Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents

This paper is concerned with the critical quasilinear Schrodinger systems in \begin{document}$ {\Bbb R}^N: $\end{document} \begin{document}$ \left\{\begin{array}{ll}-\Delta w+(\lambda a(x)+1)w-(\Delta|w|^2)w = \frac{p}{p+q}|w|^{p-2}w|z|^q+\frac{\alpha}{\alpha+\beta}|w|^{\alpha-2}w|z|^\beta\\\ -\Delta z+(\lambda b(x)+1)z-(\Delta|z|^2)z = \frac{q}{p+q}|w|^p|z|^{q-2}z+\frac{\beta}{\alpha+\beta}|w|^\alpha|z|^{\beta-2}z, \ \end{array}\right. $\end{document} where \begin{document}$ \lambda>0 $\end{document} is a parameter, \begin{document}$ p>2, q>2, \alpha>2, \beta>2, $\end{document} \begin{document}$ 2\cdot(2^*-1) and \begin{document}$ \alpha+ \beta = 2\cdot2^*. $\end{document} By using variational method, we prove the existence of positive ground state solutions which localize near the set \begin{document}$ \Omega = int \left\{a^{-1}(0)\right\}\cap int \left\{b^{-1}(0)\right\} $\end{document} for \begin{document}$ \lambda $\end{document} large enough.


1.
Introduction. Because of its deep applications in physics, in recent years, much attention has been devoted to the following quasilinear Schrödinger equation − ε 2 ∆u + V (x)u − ε 2 k(∆|u| 2 )u = g(x, u), in R N , (1.1) where V (x) is a given potential function, k is a real constant and g is a real function. Equation (1.1) appears more naturally in mathematical physics and can model certain physical phenomena (see [4,5,6,23]). Formally, this equation (1.1) has a variational structure, however we face the lack of a suitable working space such that the variational functional is smooth and has some compactness condition. Hence the standard critical point theory does not seem to apply directly. To overcome this difficulty, in the last decades, several approaches have been successfully employed in the literature to deal with the quasilinear problems, such as change of variable approach ( see Liu, Wang and Wang [19] and Colin and Jeanjean [7]), Nehari manifold method (see Liu, Wang and Wang [20]), regularization approach ( see Liu, Liu and Wang [21] ) and so on. And a lot of results have been obtained by using these methods. First, we list some results obtained by change of variables. Bezerra doÓ, Miyagaki and Soares [3] showed the the existence of a positive solution for the critical quasilinear equations. Wang and Zou [27], studied the concentration of positive bound states as ε → 0. For the case ε = 1 and V (x) = λ > 0, under different conditions, Adachi, Shibata and Watanbe [1], Wang and Shen [25], proved that the solution of (1.1) would converge to a ground state solution of a semilinear Schrödinger equation as k → 0. Moreover, Zeng, Zhang and Zhou [30] studied the positive solutions of a quasilinear Schrödinger equation with Hardy potential and critical exponent. For more information about change of variable approach, we refer to [13,24,29,8,26,14] and references therein. However, this method depends heavily on the special structure of the quasilinear term and can not be generalized to treating more general quasilinear problems. In this case, Nehari manifold method and regularization approach, especially the latter, are robust to deal with general quasilinear problems. For example, Liu, Liu and Wang [16] obtained mutiple sign-changing solutions by regularization approach. Liu, Wang and Guo [17], also by regularization approach, studied the mutibump solutions. For more information about the regularization approach, we refer to [22,15,18].
On the other hand, in some practical physical problem, when several pulses with different frequencies are transmitted in a fiber, then we have to consider the mutual interaction of the different frequencies. Thus, from the view point of mathematics, we are lead to consider the quasilinear systems. For example, Guo and Li [9] proved the existence of a ground state positive solution for a class of quasilinear Schrodinger system with critical exponents by perturbation method. We also refer to [11,10] and references therein for more related results on the quasilinear systems.
