Least energy solutions for a quasilinear Schrödinger equation with potential well
© Jiao; licensee Springer. 2013
Received: 23 October 2012
Accepted: 5 January 2013
Published: 21 January 2013
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.
Keywordsquasilinear Schrödinger equation least energy solution Orlicz space concentration compactness method variational method
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.
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).
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
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.
Then is a Banach space.
where is the positive part of v.
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.
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 2.1 Assume that (), () and (), (), () are satisfied. Then for λ large, is achieved by a critical point of such that is a least energy solution of (). Furthermore, for any sequence , has a subsequence converging to v such 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 .
for any .
Lemma 3.2 The map: from into is continuous for .
Now we consider the functional defined on by (2.2), the following Proposition 3.3 is due to .
is well defined on ;
is continuous in ;
- (iii)is Gateaux differentiable, the Gateaux derivative for is a linear functional and is continuous in v in the strong-weak topology, that is, if strongly in , then weakly. Moreover, the Gateaux derivative has the form(3.2)
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 3.4 Any of (PS) c sequence for is bounded.
Proof Suppose that is a (PS) c sequence of . We have and in the space .
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 3.5 There exists which is independent of λ such that for all and .
and we can easily deduce the desired result. □
and either or if is a (PS) c sequence for , where is the constant in Lemma 3.1.
hence, and . Therefore, we have proved that there exists a constant such that either or . □
if is a (PS) c sequence of with , , where .
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 .
where as .
Let , then is bounded in for , by the continuity of g, we have, up to a subsequence, in .
which is equivalent to , that is, . □
4 Proof of the main result
The following Lemma 4.1 is a counterpart of Lemma 3.1.
for any .
We recall that is a least energy solution of (D) if such that is achieved.
Lemma 4.2 Suppose . Then .
for any , thus we have proved that is monotone increasing for .
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 .
Moreover, there exists such that for any .
This contradiction shows that and so does v.
which shows that in for . In the above proof, we have used the fact that as and the bounded property of .
hence . A contradiction. Thus we have proved that as . □
Similar to above, there exists such that and . A contradiction, and thus we complete the proof of Theorem 2.1. □
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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