In this paper, we consider the existence of least energy solutions for the following quasilinear Schrödinger equation:
with having a potential well, where and is a parameter. Under suitable hypotheses, we obtain the existence of a least energy solution of () which localizes near the potential well for λ large enough by using the variational method and the concentration compactness method in an Orlicz space.
MSC: 35J60, 35B33.
Keywords:quasilinear Schrödinger equation; least energy solution; Orlicz space; concentration compactness method; variational method
Let us consider the following quasilinear Schrödinger equation:
Condition () is very weak in dealing with the operator on , which was firstly used by Bartsch and Wang  in dealing with the semilinear Schrödinger equation.
where is a given potential, k is a real constant and f, h are real functions. We would like to mention that (1.1) 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 was used for the superfluid film equation in plasma physics by Kurihara  (see also ); in the case of , (1.1) was used as a model of the self-changing of a high-power ultrashort laser in matter (see [4-7] and references therein).
In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:
For example, by using a constrained minimization argument, the existence of positive ground state solution was proved by Poppenberg, Schmitt and Wang . Using a change of variables, Liu, Wang and Wang  used an Orlicz space to prove the existence of soliton solution of (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 (1.2) from the classical results given by Berestycki and Lions . By using the Nehari manifold method and the concentration compactness principle (see ) in the Orlicz space, Guo and Tang  considered the following equation:
with having a potential well and , where is the critical Sobolev exponent, and they proved the existence of a ground state solution of (1.3) which localizes near the potential well for λ large enough. In , Guo and Tang also considered ground state solutions of the corresponding quasilinear Schrödinger systems for (1.3) by the same methods and obtained similar results. For the stability and instability results for the special case of (1.2), one can also see the paper by Colin, Jeanjean and Squassina .
It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.3) for has been extensively studied. One can see Bartsch and Wang , Ambrosetti, Badiale and Cingolani , Ambrosetti, Malchiodi and Secchi , Byeon and Wang , Cingolani and Lazzo , Cingolani and Nolasco , Del Pino and Felmer [21,22], Floer and Weinstein , Oh [24,25] and the references therein.
In this paper, based on the idea from Liu, Wang and Wang , we consider the more general equation (), the existence of least energy solutions for equation () with a potential well for λ large is proved under the conditions (), () and (), (), ().
The paper is organized as follows. In Section 2, we describe our main result (Theorem 2.1). In Section 3, we give some preliminaries that will be used for the proof of the main result. Finally, Theorem 2.1 will be proved in Section 4.
Throughout this paper, we use the same C to denote different universal constants.
2 Main result
We follow the idea of  and make the following change of variable.
Now we introduce the Orlicz space (see )
equipped with the norm
equipped with the norm
be the Nehari manifold and let
Note that under our assumptions, for λ large enough, the following Dirichlet problem is a kind of a ‘limit’ problem:
Our main result is as follows.
Theorem 2.1Assume that (), () and (), (), () are satisfied. Then forλlarge, is achieved by a critical pointofsuch thatis a least energy solution of (). Furthermore, for any sequence, has a subsequence converging tovsuch thatis a least energy solution of (D).
In order to obtain the compactness of the functional , we recall the following Lemmas 3.1 and 3.2 which can be found in .
Now we consider the functional defined on by (2.2), the following Proposition 3.3 is due to .
(iii) is Gateaux differentiable, the Gateaux derivativeforis a linear functional andis continuous invin the strong-weak topology, that is, ifstrongly in, thenweakly. Moreover, the Gateaux derivativehas the form
Recall that is called a Palais-Smale sequence ((PS)c sequence in short) for if and in , the dual space of . We say that the functional satisfies the (PS)c condition if any of (PS)c sequence (up to a subsequence, if necessary) converges strongly in .
It follows from Lemma 3.1 that
and we can easily deduce the desired result. □
It follows from (3.5) that
On the other hand, by the Hölder inequality and interpolation inequality, we have
By using the Gagliardo-Nirenberg inequality, we obtain
Let λ and R be large enough, from (3.8) and (3.9), we get the desired result. □
Proof By the definition of and the Ekeland variational principle, there exists a (PS)c sequence , by Lemma 3.4, we know that is bounded. Hence (up to a subsequence) we have in , in , a.e. in , in for .
it follows that
4 Proof of the main result
with the norm
with the norm
The following Lemma 4.1 is a counterpart of Lemma 3.1.
In the following, we will prove that
and hence it is easy to check that
On the other hand,
Now we show that
Suppose that (4.3) is not true, then by the concentration compactness principle of Lions (see ), there exist , and with such that
In the following, we will prove that
Proof of Theorem 2.1 Suppose that is a sequence such that , , by the proof of Lemma 3.2, we have in , in for and . Moreover, , and if , then . Hence, in the following, we need only to prove that . To do this, it is sufficient to prove that
In fact, if one of the above three limits does not hold, by the Fatou lemma, we have
The author declares that she has no competing interests.
The author would like to thank the referee for some valuable comments and helpful suggestions. This study was supported by the National Natural Science Foundation of China (11161041, 31260098) and the Fundamental Research Funds for the Central Universities (zyz2012080, zyz2012074).
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