Advances in Mathematical Physics
Volume 2018 (2018), Article ID 3615085, 11 pages
https://doi.org/10.1155/2018/3615085

Research Article

Existence of Nontrivial Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

Quanqing Li,¹ Kaimin Teng,² and Xian Wu³

¹Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China
²Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
³Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China

Correspondence should be addressed to Kaimin Teng; tengkaimin2013@163.com

Received 12 September 2017; Revised 12 November 2017; Accepted 16 November 2017; Published 3 January 2018

Academic Editor: Ciprian G. Gal

Copyright © 2018 Quanqing Li et al. 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 following generalized quasilinear Schrödinger equations with critical growth where , , is a even function, , and for all , where . Under some suitable conditions, we prove that the equation has a nontrivial solution by variational method.

1. Introduction and Preliminaries

Consider the following generalized quasilinear Schrödinger equations with critical growth:

where , , is a even function, , and for all .

The equations are related to the existence of solitary wave solutions for quasilinear Schrödinger equations

where , is a given potential, , and are suitable functions. The form of (2) has been derived as models of several physical phenomena corresponding to various types of . For instance, the case models the time evolution of the condensate wave function in superfluid film [1, 2] and is called the superfluid film equation in fluid mechanics by Kurihara [1]. In the case , problem (2) models the self-channeling of a high-power ultra short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optical index depending nonlinearly on the light intensity, and this leads to interesting new nonlinear wave equations; see [3–6]. For more physical motivations and more references dealing with applications, we can refer to [7–14] and references therein.

Set , where and is a real function. Then (2) can be reduced to the corresponding equation of elliptic type (see [15]):

where . If we take

then (1) turns into (3) (see [16]).

Moreover, problem (3) also arises in biological models and propagation of laser beams when is a positive constant (see [17, 18]). In (3), if we set , that is, , then we get the superfluid film equation in plasma physics:

if we set , that is, , then we get the equation

which models the self-channeling of a high-power ultrashort laser in matter.

In the past, the research on the existence of solitary wave solutions of Schrödinger equations (2) is for some given special function . In this paper, we will use a unified new variable replacement to study (2), constructed by Shen and Wang in [16]. Define the energy functional associated with (1) by

where . However, is not well defined in because of the term . To overcome this difficulty, we make a change of variable constructed by Shen and Wang in [16]: . Then we obtain

If is a nontrivial solution of (1), then

for all . Let . By [16] we know that (9) is equivalent to

for all . Therefore, in order to find the nontrivial solution of (1), it suffices to study the existence of the nontrivial solutions of the following equations:

Recently, the authors studied generalized quasilinear Schrödinger equations with subcritical growth [19, 20], critical growth [21], and supercritical growth [22].

In order to reduce the statements for main results, we list the assumptions as follows:

for all . and for all . and there exists such that

        for all . uniformly in as . for all and . for all . uniformly in .

Set with the norm

It is easy to prove that is well defined on and under our assumptions and its Gateaux derivative is given by

for all .

Our main result of this paper is as follows.

Theorem 1. Suppose that , , and – are satisfied. Then if , (1) admits a nontrivial solution for all ; if , (1) admits a nontrivial solution for large .

Remark 2. Condition is weaker than the following condition .

is nonincreasing on and nondecreasing on .

Indeed, set , . Then

If holds, then

whenever or . Hence is nondecreasing on , and hence for all . Consequently, implies that

for all ; that is, the condition holds.

From Remark 2 we obtain Corollary 3.

Corollary 3. Suppose that , , -, -, and are satisfied. Then if , (1) admits a nontrivial solution for all ; if , (1) admits a nontrivial solution for large .

Remark 4. In [16], Shen and Wang studied the existence of nontrivial solutions for generalized quasilinear Schrödinger equations

where is a subcritical nonlinearity satisfying the following conditions:

if . as .There exists such that

    for all .There exists such that, for any , there holds

As mentioned above, if we set , then we get the superfluid film equation in plasma physics

whose nontrivial solutions were studied in [23]. But our problem (1) is elliptic problem involving the critical exponent, so our result extends the results of the work [16, 23] to a critical setting. Moreover, the assumptions about the nonlinearity in this paper are different from the assumptions about the nonlinearity in [16, 23].

Remark 5. In [24], Deng et al. studied problem (1) and their result based on more harsh conditions:

is differentiable with respect to for all and continuous with respect to for all . Moreover, for all .There exists such that, for any , there holds , which implies that there exists such that for all .

In this paper, we just assume that is a continuous function. Moreover, there are functionals satisfying but not satisfying the above Ambrosetti-Rabinowitz type condition (see Remark   in [25]). Hence, our result is different from the result there.

2. Proof of Theorem 1

To begin with, we give some lemmas.

Lemma 6. For the functions , , and , the following properties hold:

(1)the functions and are strictly increasing and odd;(2) for all ; for all ;(3) for all ;(4) is decreasing on and increasing on ;(5) for all ;(6) for all ;(7) for all ;(8) and

Proof. Properties (1)–(3) are obvious. By (2), we have

for all and

for all . Consequently, we obtain (4). By mean value theorem and (3), one has

for all , where ; that is, (5) is proved. Obviously, (6) is a consequence of (3) and (5). Moreover, (7) is a consequence of (2). Finally, using L’ Hospital’s rule, we know that (8) is satisfied. This completes the proof.

Denote

Then

Consequently,

Lemma 7. The functions and enjoy the following properties under –:

(1) and uniformly in ;(2) and uniformly in ;(3) for all and ;(4) for all ;(5) uniformly in .

