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

Li, Quanqing; Teng, Kaimin; Wu, Xian

doi:https://doi.org/10.1155/2018/3615085

Advances in Mathematical Physics

On this page

Abstract Introduction and Preliminaries Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 3615085 | https://doi.org/10.1155/2018/3615085

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

Quanqing Li,¹Kaimin Teng,²and Xian Wu³

Academic Editor: Ciprian G. Gal

Received12 Sept 2017

Revised12 Nov 2017

Accepted16 Nov 2017

Published03 Jan 2018

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.

References

S. Kurihara, “Large-Amplitude Quasi-Solitons in Superfluid Films,” Journal of the Physical Society of Japan, vol. 50, no. 10, pp. 3262–3267, 1981.
View at: Publisher Site | Google Scholar
E. W. Laedke, K. H. Spatschek, and L. Stenflo, “Evolution theorem for a class of perturbed envelope soliton solutions,” Journal of Mathematical Physics, vol. 24, no. 12, pp. 2764–2769, 1983.
View at: Publisher Site | Google Scholar | MathSciNet
H. S. Brandi, C. Manus, G. Mainfray, T. Lehner, and G. Bonnaud, “Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. I. Paraxial approximation,” Physics of Fluids, vol. 5, no. 10, pp. 3539–3550, 1993.
View at: Publisher Site | Google Scholar
X. L. Chen and R. N. Sudan, “Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma,” Physical Review Letters, vol. 70, no. 14, pp. 2082–2085, 1993.
View at: Publisher Site | Google Scholar
A. de Bouard, N. Hayashi, and J.-C. Saut, “Global existence of small solutions to a relativistic nonlinear Schrödinger equation,” Communications in Mathematical Physics, vol. 189, no. 1, pp. 73–105, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
B. Ritchie, “Relativistic self-focusing and channel formation in laser-plasma interactions,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 50, no. 2, pp. R687–R689, 1994.
View at: Publisher Site | Google Scholar
L. Brüll and H. Lange, “Solitary waves for quasilinear Schrödinger equations,” Expositiones Mathematicae. International Journal for Pure and Applied Mathematics, vol. 4, no. 3, pp. 279–288, 1986.
View at: Google Scholar | MathSciNet
F. G. Bass and N. N. Nasonov, “Nonlinear electromagnetic-spin waves,” Physics Reports, vol. 189, no. 4, pp. 165–223, 1990.
View at: Publisher Site | Google Scholar
V. G. Makhankov and V. K. Fedyanin, “Non-linear effects in quasi-one-dimensional models of condensed matter theory,” Physics Reports, vol. 104, no. 1, pp. 1–86, 1984.
View at: Publisher Site | Google Scholar
R. W. Hasse, “A general method for the solution of nonlinear soliton and kink Schrödinger equations,” Zeitschrift für Physik. B. Condensed Matter and Quanta, vol. 37, no. 1, pp. 83–87, 1980.
View at: Publisher Site | Google Scholar | MathSciNet
H. Lange, M. Poppenberg, and H. Teismann, “Nash-Moser methods for the solution of quasilinear Schrödinger equations,” Communications in Partial Differential Equations, vol. 24, no. 7-8, pp. 1399–1418, 1999.
View at: Publisher Site | Google Scholar | MathSciNet
A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, “Magnetic Solitons,” Physics Reports, vol. 194, no. 3-4, pp. 117–238, 1990.
View at: Publisher Site | Google Scholar
M. Poppenberg, K. Schmitt, and Z.-Q. Wang, “On the existence of soliton solutions to quasilinear Schrödinger equations,” Calculus of Variations and Partial Differential Equations, vol. 14, no. 3, pp. 329–344, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
G. R. Quispel and H. W. Capel, “Equation of motion for the Heisenberg spin chain,” Physica. A. Europhysics Journal, vol. 110, no. 1-2, pp. 41–80, 1982.
View at: Publisher Site | Google Scholar | MathSciNet
S. Cuccagna, “On instability of excited states of the nonlinear Schrödinger equation,” Physica D: Nonlinear Phenomena, vol. 238, no. 1, pp. 38–54, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Shen and Y. Wang, “Soliton solutions for generalized quasilinear Schrödinger equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 80, pp. 194–201, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
W. H. Fleming, “A selection-migration model in population genetics,” Journal of Mathematical Biology, vol. 2, no. 3, pp. 219–233, 1975.
View at: Publisher Site | Google Scholar | MathSciNet
P. L. Kelley, “Self-focusing of optical beams,” Physical Review Letters, vol. 15, no. 26, pp. 1005–1008, 1965.
View at: Publisher Site | Google Scholar
Q. Li, K. Tang, and X. Wu, “Ground state solutions and geometrically distinct solutions for generalized quasilinear Schrödinger equation,” Mathematical Methods in the Applied Sciences, vol. 40, no. 6, pp. 2165–2176, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
Q. Li and X. Wu, “Multiple solutions for generalized quasilinear Schrödinger equations,” Mathematical Methods in the Applied Sciences, vol. 40, no. 5, pp. 1359–1366, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
Q. Li and X. Wu, “Existence, multiplicity, and concentration of solutions for generalized quasilinear Schrödinger equations with critical growth,” Journal of Mathematical Physics, vol. 58, no. 4, 041501, 30 pages, 2017.
View at: Google Scholar | MathSciNet
Q. Li and X. Wu, “Existence of nontrivial solutions for generalized quasilinear Schrödinger equations with critical or supercritical growths,” Acta Mathematica Scientia B, vol. 37, no. 6, pp. 1870–1880, 2017.
View at: Google Scholar | MathSciNet
M. Colin and L. Jeanjean, “Solutions for a quasilinear Schrödinger equation: a dual approach,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 56, no. 2, pp. 213–226, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Deng, S. Peng, and S. Yan, “Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth,” Journal of Differential Equations, vol. 258, no. 1, pp. 115–147, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Ye and C.-L. Tang, “Infinitely many solutions for fourth-order elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 394, no. 2, pp. 841–854, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
J. F. Aires and M. A. Souto, “Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials,” Journal of Mathematical Analysis and Applications, vol. 416, no. 2, pp. 924–946, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
M. Willem, Minimax Theorems, Birkhäuser, Boston, Mass, USA, 1996.
View at: Publisher Site | MathSciNet
J. M. Bezerra do Ó, O. m. Miyagaki, and S. H. Soares, “Soliton solutions for quasilinear Schrödinger equations with critical growth,” Journal of Differential Equations, vol. 248, no. 4, pp. 722–744, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
L. Jeanjean and K. Tanaka, “A remark on least energy solutions in R^N,” Proceedings of the American Mathematical Society, vol. 131, no. 8, pp. 2399–2408, 2003.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

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.

PDF Download Citation

Download other formats

Order printed copies

Views

852

Downloads

809

Citations