Keywords:quasilinear elliptic system; positive solution; negative solution; perturbation method
Let us consider the following quasilinear elliptic system:
which is referred to as the so-called modified nonlinear Schrödinger system.
In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:
See, for example,  where Poppenberg et al. proved the existence of a positive ground state solution by using a constrained minimization argument. Using a change of variables, Liu et al. used an Orlicz space to prove the existence of a soliton solution for equation (1.2) via the mountain pass theorem. Colin and Jeanjean  also made use of a change of variables but worked in the Sobolev space . They proved the existence of a positive solution for equation (1.2) from the classical results given by Berestycki and Lions . Liu et al. established the existence of both one-sign and nodal ground states of soliton-type solutions for equation (1.2) by the Nehari method. By using the Nehari manifold method and the concentration compactness principle (see ) in the Orlicz space, Guo and Tang  considered the following quasilinear Schrödinger system:
with , having a potential well and , , , and they proved the existence of a ground state solution for system (1.3) which localizes near the potential well for λ large enough. Guo and Tang  considered also ground state solutions of the single quasilinear Schrödinger equation corresponding to system (1.3) by the same methods and obtained similar results. In particular, by the perturbation method, Liu et al. considered the existence and multiplicity of solutions for the following quasilinear equation of the form
under suitable assumptions.
It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.2) for has been extensively studied. One can see Bartsch and Wang , Ambrosetti et al., Ambrosetti et al., Byeon and Wang , Cingolani and Lazzo , Cingolani and Nolasco , Del Pino and Felmer [16,17], Floer and Weinstein , Oh [19,20] and the references therein.
Motivated by the single equation (1.4), the purpose of this paper is to study the existence of both positive and negative solutions for the coupled quasilinear system (1.1). We mainly follow the idea of Liu et al. to perturb the functional and obtain our main results. We point out that the procedure to system (1.1) is not trivial at all. Since the appearance of the quasilinear terms and , 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 2.1 and Theorem 2.2). In Section 3, we verify the Palais-Smale condition for the perturbed functional. Section 4 is devoted to some asymptotic behavior of the sequences and satisfying some conditions. Finally, our main results will be proved in Section 5.
2 Perturbation of the functional and main results
In order to obtain the desired existence of solutions for system (1.1), in this section, we introduce a perturbation of the functional and give our main results.
The weak form of system (1.1) is
When we consider system (1.1) by using the classical critical point theory, we encounter the difficulties due to the lack of an appropriate working space. In general, it seems that there is no suitable space in which the variational functional possesses both smoothness and compactness properties. For smoothness, one would need to work in a space smaller than to control the term involving the quasilinear term in system (1.1), but it seems impossible to obtain bounds for sequence in this setting. Several ideas and approaches, such as minimizations [1,21], the Nehari method  and change of variables [2,3], have been used in recent years to overcome the difficulties. In this paper, we consider the perturbed functional
for all . The idea of this paper is to obtain the existence of the critical points of for small and establish suitable estimates for the critical points as so that we may pass to the limit to get the solutions for the original system (1.1).
Our main results are as follows.
3 Compactness of the perturbed functional
To give the proof of Proposition 3.1, we need the following lemma firstly.
Proof It follows from (3.1) and (3.2) that
Thus we have
This completes the proof of Lemma 3.2. □
Now we give the proof of Proposition 3.1.
We may estimate the terms involved as follows:
Returning to (3.3), we have
4 Some asymptotic behavior
The following proposition is the key of this section.
Proof Similar to the proof of Lemma 3.2, by (4.1)-(4.3), we have
for some C independent of n. Then, up to a subsequence, we have
5 Proof of main results
In this section, we give the proof of our main results. Firstly, we prove Theorem 2.1.
By Moser’s iteration, we have
for some C independent of n. To show that is a critical point of , we use some arguments in [22,23] (see more references therein). In (5.1), we choose , , where , , , and is a constant. Substituting into (5.1), we have
Similarly, we may obtain an opposite inequality. Thus we have
In particular, we have
Let us consider the functional
for ε, ρ small. Thus we have
From the mountain pass theorem we obtain that
Finally, we give the proof of Theorem 2.2.
Proof of Theorem 2.2 For a positive solution of system (1.1), the proof follows from Proposition 5.1 and Theorem 2.1. A similar argument gives a negative solution of system (1.1). This completes the proof of Theorem 2.2. □
The authors declare that they have no competing interests.
All the authors were involved in carrying out this study. All authors read and approved the final manuscript.
This paper was finished while the first author was a visiting fellow at the School of Mathematical Sciences of Beijing Normal University, and the first author would like to express her gratitude for their hospitality during her visit. This work is supported by the National Science Foundation of China (11061031), Fundamental Research Funds for the Central Universities (31920130004) and Fundamental Research Funds for the Gansu University.
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