Existence of Standing Wave Solutions for Coupled Quasilinear Schrödinger Systems with Critical Exponents in R

This paper is concerned with the following quasilinear Schrödinger system in R N : −ε 2 ∆u + V 1 (x)u − ε 2 ∆(u 2)u = K 1 (x)|u| 22 * −2 u + h 1 (x, u, v)u, −ε 2 ∆v + V 2 (x)v − ε 2 ∆(v 2)v = K 2 (x)|v| 22 * −2 v + h 2 (x, u, v)v, where N ≥ 3, V i (x) is a nonnegative potential, K i (x) is a bounded positive function, i = 1, 2. h 1 (x, u, v)u and h 2 (x, u, v)v are superlinear but subcritical functions. Under some proper conditions, minimax methods are employed to establish the existence of standing wave solutions for this system provided that ε is small enough, more precisely, for any m ∈ N, it has m pairs of solutions if ε is small enough. And these solutions (u ε , v ε) → (0, 0) in some Sobolev space as ε → 0. Moreover, we establish the existence of positive solutions when ε = 1. The system studied here can model some interaction phenomena in plasma physics.


Introduction
In this article we discuss the following coupled quasilinear Schrödinger system with critical exponents in (1.1) In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the form: where ε > 0 is a small parameter (e.g.see [28,31]).Part of the interest is due to the fact that the solution of (1.2) is closely related to the existence of solitary wave solutions for the following equation: where w : R × R N → C, V(x) is a given potential, k is a real constant, f , h are suitable functions.In fact, the quasilinear equation (1.3) has been derived as models of several physical phenomena.For example, it models the superfluid film equation in plasma physics [20], in self-channeling of a high-power ultra short laser in matter [3,6,24], in condensed matter theory [22] etc.It is worth pointing out that the related semilinear Schrödinger equation arises in many mathematical physics problems and has been extensively studied.We only mention [9,11,19,23] and the references therein.Also, there are more and more papers being concerned with semilinear Schrödinger system involving two condensate amplitudes w 1 , w 2 .For example, Chen and Zhou [7] proved the uniqueness of positive solutions under some conditions for a coupled Schrödinger system.Tang [27] was concerned with multi-peak solutions to coupled Schrödinger systems with Neumann boundary conditions in a bounded domain of R N for N = 2, 3 and proved that all peaks locate either near the local maxima or near the local minima of the mean curvature at the boundary of the domain.Yang, Wei and Ding [30] studied a Schrödinger system with nonlocal nonlineatities of Hartree type.Ye and Peng [32] considered a coupled Schrödinger system with doubly critical exponents on R N , which can be seen as a counterpart of the Brezis-Nirenberg problem.
Recently quasilinear systems also have been the focus for some researchers (e.g.[16,17,25]).But compared with semilinear systems, only a few papers are known for them.Guo and Tang [17] proved the existence of a ground state solution by using Nehari manifold and concentration compactness principle in a Orlicz space.Severo and Silva [25] established the existence of standing wave solutions for quasilinear Schrödinger systems involving subcritical nonlinearities in Orlicz spaces.By referring to some arguments and methods in [11,25,30,31], we consider the quasilinear Schrödinger systems (1.1) with critical nonlinearities and discuss the existence of a positive solution and multiple solutions as ε is small.Of particular interest to our paper is the results in [31], where the authors investigated the quasilinear Schrödinger equation (1.2) with critical exponent h(x, u) = K(x)|u| 22 * −2 + H u (x, u) and proved it has at least one positive solution and multiple solutions when ε is small,where H u (x, u) is a superlinear but subcritical function and satisfies some suitable conditions.The difficulty is caused by the usual lack of compactness since these problems involve critical exponents and are dealt with in the whole R N .We remark that most papers above use the Cerami condition.But in this paper we prove that (PS) c condition also holds.We suppose that the following assumptions are satisfied, where i = 1, 2: where m denotes the Lebesgue measure; and satisfy the following conditions.
(H 1 ) There is a constant 4 Notations.We collect below a list of the main notation used throughout this paper.
• C will denote various positive constants whose value may change from line to line.
• If the functions f and g satisfy • The domain of integration is R N by default.

