Existence of Solutions for a Modified Nonlinear Schrödinger System

We are concerned with the followingmodified nonlinear Schrödinger system: −Δu+u−(1/2)uΔ(u2) = (2α/(α+β))|u||V|u, x ∈ Ω, −ΔV+V−(1/2)VΔ(V2) = (2β/(α+β))|u||V|V, x ∈ Ω, u = 0, V = 0, x ∈ ∂Ω, whereα > 2, β > 2, α+β < 2⋅2, 2∗ = 2N/(N−2) is the critical Sobolev exponent, andΩ ⊂ RN (N ≥ 3) is a bounded smooth domain. By using the perturbationmethod, we establish the existence of both positive and negative solutions for this system.


Introduction
Let us consider the following modified nonlinear Schrödinger system: where  > 2,  > 2,  +  < 2 ⋅ 2 * , 2 * = 2/( − 2) is the critical Sobolev exponent, and Ω ⊂ R  ( ≥ 3) is a bounded smooth domain.Solutions for the system (1) are related to the existence of the standing wave solutions of the following quasilinear Schrödinger equation: where () is a given potential,  is a real constant, and , ℎ are real functions.We would like to mention that (2) appears more naturally in mathematical physics and has been derived as models of several physical phenomena corresponding to various types of ℎ.For instance, the case ℎ() =  was used for the superfluid film equation in plasma physics by Kurihara [1] (see also [2]); in the case of ℎ() = (1 + ) 1/2 , (2) was used as a model of the self-changing of a high-power ultrashort laser in matter (see [3][4][5][6] and references therein).
In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form: See, for example, by using a constrained minimization argument, the existence of positive ground state solution was proved by Poppenberg et al. [7].Using a change of variables, Liu et al. [8] used an Orlicz space to prove the existence of soliton solution for (3) via mountain pass theorem.Colin and Jeanjean [9] also made use of a change of variables but worked in the Sobolev space  1 (  ); they proved the existence of positive solution for (3) from the classical results given by Berestycki and Lions [10].Liu et al. [11] established the existence of both one-sign and nodal ground states of soliton type solutions for (3) by the Nehari method.In particular, in [12], by using Nehari manifold method and concentration compactness principle (see [13]) in the Orlicz space, Guo and Tang considered the following quasilinear Schrödinger system: with () ≥ 0, () ≥ 0 having a potential well and  > 2,  > 2,  +  < 2 ⋅ 2 * , where 2 * = 2/( − 2) is the critical Sobolev exponent, and they proved the existence of a ground state solution for the system (4) which localizes near the potential well int  −1 (0) for  large enough.Guo and Tang [14] also considered ground state solutions of the single quasilinear Schrödinger equation corresponding to the system (4) by the same methods and obtained similar results.It is worth pointing out that the existence of one-bump or multibump bound state solutions for the related semilinear Schrödinger equation ( 3) for  = 0 has been extensively studied.One can see Bartsch and Wang [15], Ambrosetti et al. [16], Ambrosetti et al. [17], Byeon and Wang [18], Cingolani and Lazzo [19], Cingolani and Nolasco [20], Del Pino and Felmer [21,22], Floer and Weinstein [23], and Oh [24,25] and the references therein.
The system (1) is a kind of "limit" problem of the system (4) as  → ∞.The existence of solutions for the system (1) has important physical interest.The purpose of this paper is to study the existence of both positive and negative solutions for the system (1).We mainly follow the idea of Liu et al. [26] to perturb the functional and obtain our main results.We point out that the procedure to the system (1) is not trivial at all.Since the appearance of the quasilinear terms Δ( 2 ) and VΔ(V 2 ), we need more delicate estimates.
The paper is organized as follows.In Section 2, we introduce a perturbation of the functional and give our main results (Theorem 1 and 2).In Section 3, we verify the Palais-Smale condition for the perturbed functional.Section 4 is devoted to some asymptotic behavior of sequence {(  , V  )} ⊂  1,4  0 (Ω) ×  1,4 0 (Ω) and {  } ⊂ (0, 1] satisfying some conditions.Finally, our main results will be proved in Section 5. Throughout this paper, we will use the same  to denote various generic positive constants, and we will use (1) to denote quantities that tend to 0.

Perturbation of the Functional and Main Results
In order to obtain the desired existence of solutions for the system (1), in this section, we introduce a perturbation of the functional and give our main results.
The weak form of the system (1) is which is formally the variational formulation of the following functional: We may define the derivative of  0 at (, V) in the direction of (, ) ∈  ∞ 0 (Ω) ×  ∞ 0 (Ω) as follows: We call that That is, (, V) is a weak solution for the system (1).
When we consider the system (1) by using the classical critical point theory, we encounter the difficulties due to the lack of an appropriate working space.In general it seems there is no suitable space in which the variational functional  0 possesses both smoothness and compactness properties.For smoothness one would need to work in a space smaller than  1,2 0 (Ω) to control the term involving the quasilinear term in the system (1), but it seems impossible to obtain bounds for ()  sequence in this setting.There have been several ideas and approaches used in recent years to overcome the difficulties such as by minimizations [7,27], the Nehari method [11], and change of variables [8,9].In this paper, we consider a perturbed functional where  ∈ (0, 1] is a parameter.Then it is easy to see that   is a  1 -functional on  1,4 0 (Ω)× 1,4 0 (Ω).We also can define the derivative of   at (, V) in the direction of (, ) as follows: for all (, ) ∈  ∞ 0 (Ω) ×  ∞ 0 (Ω).The idea is to obtain the existence of the critical points of   for  > 0 small and to establish suitable estimates for the critical points as  → 0 so that we may pass to the limit to get the solutions for the original system (1).
Our main results are as follows.
as  → ∞ and (, V) is a critical point of  0 .
Using Theorem 1, we have the following existence result.
Proof.It follows from (11) that Thus we have lim sup This completes the proof of Lemma 4.
Now we give the proof of Proposition 3.
Proof.Similar to the proof of Lemma 4, by (19), we have Thus for some  independent of .Then, up to a subsequence, we have as  → ∞.This completes the proof of Proposition 5.

Proof of Main Results
In this section, we give the proof of our main results.Firstly, we prove Theorem 1.
In particular, we have as  → ∞.This completes the proof of Theorem 1.