then any sequence {zn} = {(un , vn)} ⊂ E satisfying (3.6) is bounded in E.
Proof
In order to prove the boundedness of {zn} = {(un , vn)}, arguing by contradiction we suppose that
By (W1), (2.3) and (3.6) we obtain
and
Computing directly, we have
which implies that
Note that
which implies that
then by Lions’s concentration compactness principle [24, Lemma 1.21] we have (a11+a12)φn +(a21+a22)ψn → 0 in Ls for 2 < s < 2*. Set θ ∈ (0, 1) and
Hence, by (W1), (W2), (W3′) and (W4′), it follows from (3.9) and Hölder inequality that
On the other hand, set q′ = q/(q − 1), then 2 < 2q′ < 2*, by (W5′), (3.8), (3.9) and Hölder inequality we obtain
Combining (3.11) with (3.12), and using (2.2) and (3.9), we have
This contradiction shows that the vanishing case does not occur. Therefore, we get δ ≠ 0.
If necessary going to a subsequence, we may assume the existence of kn ∈ ZN such that
Since
we can obtain
Let us define
Now we define
in E,
in
a.e. on ℝN. Obviously, it follows from (3.14) that
Set ηn = η(x + kn) for each
and hence
Observe that
Using the embedding theorem and Hölder inequality we have
Moreover, it follows from (W4′) and (3.15) that
Combining the above fact, and letting n → ∞in (3.16) we have
for each
Set
Hence, from (3.17) we can duduce that (φ, ψ) is an eigenfunction of
Proof of Theorem 1.1
Employing Lemma 2.4, there exists a sequence {zn} = {(un , vn)} ⊂ E of Φ such that
By virtue of Lemma 2.5, we know that {zn} is bounded in E, namely, there exists a positive constant c such that
Acknowledgement
This work was supported by the NNSF (11701487, 12071395), Scientific Research Fund of Hunan Provincial Education Department (18B342, 19C1700).
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Conflict of interest:
Authors state no conflict of interest.
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© 2021 Fangfang Liao and Wen Zhang, published by De Gruyter
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