Positive ground state of coupled planar systems of nonlinear Schrödinger equations with critical exponential growth

. In this paper, we prove the existence of a positive ground state solution to the following coupled system involving nonlinear Schrödinger equations: where λ , V 1 , V 2 ∈ C ( R 2 , ( 0, + ∞ )) and f 1 , f 2 : R 2 × R → R have critical exponential growth in the sense of Trudinger–Moser inequality. The potentials V 1 ( x ) and V 2 ( x ) satisfy a condition involving the coupling term λ ( x ) , namely 0 < λ ( x ) ≤ λ 0 (cid:112) V 1 ( x ) V 2 ( x ) . We use non-Nehari manifold, Lions’s concentration compactness and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap regularity lifting argument and L q -estimates we get regularity and asymptotic behavior. Our results improve and extend the previous results.

Solutions of system (1.1) are related with standing waves of the following two-component system: where i denotes the imaginary unit. Such class of systems arise in various branches of mathematical physics and nonlinear optics, see [1]. For instance, solutions of (1.1) are related to the existence of solitary wave solutions for nonlinear Schrödinger equations and Klein-Gordon equations, see [4]. For system (1.2), a solution of the form (ψ(x, t), φ(x, t)) = (e −iMt u(x), e −iMt v(x)), where M is some real constant, called standing wave solution.
In order to motivate our results, we begin by giving a brief survey on this subject. Let us consider the scalar case. Notice that if λ ≡ 0, V 1 ≡ V 2 = V(x), f 1 ≡ f 2 = f and u ≡ v, system (1.1) reduces to the scalar equation This class of nonlinear Schrödinger equation has been widely studied by many researchers, under various hypotheses on the potential V(x) and nonlinear term f (x, u). Such as coercive potential, axially symmetric potential, positive potential and periodic potential. In particular, Chen and Tang [8] developed a direct approach to get nontrivial solutions and ground state solutions when they considered the equation (1.3) in R 2 where V(x) was a 1-periodic function with respect to x 1 and x 2 , 0 lies in the gap of −∆ + V, and the nonlinear term was of Trudinger-Moser critical exponential growth. Using the generalized linking theorem to obtain a Cerami sequence, they showed that the Cerami sequence was bounded and the minimax-level was less than the threshold value by virtue of Moser type functions. Furthermore, they obtained that the Cerami sequence was nonvanishing, which extended and improved the results of [2,17].
For the system of nonlinear Schrödinger equations, there are some results on the linearly coupled system in subcritical and critical case. Chen and Zou [9] studied the following system −∆u + u = f (x, u) + λv, x ∈ R N , −∆v + v = g(x, v) + λu, x ∈ R N , (1.4) where 0 < λ < 1. They discussed the system for non-autonomous and autonomous nonlinearities of subcritical growth respectively. When N ≥ 2, f (x, u) = (1 + a(x))|u| p−1 u and g(x, v) = (1 + b(x))|v| p−1 v, they improved the results of [3] for establishing energy estimates of the ground states. Under some assumptions of potential a(x) and b(x), they obtained not only the existence of positive bound states, but also a precise description of the limit behavior of the bound states as the parameter λ goes to zero. When , and Berestycki-Lions type assumptions were satisfied, they proved system (1.4) had a positive radial ground state, moreover, the behavior and energy estimates of the bound states as λ → 0 were also obtained. Later, Chen and Zou [10] investigated the following coupled systems with critical powertype nonlinearity: where 0 < λ < √ µν, 1 < p < 2 * − 1 and N ≥ 3. They proved the existence of positive ground states for system (1.5) when 0 < µ ≤ µ 0 , where µ 0 ∈ (0, 1) was some critical value. When µ and λ were both large, system (1.5) had a positive ground state also. While, when µ was large but λ was small, the system (1.5) had no ground state solutions. In addition, when p = 2 * − 1, system (1.5) had no nontrivial solutions by the Pohozaev identity. Motivated by [10], Li and Tang [14] considered system (1.5) were continuous functions, 1-periodic in each of x 1 , x 2 , . . . , x N , and satisfied λ(x) < a(x)b(x), they proved system (1.5) had a Nehari-type ground state solution when 0 < a(x) < µ 0 for some µ 0 ∈ (0, 1). Some related linearly coupled systems were also studied in [3,11,12] and the references therein.
In the above references we refer to, it is noticed that the nonlinearities were only considered the polynominal growth of subcritical or critical type in terms of the Sobolev embedding. As we all know, the Trudinger-Moser inequality in R 2 with critical exponential growth instead of the Sobolev inequality in R N with critical polynominal growth, which was first established by Cao in [5], reads as follows.
. then there exists a constant C(M, α), which depends only on M and α, such that By virtue of the Trudinger-Moser inequality, do Ó and de Albuquerque [16] investigated the following linear coupled system with constant potential in R 2 , (1.6) By using the minimization technique over the Nehari manifold and strong maximum principle, the existence of positive ground state solution and the corresponding asymptotic behavior were obtained. In the paper [15], do Ó and de Albuquerque used the same idea as [16] to investigate the existence of positive ground state solution and asymptotic behaviors for the coupled system (1.1) with nonnegative variable potentials. The main problem they faced was to overcome the difficulty originated from the lack of compactness when the nonlinear terms had critical exponential growth in R 2 . Based on this, they considered the following weighted Sobolev space defined by They assumed the following conditions on the potential V i (x), i = 1, 2.