Recently, Guo and Tang [12] studied the existence and the concentration behavior of the ground state solutions for systems (1.2) but with subcritical exponent. The aim of the present paper is to extend the results in [12] to the critical exponent case. Recall that the critical exponent for system (1.2) is 2 · 2 * . To do this, besides the difficulty caused by the lack of suitable working space, we have to overcome the loss of compactness caused by the critical exponent 2 · 2 * and the whole space R N . We will follow the method used in Colin and Jeanjean [7] to deal with these difficulties. We first choose H 1 (R N )×H 1 (R N ) as the working space, then introduce a limit functional of the original energy functional, combined the method of change of variables, we can prove the infimum of the original energy functional on the Nehari manifold can be achieved, which in turn lead to the results that we expect.
We assume: one of the last two inequalities is strict on a subset of positive measure in R N .
implies that the sets Ω a and Ω b consists of finite connected bounded domains in R N . Moreover, any two connected components of Ω a and Ω b are either overlapped or isolated.
We denote by · and | · | q the norms of H 1 (R N ) and L q (R N ) respectively, that is We observe that the natural energy functional associated with problem (1.2) is given by: where A λ (x) = λa(x) + 1 and B λ (x) = λb(x) + 1.
HoweverJ λ (w, z) is not well defined on X. We follow the idea of [7] and make the following change of variable u = f −1 (w) and v = f −1 (z), where f is defined by After the change of variables, we obtain the following functional: which is well defined on X and is of class C 1 . And for any (φ, ψ) ∈ X, we have We define the Nehari manifold N λ by: We also define the functional Φ λ∞ on X by: where A λ∞ = λa ∞ + 1 and B λ∞ = λb ∞ + 1, and the Nehari manifold N λ∞ by: We point out that under our assumptions, for λ large enough, the following Drichlet problem is a kind of "limit" problem: (1.5) Here are our main results : ) is a least engergy solution of (1.2). Furthermore, for any sequence λ n → +∞, (u λn , v λn ) has a subsequence converging to (u, v) such that (f (u), f (v)) is a least energy solution of (1.5).
The paper is organized as follows: In section 2, we give some preliminaries. In section 3, behaviors of (P S) c sequence and some energy levels are studied. In section 4, the existence of positive ground state solution for (1.2) is proved. Section 5 is devoted to the discussion of "limit problem". In section 6, the proof of Theorem 1.2 is given . Throughout of the paper, we will use the same C to denote different constants unless otherwise specified.
2. Preliminaries and estimates in X. In this section, we give some properties about f and make some estimates about the norm in X = H 1 (R N ) × H 1 (R N ). The following properties were proved in [7] and [27].
Lemma 2.1. The function f (t) has the following properties: (1) f is uniquely defined, C ∞ and invertible.
There exists a positive constant C such that Lemma 2.2. There exists two constant C 1 > 0 independent of λ, C 2 > 0 such that for any (u, v) ∈ X, Proof. First, by(A 2 ) and (3) of Lemma 2.1, there exists C 2 > 0, such that On the other hand, by (6) of Lemma2.1, Let a = R N |∇u| 2 + A λ f 2 (u)dx, then by the above iequality, we have If a ≥ 1, then we have N +4 . If a ≤ 1, then we have u 2 ≤ 2Ca, thus a ≥ 1 2C u 2 . Therefore, there exists C 1 > 0 independent of λ such that Thus, Similarly, we have From the above two estimates, we can easily get Thanks to the inequality: On the other hand, using (8) and (9) of Lemma2.1 , we have 3. Behaviors of (P S) c sequence and some energy levels. In this section, we study the boundedness of (P S) c sequence and give some energy levels about associated functional. Recall that we say It follows that Proof. For any (u, v) ∈ N λ \ {(0, 0)} , without loss of of generality, we may assume that (u, v) ≤ 1 ( otherwise the conclusion is also true up to a constant ). Then, by (4) of Lemma 2.1, we have 1 From Lemma 2.2 and (2.7), we have Since (u, v) ≤ 1, one can easily deduce the first desired result. On the other hand, by (4) of Lemma 2.1, we have Thus, This completes the proof of the lemma. Let which can be written as: From (10) of Lemma 2.1 and Lemma 3.1 of [12], we know that the right side of the above equality is increasing about t, if there exists t (u,v) that makes it hold, then t (u,v) must be unique, now we turn to the existence of t (u,v) . Firstly, by Lemma 2.2 and (2.7), when t is small enough, we have Then, by Lemma 2.2 and (6) of Lemma 2.1, we get When t is large enough, we have Hence, there exists a unique t (u,v) > 0 such that t (u,v) (u, v) ∈ N λ . Moveover, g (t) > 0 in (0, t (u,v) ) and g (t) < 0 in (t (u,v) , ∞), we get Φ λ (t (u,v) (u, v)) = sup t≥0 Φ λ (t(u, v)).
