## Boundary Value Problems

Impact Factor 1.014

Open Access

# Least energy solutions for a quasilinear Schrödinger equation with potential well

Boundary Value Problems20132013:9

DOI: 10.1186/1687-2770-2013-9

Accepted: 5 January 2013

Published: 21 January 2013

## Abstract

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

## 1 Introduction

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 [1] 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:
(1.1)

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 [2] (see also [3]); in the case of , (1.1) was used as a model of the self-changing of a high-power ultrashort laser in matter (see [47] and references therein).

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:
(1.2)
For example, by using a constrained minimization argument, the existence of positive ground state solution was proved by Poppenberg, Schmitt and Wang [8]. Using a change of variables, Liu, Wang and Wang [9] used an Orlicz space to prove the existence of soliton solution of (1.2) via the mountain pass theorem. Colin and Jeanjean [10] 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 [11]. By using the Nehari manifold method and the concentration compactness principle (see [12]) in the Orlicz space, Guo and Tang [13] considered the following equation:
(1.3)

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 [14], 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 [15].

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 [1], Ambrosetti, Badiale and Cingolani [16], Ambrosetti, Malchiodi and Secchi [17], Byeon and Wang [18], Cingolani and Lazzo [19], Cingolani and Nolasco [20], Del Pino and Felmer [21, 22], Floer and Weinstein [23], Oh [24, 25] and the references therein.

In this paper, based on the idea from Liu, Wang and Wang [9], 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:
(2.1)

for . Note that under our assumptions, the functional is not well defined on X.

We follow the idea of [9] 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 [26])
equipped with the norm

Then is a Banach space.

Let
equipped with the norm
Using the change of variable, we define the functional on by
(2.2)

where is the positive part of v.

Let
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:

where .

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).

## 3 Preliminaries

In order to obtain the compactness of the functional , we recall the following Lemmas 3.1 and 3.2 which can be found in [13].

Lemma 3.1 There exist two constants , such that
(3.1)

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 [9].

Proposition 3.3
1. (i)

is well defined on ;

2. (ii)

is continuous in ;

3. (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 .

Taking , then , we have , thus
(3.3)
and
(3.4)
Taking yields
Note that
we have
It follows from Lemma 3.1 that
(3.5)

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 .

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 3.6 There exists a positive constant such that

and either or if is a (PS) c sequence 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
(3.6)
Thus, there exists () such that
(3.7)
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 3.7 Let be a constant. Then for any , there exist , such that

if is a (PS) c sequence of with , , where .

Proof For all , let
We have
(3.8)
On the other hand, by the Hölder inequality and interpolation inequality, we have
(3.9)
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,
(3.10)
it follows that
Let , since strongly in for , by Proposition 3.7, there exist , such that for , ,
thus

Hence .

Now we prove . Indeed, since is a (PS) c sequence, we have
(3.11)

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 4.1 There exist two constants , such that

for any .

We denote by the closure of in . We define the functional on by
(4.1)
and we define the Nehari manifold by
Let

We recall that is a least energy solution of (D) if such that is achieved.

Lemma 4.2 Suppose . 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,
(4.2)
We first prove that there exists such that . This is sufficient to prove that
That is,
Let
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
Obviously,
and hence it is easy to check that
On the other hand,
by , it is easy to check that for any ,
which implies

for any , thus we have proved that is monotone increasing for .

Now we consider the function defined by
Then
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
hence,

This contradiction shows that and so does v.

Now we show that
(4.3)
Suppose that (4.3) is not true, then by the concentration compactness principle of Lions (see [12]), 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
hence,
(4.4)
In the following, we will prove that
Indeed,
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
and
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. □

## Declarations

### Acknowledgements

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).

