Abstract and Applied Analysis

Volume 2013 (2013), Article ID 256324, 13 pages

http://dx.doi.org/10.1155/2013/256324

Research Article

Existence of Nontrivial Solutions and High Energy Solutions for a Class of Quasilinear Schrödinger Equations via the Dual-Perturbation Method

Yu Chen¹ and Xian Wu²

¹Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

²Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China

Received 23 June 2013; Accepted 12 September 2013

Academic Editor: Mihai Mihǎilescu

Copyright © 2013 Yu Chen and Xian Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the quasilinear Schrödinger equation of the form , . Under appropriate assumptions on and , existence results of nontrivial solutions and high energy solutions are obtained by the dual-perturbation method.

1. Introduction and Preliminaries

In this paper we consider the quasilinear Schrödinger equation of the form

where and . Solutions of (1) are standing waves of the following quasilinear Schrödinger equation:

where is a given potential, is a real constant, and and are real functions. The quasilinear Schrödinger equations (2) are derived as models of several physical phenomena; for example, see [1–5]. Several methods can be used to solve (1). For instance, the existence of a positive ground state solution has been proved in [6, 7] by using a constrained minimization argument; the problem is transformed to a semilinear one in [8–11] by a change of variables (dual approach); Nehari method is used to get the existence results of ground state solutions in [12, 13].

Recently, some new methods have been applied to these equations. In [14], the authors prove that the critical points are functions by the Moser’s iteration; then the existence of multibump type solutions is constructed for this class of quasilinear Schrödinger equations. In [15], by analysing the behavior of the solutions for subcritical case, the authors pass to the limit as the exponent approaches to the critical exponent in order to establish the existence of both one-sign and nodal ground state solutions. Another new method which works for these equations is perturbations. In [16] 4-Laplacian perturbations are led into these equations; then high energy solutions are obtained on bounded smooth domain.

In this paper, the perturbation, combined with dual approach, is applied to search the existence of nontrivial solution and sequence of high energy solutions of (1) on the whole space . For simplicity we call this method the dual-perturbation method.

We need the following several notations. Let be the collection of smooth functions with compact support. Let

with the inner product

and the norm

Let the following assumption hold:

satisfies and .

Set

with the inner product

and the norm

Then both and are Hilbert spaces.

By the continuity of the for we know that, for each , there exists constant such that

where denotes the -norm. In the following, we use or to denote various positive constants. Moreover, we need the following assumptions:

there if and if such that

uniformly in , there exist and such that

for all and , where .

By Lemma 3.4 in [17] we know that, under the assumption , the embedding is compact for each .

Equation (1) is the Euler-Lagrange equation of the energy functional

where . Due to the presence of the term , is not well defined in . To overcome this difficulty, a dual approach is used in [9, 10]. Following the idea from these papers, let be defined by

on , and on . Then has the following properties:

is uniquely defined function and invertible; for all ; for all ;;, ; for all and for all ; for all ; the function is strictly convex; there exists a positive such that

there exist positive constants and such that

for all ; for all ; for each , there exists such that .

The properties have been proved in [8–11]. It suffices to prove .

Indeed, by , , and , there exist and such that, for ,

and for ,

Since there exists a such that (see [10]), we can assume that . For , we have , and hence

for , one has , and hence

and for , there exist and such that and . Then we have

Hence , where .

After the change of variable, can be reduced to

From [8, 9, 11] we know that if is a critical point of , that is,

for all , then is a weak solution of (1). Particularly, if is a critical point of , then is a classical solution of (1).

A sequence is called a Cerami sequence of if is bounded and in . We say that satisfies the Cerami condition if every Cerami sequence possesses a convergent subsequence.

2. Some Lemmas

Consider the following perturbation functional defined by

where . We have the following lemmas.

Lemma 1. If assumptions , , and hold, then the functional is well defined on and .

Proof. By conditions and , the properties , , , and imply that there exists such that

Hence

for all . By (26) and the continuity of the embedding (),

Hence is well defined in .

Now, we prove that . It suffices to prove that

For any and , by the mean value theorem, (25) and -, we have

The Hölder inequality implies that

Hence, by the Lebesgue theorem, we have

for all . Now, we show that , , are continuous. Indeed, if in , then in for all .

On the space , we define the norm

Then

Moreover, on the space , we define the norm

By (25), we have

where and . Then Theorem A.4 in [18] implies

as . If with and , one has

Hence

and hence

as . Therefore, .

Define

with the norm . On the space , we define the norm

On the space , we define the norm

From in , one has and

as . Since , by the following Lemma 2, we have

If with and , one has

Hence

and hence

as . Therefore, . This completes the proof.

Lemma 2. Assume that , and

Then, for every , , and the operator

is continuous.

Proof. Let be a smooth cut-off function such that for and for . Define

We can assume that . Hence

for all . Assume in . Then in and in . As in the proof of Lemma A.1 in [18], there exists a subsequence of and such that and for a.e. . Hence, from (51), one has

a.e. on . It follows from the Lebesgue theorem that in . Consequently, in . Similarly, we can prove in . Since

it follows that in . This completes the proof.