Proof. To prove the lemma, we first prove Thus, We divide the proof into the following three steps.
Step3. c * * ≥ c λ . The manifold N λ separates X into two components. By Remark 2, the component contains the origin also contains a small ball around the ori- . Thus every γ ∈ Γ has to cross N λ and c * * ≥ c λ .
The following lemma can be found in [2].
Especially, when a + b = 2 * , we have S a+b (Ω) = S 2 * (Ω) = S 2 * (R N ) = S, where S is the Sobolev constant, and S a,b (Ω) is independent of the domain Ω, so we denote S a,b instead of S a,b (Ω).

Remark 3.
It is easy to know that the related results from Lemma 2.2 to Lemma 3.6 also hold for N λ∞ and Φ λ∞ .
4. the existence of positive ground state solution for (1.2). In this secion, we prove the existence of positive ground state solution for (1.2).
Proof. The proof is almost the same with Lemma 4.2, we omit it.
Proof. By Lemma 3.4 and Remark 2, there exits a sequence {(u n , v n )} which is a (P S) c λ sequence of Φ λ . By Lemma 3.1 , we know that {(u n , v n )} is bounded in X.
In fact, for R > 0 , For any (φ, ψ) ∈ X, we have When R large enough, by (A 2 ) and u n 0, v n 0 in H 1 , we have Claim1. Claim2: Let u n (x) = u n (x−y n ), v n (x) = v n (x−y n ), then there exists ( u, v) ∈ X such that ( u, v) is a critical point of Φ λ∞ and ( u, v) = (0, 0) . By Claim 1, it is easy to prove that Moreover, ( u n , v n ) ≤ C, then there exist ( u, v) ∈ X such that u n u, v n v in H 1 and u n → u, v n → v a.e in R N . By (4.15), we get ( u, v) = (0, 0). Using a standard argument, we have Φ λ∞ ( u, v) = 0 and ( u, v) ∈ N λ∞ .

CONCENTRATION SOLUTIONS FOR QUASILINEAR SCHRÖDINGER SYSTEMS 2709
Using Hölder inequality and (2.7), there exists 0 < θ < 1 such that Thus, Without loss of generality, we assume v n 0 in H 1 (R N ). From the above inequality, we have f (v n ) → 0 in L p+q (R N ) as n → ∞. Thus, for 4 < p + q < 2 · 2 * , we get 5. Discussion of the "limit problem". In this section, we discuss the existence of ground state for the "limit problem" and make some comparison among the least energy.
We define the Nehari manifold N Ω by We say that (f (u), f (v)) is a least energy solution of (5.16) if (u, v) is a critical point of Φ Ω such that c Ω is achieved. By using the similar arguments used in the proof of those Lemmas in Section 2, we have the following Lemmas.
Proof. Similar to Lemma 4.2 , there exits a sequence {(u n , v n )} which is a (P S) cΩ sequence of Φ Ω ( up to a subsequence ) such that Assume
By A λ = B λ = 1 in Ω and u = v = 0 in R N \ Ω, the above equality can be written as On the other hand, Therefore c λ ≤ c Ω .
6. Proof of the main Theorem. In this section, we give a proof of our main results, that is, the least engergy solution of (1.2) converging to the least energy solution of (1.5).
Proof of Theorem 1.2. By Lemma 4.2, suppose that {(u n , v n )} ⊂ N λn is a sequence such that Φ λn (u n , v n ) = c λn , Φ λn (u n , v n ) = 0, by Lemma 3.1 and Lemma 5.4, there exists C > 0, such that (u n , v n ) ≤ C. Thus, there exists (u, v) ∈ X, such that We claim that u| Ω c a = 0 and v| Ω c b = 0, where Ω c a =: x|x ∈ R N \ Ω a and Ω c b =: Moreover, there exists 0 > 0 such that a(x) ≥ 0 for any x ∈ F.