## Authors’ Affiliations

(1)
College of Mathematics and Computer Science, Northwest University for Nationalities
(2)
College of Mathematics and Statistics, Northwest Normal University

## References

1. Bartsch T, Wang Z: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 2000, 51: 366-384.
2. Kurihura S: Large-amplitude quasi-solitons in superfluid. J. Phys. Soc. Jpn. 1981, 50: 3262-3267. 10.1143/JPSJ.50.3262View Article
3. Laedke EW, Spatschek KH, Stenflo L: Evolution theorem for a class of perturbed envelop soliton solutions. J. Math. Phys. 1983, 24: 2764-2769. 10.1063/1.525675
4. Brandi HS, Manus C, Mainfray G, Lehner T, Bonnaud G: Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys. Fluids B 1993, 5: 3539-3550. 10.1063/1.860828View Article
5. Chen XL, Sudan RN: Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse. Phys. Rev. Lett. 1993, 70: 2082-2085. 10.1103/PhysRevLett.70.2082View Article
6. De Bouard A, Hayashi N, Saut JC: Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Commun. Math. Phys. 1997, 189: 73-105. 10.1007/s002200050191
7. Ritchie B: Relativistic self-focusing channel formation in laser-plasma interactions. Phys. Rev. E 1994, 50: 687-689. 10.1103/PhysRevE.50.R687View Article
8. Poppenberg M, Schmitt K, Wang Z: On the existence of solutions to quasilinear Schrödinger equation. Calc. Var. Partial Differ. Equ. 2002, 14: 329-344. 10.1007/s005260100105
9. Liu J, Wang Y, Wang Z: Soliton solutions for quasilinear Schrödinger equation II. J. Differ. Equ. 2003, 187: 473-493. 10.1016/S0022-0396(02)00064-5View Article
10. Colin M, Jeanjean L: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 2004, 56: 213-226. 10.1016/j.na.2003.09.008
11. Berestycki H, Lions PL: Nonlinear scalar field equations I. Arch. Ration. Mech. Anal. 1983, 82: 313-346.MathSciNet
12. Lions PL: The concentration-compactness principle in the calculus of variations. The locally compact case. Part I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 109-145.
13. Guo Y, Tang Z: Ground state solutions for the quasilinear Schrödinger equation. Nonlinear Anal. 2012, 75: 3235-3248. 10.1016/j.na.2011.12.024
14. Guo Y, Tang Z: Ground state solutions for the quasilinear Schrödinger systems. J. Math. Anal. Appl. 2012, 389: 322-339. 10.1016/j.jmaa.2011.11.064
15. Colin M, Jeanjean L, Squassina M: Stability and instability results for standing waves of quasi-linear Schrödinger equations. Nonlinearity 2010, 23: 1353-1385. 10.1088/0951-7715/23/6/006
16. Ambrosetti A, Badiale M, Cingolani S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 1997, 140: 285-300. 10.1007/s002050050067
17. Ambrosetti A, Malchiodi A, Secchi S: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 2001, 159: 253-271. 10.1007/s002050100152
18. Byeon J, Wang Z: Standing waves with a critical frequency for nonlinear Schrödinger equations II. Calc. Var. Partial Differ. Equ. 2003, 18: 207-219. 10.1007/s00526-002-0191-8
19. Cingolani S, Lazzo M: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 2000, 160: 118-138. 10.1006/jdeq.1999.3662
20. Cingolani S, Nolasco M: Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc. R. Soc. Edinb. 1998, 128: 1249-1260. 10.1017/S030821050002730X
21. Del Pino M, Felmer P: Semi-classical states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré 1998, 15: 127-149.
22. Del Pino M, Felmer P: Multi-peak bound states for nonlinear Schrödinger equations. J. Funct. Anal. 1997, 149: 245-265. 10.1006/jfan.1996.3085
23. Floer A, Weinstein A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 1986, 69: 397-408. 10.1016/0022-1236(86)90096-0
24. Oh YG: On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 1990, 131: 223-253. 10.1007/BF02161413View Article
25. Oh YG:Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View Article
26. Rao M, Ren Z: Theory of Orlicz Space. Dekker, New York; 1991.