In this paper, we consider the following quasilinear elliptic system:
where , , , , , , is the critical Sobolev exponent and ( ) is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.
MSC: 35J60, 35B33.
Keywords:quasilinear elliptic system; positive solution; negative solution; perturbation method
Let us consider the following quasilinear elliptic system:
where , , , , , , is the critical Sobolev exponent and ( ) is a bounded smooth domain. This system includes the following special class of system with , , i.e.,
which is referred to as the so-called modified nonlinear Schrödinger system.
Our assumptions on the functions and are as follows.
(A1) The functions , , , , .
(A2) There exist constants , , , satisfying , , and such that
for , , .
for , , .
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.
Throughout this paper, we will use the same C to denote various generic positive constants, and we will use to denote quantities that tend to 0.
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
for all , which is formally the variational formulation of the following functional:
We may define the derivative of at in the direction of as follows:
We call a critical point of if , , and for all . That is, is a weak solution for system (1.1).
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
where is a parameter. Then it is easy to see that is a -functional on . We can define also the derivative of at in the direction of as follows:
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.
Theorem 2.1Assume that (A1)-(A3) hold, , and . Let and let be a sequence of critical points of satisfying and for someCindependent ofn. Then, up to a subsequence,
as , and is a critical point of .
Theorem 2.2Assume that (A1)-(A3) hold, , and . Then has a positive critical point and a negative critical point , and (resp., ) converges to a positive (resp., negative) solution for system (1.1) as .
Notation We denote by the norm of and by the norm of ( ).
3 Compactness of the perturbed functional
In this section, we verify the Palais-Smale condition ( condition in short) for the perturbed functional . We have the following proposition.
Proposition 3.1For fixed, the functional satisfies condition for all . That is, any sequence satisfying, for ,
has a strongly convergent subsequence in , where is the dual space of .
To give the proof of Proposition 3.1, we need the following lemma firstly.
Lemma 3.2Suppose that a sequence satisfies (3.1) and (3.2). Then
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.
Proof of Proposition 3.1 From Lemma 3.2 , we know that is bounded in . So there exists a subsequence of , still denoted by , such that
Now we prove that in . In (2.5), choosing , we have
We may estimate the terms involved as follows:
Returning to (3.3), we have
which implies that , i.e., in . This completes the proof of Proposition 3.1. □
4 Some asymptotic behavior
Proposition 3.1 enables us to apply minimax argument to the functional . In this section, we also study the behavior of the sequences and satisfying
The following proposition is the key of this section.
Proposition 4.1Assume that the sequences and satisfy (4.1)-(4.3). Then, after extracting a sequence, still denoted byn, we have
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
as . This completes the proof of Proposition 4.1. □
5 Proof of main results
In this section, we give the proof of our main results. Firstly, we prove Theorem 2.1.
Proof of Theorem 2.1 Note that satisfies the following equation:
for all . Since
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
Note that , are positive for M large enough. By Fatou’s lemma, the weak convergence of and the fact that is bounded, we have
Let , . We may choose , such that , and . Then we obtain
for all , .
Similarly, we may obtain an opposite inequality. Thus we have
for all . That is, is a critical point of and a solution for system (1.1). By doing approximations, we have in the place of of (5.7)
Setting in (5.1), we have
Using as , (5.8), (5.9) and lower semi-continuity, we obtain
In particular, we have
as . This completes the proof of Theorem 2.1. □
Next, we apply the mountain pass theorem to obtain the existence of critical points of . Set
Let us consider the functional
Here and in what follows, we denote . The functional satisfies condition. Similarly, we may verify that satisfies condition. By the ε-Young inequality, for any , there exists such that
for ε, ρ small. Thus we have
for and for small enough. Choose , and . Define a path by . When T is large enough, we have
for some m independent of .
From the mountain pass theorem we obtain that
is a critical value of .
Let be a critical point corresponding to . We have . Thus is a positive critical point of by the strong maximum principle. In summary, we have the following.
Proposition 5.1There exist positive constantsρandmindependent ofμsuch that has a positive critical point satisfying
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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