•
f (x)dx will be represented by f (x).
• We use L s (R N ), 1 ≤ s ≤ ∞, to denote the usual Lebesgue spaces with the norms • S denotes the best Sobolev constant for H 1 (R N ).
Theorem 1.1.Assume that (V 1 )-(V 2 ), (K) and (H 1 )-(H 5 ) are satisfied.Then for any σ > 0, there is τ σ > 0 such that if ε ≤ τ σ , system (1.1) has at least one positive solution u ε = (u ε , v ε ).Moreover, for any m ∈ N and σ > 0, there is τ σm > 0 such that if ε ≤ τ σm , system (1.1) has at least m pairs of solutions The existence and multiplicity of solutions for system (1.1) depends on the small parameter ε.If the parameter ε is not small enough, such as ε ≡ 1, we cannot get the similar results as Theorem 1.1 unless we add some suitable conditions, where i = 1, 2: and there is a constant a 0 > 0 such that and there is a point Theorem 1.2.Let ε = 1.Assume that (V 3 ), (K ), (H 1 )-(H 4 ) and (H 6 ) are satisfied.Then system (1.1) has at least one positive solution u = (u, v) if N and q satisfy one of the following two conditions: Remark 1.1.Guo and Li in [18] discussed a class of modified nonlinear Schrödinger systems where F(u, v) = |u| α |v| β + |u| p |v| q , α, β, p, q > 1, α + β = 22 * and 4 < p + q < 22 * , and they proved the existence of a ground state positive solution by using a perturbation method.For the special case of a ij (s) = (1 + 2s 2 )δ ij , system (1.4) can be rewritten as Comparing with (1.5), the coupling term in the present paper is not critical growth, but is more general than the coupling subcritical term of (1.5).The subcritical nonlinearities of (1.5) do not satisfy our condition (H 4 ).Hence, the proof in this paper is different from the one in [18].
The organization of this paper is as follows.In Section 2, we introduce the variational framework and restate the problem in a equivalent form by replacing ε −2 with λ.Furthermore, we reduce the quasilinear problem into a semilinear one by making change of variables and show some preliminary results.In Section 3, we prove the behaviors of the bounded (PS) c sequences and then show that the energy functional satisfies the (PS) c condition under some suitable conditions.In Section 4, we verify the geometry of the mountain pass theorem and estimate the minimax values.In Section 5, we complete the proof of Theorem 1.1.In the final section, we prove Theorem 1.2.

An equivalent variational problem
To prove the existence of standing wave solutions of system (1.1) for small ε, we rewrite (1.1) in a equivalent form.Let λ = ε −2 .Then system (1.1) can be rewritten as for λ → +∞.
We introduce the Hilbert spaces and the associated norms We shall work in the product space E = E 1 × E 2 with elements u = (u, v).Thus, the norm in E can be defined as ) and (V 2 ) that E i embeds continuously in H 1 (R N ) (e.g.see [12]) and consequently E embeds continuously in Associated to system (2.1), the energy functional is To save from this trouble, we make use of a change of variables u [8,10,13,21]), where f is defined by We list some properties of f .Their proofs may be found in the above references.
Lemma 2.1.The function f satisfies the following properties: (i) f is uniquely defined, C ∞ and invertible; (x) there exists a positive constant A such that After the change of variables, we obtain the following functional Then Φ λ is well-defined on E and belongs to C 1 under hypotheses (V 1 ), (V 2 ), (K) and (H 3 ).Furthermore, we can check that it is a weak solution of the following system associated with the functional ) is a weak solution of system (2.1) (cf.[8]).Theorem 1.1 can be restated as Theorem 2.1.Assume that (V 1 )-(V 2 ), (K) and (H 1 )-(H 5 ) are satisfied.Then for any σ > 0, there is 2) has at least one positive solution u λ = (u λ , v λ ).Moreover, for any m ∈ N and σ > 0, there is Remark 2.1.In order to get the positive solution, we introduce where u + := max{u, 0}, v + := max{v, 0}.Then Φ + λ ∈ C 1 and the critical points of Φ + λ are the positive solutions of system (2.2).