Proof. By -, for any , there exists such that

for all . Set . Then Lemma 6(8) implies that

uniformly in . Moreover, by Lemma 6(6) one has

uniformly in . Similarly, we have

uniformly in and

uniformly in . Hence, (1) and (2) hold.
In the following, we set . If , by Lemma 6(2) and for , we have

for , which implies that

for all . Let . Then

and hence

for . Consequently,

for all , that is, is increasing with respect to . Hence for all and ; that is,

for all and . Note that Lemma 6(1) implies that is an even function. Therefore, if , we easily obtain that for all and . Consequently,

for all and . Combining with , we can conclude (3). Moreover, and Lemma 6(5) imply that for all . Clearly, and Lemma 6(5) imply that (5) is satisfied. This completes the proof.

Lemma 8. Suppose that , , and - are satisfied. Then the energy functional satisfies the following conditions:

(i)There exist such that for .(ii)There exists with such that .

Proof. (i) Set . By -, Lemmas 6(6) and 7(1), and (2), for any , there exists such that

for all . Consequently, for , we have

for small and .
(ii) Take . Then

for large and small . Consequently, we can take for some large such that (ii) holds. This completes the proof.

Lemma 9. Suppose that , , and – are satisfied. Then there exists a bounded Cerami sequence for with , where

is the constant appearing in Lemma 8.

Proof. By Lemma 8 and the mountain pass theorem without condition (see Theorem   in [26]), there exists a Cerami sequence satisfying

where

is the constant appearing in Lemma 8.
Let be such that . Then is bounded from above. Indeed, without loss of the generality, we may assume that for all . Hence, by Lemma 7(3) we have

This shows that is bounded from above.
Now, we prove that is bounded in . Otherwise, if is unbounded, then, up to a subsequence, we may assume that . Set . Then there exists such that in . By , we have

Set . If , then by Lemma 7(4) and Fatou Lemma, one has

as . This is a contradiction. Hence , that is, a.e. on . For any , by we have

for sufficiently large. By (29), Lemmas 6(6) and 7(1), and (2), for any , there exists such that

for all . Consequently,

as and so as by using interpolation inequality. Moreover, (41) implies that

By the arbitrariness of , we obtain as . Hence

This contradicts the fact that is bounded from above. Consequently, is bounded in . This completes the proof of Lemma 9.

Lemma 10. Suppose that , , and – are satisfied. Then if , the minimax level satisfies for all ; if , the minimax level satisfies for large , where is the best constant of the embedding .

Proof. From the minimax characterization of we see that it is sufficient to show that there exists such that .
We follow the strategy used in [24] but need to modify some process. Given , we consider the function

which satisfies the following equations:

Moreover, satisfies

Let be such that for and for , where with . Set . Then

Since and , there exists such that . We claim that there exist two positive constants , independent of such that

for small . Indeed, by we have

By (29), Lemmas 6(6) and 7(1), and (2), for any , there exists such that

for all . Consequently,

as . Note that

as . Hence by (60) one has

as , which implies that

for small enough. On the other hand, (60) leads to

as , which implies that

for and small enough.
Since has only maximum at , one has

Notice that, for , we have

as , which combining with Lemma 7(4) and (5) implies that for any

for small enough. Note that

Consequently,

for small enough. Hence by (68)

From this, we see that for small enough and big enough if . Consequently, for all if .
In the following, we consider the case . Indeed, if the conclusion is false, then there exists a sequence with such that . Take . Then by the proof of Lemma 8, there exists a unique such that . Hence

By Lemma 6(6) and we get

which implies that is bounded. Hence, up to a subsequence, there exists such that as . If , then by and Fatou lemma we have

But, on the other hand, by Lemma 6(6) one has

a contradiction. Hence and by Lemma 7(4) we know that

as . Consequently,

a contradiction. This completes the proof.

Proof of Theorem 1. Since is a bounded Cerami sequence for at the level , there exists such that

Using a standard argument, we know that , that is, is a weak solution of (11). Indeed, for any , we have

Since in , one has

Consequently,

for all . For any , there exists a sequence such that in . Hence

Let , we get

that is, for all . Hence ; that is, is a weak solution of (11).
In the following, we prove that is nontrivial. With the aid of Lemma 10, the proof follows essentially the proof of Theorem   in [16]. For completeness, we present the proof as follows. If the conclusion is false, we may assume . We divide the proof into four steps.
Step  1. We prove that is also a Cerami sequence for the functional , where

By and in , one has

as . Similarly, we have

as . Consequently, is also a Cerami sequence of .
Step  2. There exist and such that

Indeed, by contradiction, then by Lemma   in [27], one has in for . Notice that

which combining with (51) leads to

as . Consequently, there exists a constant such that

Obviously, . Otherwise, as , which contradicts with . Hence by the definition of , we have

that is, . Therefore, (41) implies that

as , which implies that , a contradiction.
Step  3. After a translation of called , then converges weakly to a nonzero critical point of .
Set . Since is a Cerami sequence of and , arguing as in the case of , we may assume in and . So by Step we know . By Lemma 7(3) and Fatou Lemma, one has

which implies that .
Step  4. We use to construct a path which allows us to obtain a contradiction with the definition of mountain pass level .
Define the mountain pass level , where . It follows the arguments used in [28, 29], we can construct a path such that

Then . If , we have already proved Theorem 1. If but , we take the path given by above, and by , we have

a contradiction. Consequently, . This completes the proof of Theorem 1.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (11501403; 11461023; 11701322), the Shanxi Province Science Foundation for Youths under Grant 2013021001-3, the Honghe University Doctoral Research Programs (XJ17B11 and XJ17B12), the Yunnan Province Applied Basic Research for Youths, and the Yunnan Province Local University (Part) Basic Research Joint Project.

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