Here, (V1') and (V2') is assumed to ensure that H V i (R 2 ) is a Hilbert space, (V3') and (V4') play a crucial role in overcoming the lack of compactness. In terms of nonlinearities, they defined f i : Here, A i (x) was defined in (V4'). When A i (x) = 1, (F1) holds. In addition, they assumed the following hypotheses: is increasing for t > 0; (F5') There exists q > 2 such that for all x ∈ R 2 and s, t ≥ 0, ϑ > 0 is a constant.
In [15], jointly with (V4'), one can find that the growth of f i are controlled by the growth of V i (x), i = 1, 2 form (F1'). Moreover, the condition f i ∈ C 1 in (F1') plays a crucial role to obtain the Nehari-type ground state solutions for (1.1) via the Nehari manifold method. (F3') is the well-known Ambrosetti-Rabinowitz condition ((AR) condition), which ensures that the functional associated with the problem has a mountain pass geometry and guarantees the boundedness of the Palais-Smale sequence. (F4') is the Nehari monotonic condition. (F5') needs that the nonlinearities are super-q growth at zero, q > 2. It is noticed that sufficiently large ϑ in (F5') is very crucial in their arguments. In fact, by virtue of this condition, the minimax-level for the energy functional can be choosen sufficiently small, therefore the difficult arising from the critical growth of Trudinger-Moser type is easily overcome. But this result has no relationship with the exponential velocity α i 0 , i = 1, 2 , hence it does not reveal the essential characteristics with the critical growth of Trudinger-Moser type.
Recently, Wei, Lin and Tang [20] used non-Nehari manifold methods (see [19]), Lions's concentration compactness and a direct approach derived from [7] for obtaining the minimax estimate to investigate system (1.6) in the non-autonomous case. They proved that (1.6) still possessed a Nehari-type ground state solution and a nontrivial solution. Their results improved the existence results of [16] by weakening the nonlinearities to be continuous, and only needed to satisfy the weaker Nehari monotonic condition, even without (AR) condition. Additionally, since the generalized linking theorem did not work for the strongly indefinite Hamiltonian elliptic system with critical exponential growth in R 2 , Qin, Tang and Zhang [18] developed a new approach to seek Cerami sequences for the energy functional and estimated the minimax levels of these sequences. Furthermore, they used non-Nehari manifold method to obtain the existence of ground state solutions without (AR) condition.
It is interesting to ask if the existence of positive ground state solutions for linearly coupled systems with variable potentials is preserved without (AR) condition. Our aim in this paper is to prove the existence of positive Nehari-type ground state solution of (1.1) and obtain the asymptotic behaviors of ground states with some mild assumptions. This work is motivated by the results of [15,18,20]. Our main result below (Theorem 1.2) can handle the case of f i (x, t) with less restrictions, which are in the true sense of critical exponential growth, and are independent of (F5') with some large constant ϑ (see [15,Theroem 1.1]).
To this end, we emphasize that we need refinements in order to treat the different setting from the constant potentials to the variable ones. Indeed, it is easy to get the mountain pass geometry for the problem with the constant potentials, while for variable potentials, some new analysis techniques and imbedding inequalities such as (2.2) are needed. We borrow the ideas from [18,20] to look for the minimizing Cerami sequence for the energy functional associated with (1.1) by using the non-Nehari manifold approach. By means of slightly weaker monotonic conditions, we show the boundedness of the Cerami sequence. Furthermore, to recover the compactness of the minimizing Cerami sequence, we estimate an accurate threshold for the minimax-level, meanwhile, we use Lions's concentration compactness principle and the invariance of the energy functional by translation to show that the sequence does not vanish. Then by using a standard bootstrap argument and L q -estimates we get regularity and asymptotic behavior of the ground state solution.
To state our main results, in addition to (F1) and (F2), we also introduce the following assumptions: (F3) There exists M 0 > 0 and t 0 > 0 such that for every x ∈ R 2 , In view of Lemma 1.1 i), under assumption (V), (F1) and (F2), the weak solutions of (1.1) correspond to the critical points of the energy functional defined by where · is defined in Section 2, (2.3). Now our main results can be stated as follow. . For obtaining the boundedness of Cerami sequence, we only need the condition (F3) used for the exponential growth problems instead of (F3'). When it comes to the minimax level estimates of the energy functional, the authors in [15] made use of a rigorous limitation on the norm of the minimizing sequence by the the polynomial controlled condition (F5'), while we use the direct calculation argument with the exponential controlled condition (F5). Moreover, we use the weaker monotonicity condition (F4) to replace (F4').