Behavior of (PS) c sequences
At this point, we recall that a sequence (u n ) ⊂ E is a (PS) c sequence at level c ((PS) c sequence for short), if Φ λ (u n ) → c and Φ λ (u n ) → 0. Φ λ is said to satisfy the (PS) c condition if any (PS) c sequence contains a convergent subsequence.However, due to the unboundedness of the domain and the critical term, we can not prove the (PS) c condition holds in general.By establishing several lemmas, we will discuss the behaviors of (PS) c sequences.Lemma 3.1.Suppose that (V 2 ), (K) and (H 1 ) hold.Let (u n ) ⊂ E be a (PS) c sequence for Φ λ .Then c ≥ 0 and (u n ) is bounded in E.
Proof.Set (u n ) to be a (PS) c sequence: By Lemma 2.1 (vi) and (H 1 ), one sees that Hence From (3.2), we only need to prove that λ We write that Combining (V 2 ), (K), (3.2) and Lemma 2.1 (viii), we have Taking the limit in (3.1) we shows that c ≥ 0.
By the above lemma, we know that every (PS) c sequence (u n ) is bounded.We may assume up to a subsequence that u n u in E and in Clearly u is a critical point of Φ λ .Lemma 3.2.Let (u n ) be stated as in Lemma 3.1 and s ∈ [2, 2 * ).There is a subsequence (u n j ) such that for each > 0, there exists R > 0 with Proof.The proof is similar as that in [11].We omit it here.
For notational convenience, we can assume in the following that Lemma 3.2 holds for both s = 2 and s = p+1 2 with the same subsequence.Let where p ∈ [2, 22 * ].
Proof.We only show that the first equality holds.As in [29], for any fixed > 0, there exists C > 0 such that, for all a, b ∈ R ||a + b| q − |a| q | ≤ |a| q + C |b| q , 1 ≤ q < +∞.
We deduce that, by Lemma 2.1 (ix), for any fixed > 0, there exists C > 0 such that where and below θ ∈ (0, 1).Then by Lemma 2.1 (vii) The Lebesgue Dominated Convergence Theorem implies that Γ n j → 0 as j → ∞.Hence Lemma 3.4.Let (u n j ) be stated as in Lemma 3.2.Denote by We have Proof.Note that (3.3) and the local compactness of the Sobolev embedding theorem imply that, for any R > 0 uniformly for ϕ 1,λ ≤ 1.For any ε > 0, from (3.3) and the integrability of |u| s on R N , we can choose R > 0 such that lim sup Combining (H 2 ), (H 3 ) and Lemma 2.1 (ii), (iii), (vii), we get that Therefore, it follows from (3.4), (3.5), the Hölder inequality and Lemma 3.2 that lim sup uniformly for ϕ 1,λ ≤ 1.Similarly, we can get that the other equality holds.Lemma 3.5.Let (u n j ) be stated as in Lemma 3.2.One has along a subsequence: Proof.(i) Obviously, we can see We claim that By conditions (V 2 ), (K) and Lemma 3.3, we conclude that (3.6)-(3.9)hold.Similar to the proof of Lemma 3.4, it is easy to see that (3.10) holds.Using the fact Φ λ (u n j ) → c and Φ λ ( ũj ) → Φ λ (u), we get conclusion 1.
(ii) We first notice that, for any given w = (ϕ, ψ) ∈ E satisying w λ ≤ 1, where h n j (x) and g n j (x) are stated in Lemma 3.4.Noticing the boundedness of (u n j ) in E, the equality the mean value theorem, Lemma 2.1 (vii), (ix) and the Hölder inequality, we have for R > 0 We have also that Thus, for every > 0, there exists R = R > 0 such that for any ϕ 1,λ ≤ 1 On the other hand, applying the Rellich compact embedding theorem, we have Hence, by (K), we get that uniformly for w λ ≤ 1.Since Φ λ (u n j ) → 0 and Φ λ ( ũj ) = 0, we get the conclusion 2 by Lemma 3.4.
Lemma 3.6.If the conditions (V 1 ), (K) and (H 1 )-(H 3 ) hold.There is a constant α 0 > 0 being independent of λ such that, for any (PS) c sequence (u n ) for Φ λ with u n u, either Obviously, there exists a constant γ b 2 > 0 such that Then from Lemma 2.1 (vi), (vii), (3.11) and (3.12), we obtain that Additionally, (K) and (H 1 ) imply that So, we obtain that Hence, we get that is, there is α 0 > 0 being independent of the parameter λ such that From Lemma 3.6, we have the following conclusions.

The mountain pass geometry
The following lemmas imply that Φ λ possesses the mountain pass geometry.

