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:
for sufficiently large λ, where .
Our assumptions on are as follows:
( ) , the potential well is a non-empty set and ;
( ) There exists a constant such that , where μ denotes the Lebesgue measure on .
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.
Remark 1.1 can be unbounded.
For , we assume that f is continuous and satisfies the following conditions:
( ) ;
( ) for , where is a constant and , where ;
( ) There is a number such that for all , we have , where .
Hypotheses ( ), ( ) and ( ), ( ), ( ) will be maintained throughout this paper.
Solutions of ( ) are related to the existence of the standing wave solutions of the following quasilinear 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
Let . Formally, we define the following functional:
for . Note that under our assumptions, the functional is not well defined on X.
We follow the idea of  and make the following change of variable.
Let , then . Moreover, satisfies
Since , is strictly monotone and hence has an inverse function denoted by . Obviously,
Let . Then it holds that
and is convex. Moreover, there exists such that ,
Now we introduce the Orlicz space (see )
equipped with the norm
Then is a Banach space.
equipped with the norm
Using the change of variable, we define the functional on by
where is the positive part of v.
be the Nehari manifold and let
be the infimum of on the Nehari manifold , where is the Gateaux derivative (see Proposition 3.3).
We say that is a least energy solution of ( ) if such that is achieved.
Note that under our assumptions, for λ large enough, the following Dirichlet problem is a kind of a ‘limit’ problem:
Similar to the definition of the least energy solution of ( ), we can define the least energy solution of (D) which will be given in Section 4.
Our main result is as follows.
Theorem 2.1Assume that ( ), ( ) and ( ), ( ), ( ) are satisfied. Then forλlarge, is achieved by a critical point of such that is a least energy solution of ( ). Furthermore, for any sequence , has a subsequence converging tovsuch that is 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 .
Lemma 3.1There exist two constants , such that
for any .
Lemma 3.2The map: from intois continuous for .
Now we consider the functional defined on by (2.2), the following Proposition 3.3 is due to .
(i) is well defined on ;
(ii) is continuous in ;
(iii) is Gateaux differentiable, the Gateaux derivative for is a linear functional and is continuous invin the strong-weak topology, that is, if strongly in , then weakly. Moreover, the Gateaux derivative has 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 .
Lemma 3.4Any of (PS)csequence for is bounded.
Proof Suppose that is a (PS)c sequence of . We have and in the space .
Taking , then , we have , thus
It follows from Lemma 3.1 that
thus is bounded in .
Let be the critical set of . Suppose , then it is easy to check that either or in by the definition of and the strong maximum principle. □
Lemma 3.5There exists which is independent ofλsuch that for all and .
Proof Assume that for any (otherwise, the conclusion is true). From ( ), ( ), we see that for any , there is a constant such that for . We have
and we can easily deduce the desired result. □
Lemma 3.6There exists a positive constant such that
and either or if is a (PS)csequence for , where is the constant in Lemma 3.1.
Proof Since is a (PS)c sequence, we have
It follows from (3.5) that
On the other hand, for , we have
Thus, there exists ( ) such that
Taking , then we have
if . It follows from (3.6) and (3.7) that
hence, and . Therefore, we have proved that there exists a constant such that either or . □
Proposition 3.7Let be a constant. Then for any , there exist , such that
if is a (PS)csequence of with , , where.
Proof For all , let
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. □
Lemma 3.8 is achieved by some .
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 is sufficient to prove that and . In fact,
it follows that
Let , since strongly in for , by Proposition 3.7, there exist , such that for , ,
Now we prove . Indeed, since is a (PS)c sequence, we have
where as .
Let , then is bounded in for , by the continuity of g, we have, up to a subsequence, in .
Similarly, we have is bounded in . Again, by the continuity of g, we have in . Passing to the limits in (3.11), we get
which is equivalent to , that is, . □
4 Proof of the main result
Consider the following quasilinear Schrödinger equation in ( ):
We have the same change of variables and the same notation as in the previous sections. Define the corresponding Orlicz space by
with the norm
The space is defined by
with the norm
The following Lemma 4.1 is a counterpart of Lemma 3.1.
Lemma 4.1There exist two constants , such that
for any .
We denote by the closure of in . We define the functional on by
and we define the Nehari manifold by
We recall that is a least energy solution of (D) if such that is achieved.
Lemma 4.2Suppose . Then .
Proof It is easy to see that for . We claim that is monotone increasing with respect to λ. In fact, for , we assume that , are achieved for , . Obviously,
We first prove that there exists such that . This is sufficient to prove that
Then by ( ), we can obtain and
Hence, there exists such that , i.e., . Thus
In the following, we will prove that
In fact, we consider the function defined by
By for , we have . It follows that
and hence it is easy to check that
On the other hand,
by , it is easy to check that for any ,
for any , thus we have proved that is monotone increasing for .
Now we consider the function defined by
for . Therefore, is monotone increasing with respect to . Thus, we deduce that
Assume that . If , then for any sequence ( ), we have .
We assume that is such that is achieved, by Lemma 3.4, is bounded in . Since , is bounded in , as a result, we have in , in for , in for , a.e. in .
We claim that , where . Indeed, it is sufficient to prove . If not, then there exists a compact subset with such that and
Moreover, there exists such that for any .
By the choice of , we have
This contradiction shows that and so does v.
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
On the other hand, by the choice of , we have
which shows that in for . In the above proof, we have used the fact that as and the bounded property of .
Now, since is bounded, by the Fatou lemma, we obtain
But, by the choice of , we have
In the following, we will prove that
Since , one can easily see that as , and
by using in for . It follows from (4.4) that
thus, there exists such that and
hence . A contradiction. Thus we have proved that as . □
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
Similar to above, there exists such that and . A contradiction, and thus we complete the proof of Theorem 2.1. □
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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