Table of Contents Author Guidelines Submit a Manuscript
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

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; moc.361@3102nimiakgnet

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 [36]. For more physical motivations and more references dealing with applications, we can refer to [714] 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)