Remark 1.4.
There are many functions satisfying the conditions (F1)-(F5) of the nonlinearities in this paper, but not satisfying the conditions (F4') and (F5') in [15]. For example, fora 1 , a 2 > 0, The paper is organized as follows. In Section 2, we give the variational setting and preliminaries. In Section 3, we establish the minimax estimates of the energy functional. The proof of ground state solution will be stated in Section 4. Then in Section 5, we give the proof of regularity and asymptotic behavior.
Throughout the paper, we make use of the following notations: . . denote positive constants possibly different in different places.

Minimax estimates
In this section, we give a accurate estimation about the minimax level c * defined by Lemma 2.1.
At first, we define a Moser type function w n (x) supported in B √ 2/V M := B √ 2/V M (0) as follows: Proof. By (F5), we can choose ε > 0 and t ε > 0 such that and From (1.7), (3.2) and (3.3), we have There are four cases to distinguish. In the sequel, we agree that all inequalities hold for large n ∈ N without mentioning.
On the other hand, it follows from (2.20) and Fatou's lemma that We have proved that (ũ,ṽ) is a ground state solution for system (1.1). In order to seek a positive ground state, we note by assumptions (F1) and (F2) that Thus, we can deduce that Φ(|ũ|, |ṽ|) ≤ Φ(ũ,ṽ).

The regularity and asymptotic behavior
In this section, we use strong maximum principle to get a unique positive ground state solution, we will introduce methods to show that a weak solution of (1.1) is in fact smooth. Moreover, we establish a priori estimate in W 2,p for the solution of system (1.1), we show that if the functions f 1 (x, u) and f 2 (x, v) are in L p loc (R 2 ), then (u, v) ∈ W 2,p loc (R 2 ) is a strong solution of (1.1), that is, there exists a constant C such that where p i (x) can be defined as (5.3). We will establish this for a Newtonian potential, finally, we use a bootstrap regularity lifting methods to boost the regularity of solution. The bootstrap method can be found in [6, Subsection 3.3.1], which uses a lot of Sobolev imbedding to enhance the regularity of the weak solution repeatedly, finally, Schauder's estimate will lift the solution to be a classical solution. Lemma 5.1. There exists a positive ground state solution (ū,v) ∈ C 1,β for some β ∈ (0, 1) with the following asymptotic behavior Proof. Let (ũ,ṽ) ∈ E be the ground state obtained in Lemma 4.2 It follows from Lemma 2.4 that there exists a unique t 0 > 0 such that (t 0 |ũ|, t 0 |ṽ|) ∈ N . Moreover, since (ũ,ṽ) ∈ N , we point out that max t≥0 Φ(tũ, tṽ) = Φ(ũ,ṽ). Thus we have that Therefore, (t 0 |ũ|, t 0 |ṽ|) ∈ N is a nonnegative ground state solution for (1.1). Next, we denote (ū,v) = (t 0 |ũ|, t 0 |ṽ|). In order to use the strong maximum principle, we note that −ū ∈ H V 1 (R 2 ) \ {0} and take (ϕ, 0) as a test function. Here, ϕ ∈ C ∞ 0 (R 2 ), ϕ ≥ 0. Then we have Now suppose by contradiction that there exists x 0 ∈ R 2 such thatū(x 0 ) = 0. Thus, since −ū ≤ 0 in R 2 , for any R > 0 we have that By the strong maximum principle we conclude that −ū ≡ 0 in R 2 , which is a contradiction. Thereforeū > 0 in R 2 . Similarly, we can prove thatv > 0 in R 2 . Therefore, (ū,v) is positive. In order to obtain the regularity, we use a bootstrap method. The ground state solution (ū,v) is a weak solution of the restricted problem where and in the continuation B 2R = B 2R (x) ⊂ R 2 denote the ball centered in a fixed point x ∈ R 2 . Since V i (x) ∈ C(R 2 ), then V i (x), λ(x) ∈ L ∞ loc (R 2 ). Forū,v ∈ L p (R 2 ), p ≥ 2, we have that λ(x)v, V 1 (x)ū ∈ L p (B 2R ) for all p ≥ 2. By (F1) and (F2), for ε > 0, p, q ≥ 2, r > p and α > α 1 , we have that ε p |ū| p dx + C 13 B 2R C p ε (e αū 2 − 1) p |ū| p(q−1) dx ≤ C 13 ε p ū p L p (B 2R ) + C 13 B 2R C p ε (e rαū 2 − 1)|ū| p(q−1)−1 |ū|dx. (5.4) By using Hölder's inequality, it follows from Lemma 1.1 that Since the right-hand side is finite for all p ≥ 2, we have that f 1 (x,ū) ∈ L p (B 2R ) for all p ≥ 2, together with λ(x)v, V 1 (x)ū ∈ L p (B 2R ) , we have that p 1 (x) ∈ L p (B 2R ) for all p ≥ 2. Let f p 1 be the Newtonian potential of p 1 (x). In light of L p -regularity theory [6, Theorem 3.1.1], ∇(ū − f p 1 )φdx = 0, ∀φ ∈ C ∞ 0 (B 2R ).