Lemma 4.1.
There exist ρ, α > 0 such that As in [15], suppose that there is where ) is equivalent to the both limits and a.e. up to a subsequence.We consider two cases: If w 1 = 0, Fatou's lemma and Lemma 2.1 (iv) imply that
Proof.Due to (K), Lemma 2.1 (7) and the Sobolev embedding inequality, it is easy to obtain that Based on Lemma 2.1 (iii), (vii), (H 2 ) and (H 3 ), it is obvious that for all > 0, there exists C > 0 such that Therefore, combining the above inequalities and Lemma 4.1, we obtain that for every u λ = ρ.Choosing for all ∈ 0, α 2λν 2 2 and ρ sufficiently small, we derive that there exists a constant β > 0 with inf u λ =ρ Φ(u) ≥ β.

Proof of Theorem 2.1
In this section we will prove Theorem 2.1.
Proof of Theorem 2.1.Lemmas 4.2-4.3imply that for any α 0 > σ > 0 there exists Λ σ > 0 such that for each λ ≥ Λ σ , there is β > 0 and a (PS In virtue of Corollary 3.1, we get that (PS) c condition holds for Φ λ at c. Thus there is u

1).
In order to get the multiplicity of critical points, we will use the index theory defined by the Krasnoselski genus.Define the set of all symmetric (in the sense that −A = A) and closed subsets of E as Σ.For all A ∈ Σ, denote gen(A) by the Krasnoselski genus and where Γ is the set of all odd homeomorphisms h ∈ C(E, E) and S λ is the closed symmetric set Then if c λ j is finite and the (PS) condition holds for Φ λ at c λ j , we know that c λ j is a critical value for Φ λ .However, the (PS) condition does not hold in general.In order to show that Φ λ satisfies the (PS) condition for λ large enough and c λ j sufficiently small, as in [31] we will construct here small minimax levels for Φ λ when λ large enough.Similar to the proof in Lemma 4.3, for any m ∈ N, δ > 0 and j = 1, 2, . . ., m, one can choose m functions ϕ and define and for t j ≥ 0 − λ H(x, f (t j e j λ ), f (t j e j λ )) Consequently, there holds sup Choose δ > 0 so small that mCδ q q−2 λ 1− N 2 ≤ σ.Thus for any m ∈ N and σ ∈ (0, α 0 ), there exists Λ σm = Λ δm such that for each λ > Λ σm , we can choose a m-dimensional subspace H m λ with max Φ λ (H m λ ) ≤ σλ 1− N 2 .Now we can define the minimax values c λ j by It follows from Corollary 3.1 that Φ λ satisfies the (PS) condition at all levels c λ j , since c λ j < α 0 λ 1− N 2 .Then all c λ j are critical values.Hence Φ λ has at least m pairs of nontrival critical points satisfying Therefore, Φ λ has at least m pairs of solutions Taking ν = 4 gives From the above two inequalities and (H 4 ) it follows that (5.1) We claim that (5.2) In fact, from (5.1), we only need to prove that λ We write that Combining (V 2 ), (K), (5.1) and Lemma 2.1 (viii), we have Thus λ V 1 (x)u 2 λ ≤ Cσλ 1− N 2 .Similarly, we can get λ V 2 (x)v 2 λ ≤ Cσλ 1− N 2 .Then we conclude that (5.2) holds, which shows (u λ , v λ ) → (0, 0) in E as λ → ∞.Meanwhile, we also have by Lemma 2.1 (ii) which shows that ( f (u λ ), f (v λ )) → (0, 0) in E as λ → ∞.It follows from λ = ε −2 that Theorem 1.1 is completed.
Remark 5.1.The same arguments applied to Φ + λ can give the existence of multiple positive solutions for system (2.2).

Proof of Theorem 1.2
In this section, we shall give some crucial lemmas and prove Theorem 1.2 under the conditions (V 3 ), (K ), (H 1 )-(H 4 ) and (H 6 ).
Let ε = 1.We redefine the functional Similar to Lemmas 4.1-4.3,Φ satisfies the mountain pass geometry in E. And the (PS) c sequence (u n ) for Φ is bounded in E by Lemma 3.1.Some propositions and lemmas are needed and their proofs are similar as in [26].We just state them briefly and omit their proofs.  .Recall that by [1,29], {w ε } ε>0 is a family of functions at which the infimum, that defines the best constant S, for the Sobolev imbedding D 1,2 (R N ) ⊂ L 2 * (R N ), is attained.Moreover, one has