Lemma 3. Let , , and hold. Then every bounded sequence with possesses a convergent subsequence.

Proof. Since is bounded, then, by the compactness of the embedding (), passing to a subsequence, one has in , in for all , and for a.e. . By (25)

as . Similarly, as . Hence, by the property of , we have

where as . This shows that as . This completes the proof.

The following Lemma 4 has been proved in [10] (see Proposition 2.1(3) in [10]).

Lemma 4. If a.e. in and , then as .

3. Main Results

Theorem 5. Assume conditions , hold. Let be such that . Let be a critical point of with for some constant independent of . Then, up to subsequence, one has in , and is a critical point of .

Proof. By , for , there exists such that

By , for ( is the constant appearing in condition ), we have

where is the constant appearing in condition . Hence

Since , there exists such that

for all . Hence

Since is a critical point of ,

for all . Consequently, taking , by and we have

and hence

for some constant independent of . By the boundedness of , there exists such that

for all . Hence, by the Sobolev embedding theorem, one has

Next, we prove that and , where the positive constant is independent of . Setting , , define , where is a smooth function satisfying for , ; for , and is decreasing in .

This means that , for ; , for ; , for , where .

Let ; then . By (61) . Hence

where

For , . By the properties of and , the mean value theorem implies

Hence

Consequently,

Combining (67) and (68), we have

For any , by and , there exists such that

Combining (66), (72), and (73), one has

By the Hölder inequality and (65),

Moreover,

Hence

Since , . Set . Then

Take such that . Since , . Hence, from (65), we have

Since as , taking in (78) with , we have

Set . Then

Inductively, we have

where   , and

is convergent as . Let . Then as . Hence

Let ; by (65), we have

Hence, by and (85), we have

By (63) we know that is bounded, and hence is bounded in . Up to subsequence, one has in , in for , and a.e. .

Now, we show that is a critical point of . For any with , by (85), we know that . Take as the test function in (61); we have

By , one has

Since , by (63)

as . Moreover, notice that in , in for , and a.e. ; by Hölder inequality and Lebesgue theorem, we have

Hence, from (87), we have

For any with , by (85) we know that . By Theorem 2.8 in [19], there exists a sequence such that and and for a.e. . Take in (91), and let ; we have

The opposite inequality can be obtained by taking and . Consequently,

This shows that is a critical point of , and by taking , one has

Finally, taking as the test function in (61), we have

Since

by Fatou’s Lemma, (63), (94), (95), up to subsequence, one has

Hence as . Set . By , , and (85), one has

Consequently, by , (98), and Lemma 4, one has

as . Therefore, in . This completes the proof.

Theorem 6. Assume conditions , hold; then (1) has a weak solution.

Proof. First, we prove that, for each , satisfy the Cerami condition. Indeed, let be an arbitrary Cerami sequence of . Set . Then . Similar to the proof of (63), we can prove that is bounded in . Hence, by Lemma 3, the sequence possesses a convergent subsequence in . This shows that satisfy the Cerami condition.

Next, for any , by , , , and , there exists such that

for all . For small , set

Then, from (101), for ,

for small and . Moreover, by , for any with , one has

Since , there exists a constant . Hence, by , we have

uniformly in . Consequently, there exist constants such that

for all . For any finite-dimensional subspace , by the equivalency of all norms in the finite-dimensional space, there is a constant such that

By , , and (106), there exists a positive constant such that

Since , by , (107), and (108), we have

for all . Hence there exists a large such that on . Set a fixed with . For any fixed , define the path by . Then for large , by (109), one has

Hence by Theorem 2.2 with the Cerami condition in [20], possesses a critical value

where

Consequently, by Theorem 5, we know that (1) has a weak solution. This completes the proof of Theorem 6.

Remark 7. Let and . Set

instead of and , respectively. Then, under the conditions of Theorem 6, we can obtain the existence of a positive solution and a negative solution for (1).

Theorem 8. Assume conditions , hold. If is odd in , then (1) has a sequence of solutions such that .

Proof. Consider the eigenvalue of the operate . By assumption and the compactness of the embedding , we know that the spectrum of with

and as (see page in [21]). Let be the eigenfunction corresponding to . By regularity argument we know that . Set . Then we can decompose the space as for , where is orthogonal to in . For , set

By (109) there exists independent of such that

Set

where is the genus. Let

We have the following three facts (we refer the reader to [16] for their proofs).

  Fact (1). For each , if for all , then .  Fact (2). There exist constants such that and as .  Fact (3).  , are critical values of .

Consequently, Theorem 8 follows from Theorem 5 and the above Facts (2)-(3). This completes the proof.

Corollary 9. If the following conditions and are used in place of ; then the conclusions of Theorem 5, Theorem 6, and Theorem 8 hold:

uniformly in , there exist and such that

for all and .

Proof. By , there are constants and such that whenever , one has

Set . Then, by ,

for all and . Therefore, condition holds. This completes the proof.

Acknowledgment

This work was supported partially by the National Natural Science Foundation of China (11261070).

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