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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 431672, 9 pages
Existence of Solutions for a Modified Nonlinear Schrödinger System
1College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730124, China
2School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Received 8 April 2013; Accepted 18 June 2013
Academic Editor: Renat Zhdanov
Copyright © 2013 Yujuan Jiao and Yanli Wang. 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.
We are concerned with the following modified nonlinear Schrödinger system: , , , , , , , where , , , is the critical Sobolev exponent, and is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.
Let us consider the following modified nonlinear Schrödinger system: where , , , is the critical Sobolev exponent, and is a bounded smooth domain.
Solutions for the system (1) are related to the existence of the standing wave solutions of the following quasilinear Schrödinger equation: where is a given potential, is a real constant, and , are real functions. We would like to mention that (2) appears more naturally in mathematical physics and has been derived as models of several physical phenomena corresponding to various types of . For instance, the case was used for the superfluid film equation in plasma physics by Kurihara  (see also ); in the case of , (2) was used as a model of the self-changing of a high-power ultrashort laser in matter (see [3–6] and references therein).
In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form: See, for example, by using a constrained minimization argument, the existence of positive ground state solution was proved by Poppenberg et al. . Using a change of variables, Liu et al.  used an Orlicz space to prove the existence of soliton solution for (3) via mountain pass theorem. Colin and Jeanjean  also made use of a change of variables but worked in the Sobolev space ; they proved the existence of positive solution for (3) from the classical results given by Berestycki and Lions . Liu et al.  established the existence of both one-sign and nodal ground states of soliton type solutions for (3) by the Nehari method. In particular, in , by using Nehari manifold method and concentration compactness principle (see ) in the Orlicz space, Guo and Tang considered the following quasilinear Schrödinger system: with , having a potential well and , , where is the critical Sobolev exponent, and they proved the existence of a ground state solution for the system (4) which localizes near the potential well for large enough. Guo and Tang  also considered ground state solutions of the single quasilinear Schrödinger equation corresponding to the system (4) by the same methods and obtained similar results.
It is worth pointing out that the existence of one-bump or multibump bound state solutions for the related semilinear Schrödinger equation (3) for has been extensively studied. One can see Bartsch and Wang , Ambrosetti et al. , Ambrosetti et al. , Byeon and Wang , Cingolani and Lazzo , Cingolani and Nolasco , Del Pino and Felmer [21, 22], Floer and Weinstein , and Oh [24, 25] and the references therein.
The system (1) is a kind of “limit” problem of the system (4) as . The existence of solutions for the system (1) has important physical interest. The purpose of this paper is to study the existence of both positive and negative solutions for the system (1). We mainly follow the idea of Liu et al.  to perturb the functional and obtain our main results. We point out that the procedure to the system (1) is not trivial at all. Since the appearance of the quasilinear terms and , we need more delicate estimates.
The paper is organized as follows. In Section 2, we introduce a perturbation of the functional and give our main results (Theorems 1 and 2). In Section 3, we verify the Palais-Smale condition for the perturbed functional. Section 4 is devoted to some asymptotic behavior of sequence and satisfying some conditions. Finally, our main results will be proved in Section 5.
Throughout this paper, we will use the same to denote various generic positive constants, and we will use to denote quantities that tend to .
2. Perturbation of the Functional and Main Results
In order to obtain the desired existence of solutions for the system (1), in this section, we introduce a perturbation of the functional and give our main results.
The weak form of the system (1) is which is formally the variational formulation of the following functional:
We may define the derivative of at in the direction of as follows: We call that is a critical point of if , , and for all . That is, is a weak solution for the system (1).
When we consider the system (1) by using the classical critical point theory, we encounter the difficulties due to the lack of an appropriate working space. In general it seems there is no suitable space in which the variational functional possesses both smoothness and compactness properties. For smoothness one would need to work in a space smaller than to control the term involving the quasilinear term in the system (1), but it seems impossible to obtain bounds for sequence in this setting. There have been several ideas and approaches used in recent years to overcome the difficulties such as by minimizations [7, 27], the Nehari method , and change of variables [8, 9]. In this paper, we consider a perturbed functional where is a parameter. Then it is easy to see that is a -functional on . We also can define the derivative of at in the direction of as follows: for all . The idea is to obtain the existence of the critical points of for small and to establish suitable estimates for the critical points as so that we may pass to the limit to get the solutions for the original system (1).
Our main results are as follows.
Theorem 1. Assume that , and . Let and let be a sequence of satisfying and for some independent of . Then, up to a subsequence as and is a critical point of .
Using Theorem 1, we have the following existence result.
Theorem 2. Assume that , and . Then has a positive critical point and a negative critical point , and (resp., ) converges to a positive (resp., negative) solution for the system (1) as .
Notation. We denote by the norm of and by the norm of .
3. Compactness of the Perturbed Functional
In this section, we verify the Palais-Smale condition ( condition in short) for the perturbed functional . We have the following proposition.
Proposition 3. For fixed, the functional satisfies condition for all . That is, any sequence satisfying, for , has a strongly convergent subsequence in , where is the dual space of .
For giving the proof of Proposition 3, we need the following lemma firstly.
Lemma 4. Suppose that a sequence satisfies (11). Then
Now we give the proof of Proposition 3.
Proof of Proposition 3. From Lemma 4, we know that is bounded in . So there exists a subsequence of , still denoted , such that Now we prove that in . In (9), choosing , we have We may estimate the terms involved as follows: Returning to (16), we have which implies that , that is, in . This completes the proof of Proposition 3.
4. Some Asymptotic Behavior
Proposition 3 enables us to apply minimax argument to the functional . In this section, we also study the behavior of sequence and satisfying
The following proposition is the key of this section.
Proposition 5. Assume sequence and satisfy (19). Then after extracting a sequence, still denoted by , one has as .
5. Proof of Main Results
In this section, we give the proof of our main results. Firstly, we prove Theorem 1.
Proof of Theorem 1. Note that satisfies the following equation:
for all . Since
By Moser’s iteration, we have
for some independent of . To show that is a critical point of we use some arguments in [28, 29] (see more references therein). In (24) we choose , , where , , . Substituting into (24), we have
Note that , . By Fatou’s Lemma, the weak convergence of and the fact that is bounded, we have
Let , . We may choose a sequence of nonnegative functions such that in , a.e. and is uniformly bounded in . Then by approximations in (29) we may obtain
for all , .
Similarly, we may obtain an opposite inequality. Thus we have for all . That is, is a critical point of and a solution for the system (1). By doing approximations again, we have in the place of of (31) Setting in (24), we have Using as , (32), (33), and lower semicontinuity, we obtain as .
In particular, we have as . This completes the proof of Theorem 1.
Next, we apply the mountain pass theorem to obtain existence of critical points of . Set for .
Let us consider the functional Here and in the following we denote . The functional satisfies condition. Similarly, we may verify that satisfies condition. By -Young inequality, for any , there exists such that Since Then for small. Thus we have for and for small enough. Choose , and . Define a path : by . When is large enough, we have for some independent of .
Define where From the mountain pass theorem we obtain that is a critical value of .
Let be a critical point corresponding to . We have . Thus is a positive critical point of by the strong maximum principle. In summary, we have the following.
Proposition 6. There exist positive constants and independent of such that has a positive critical point satisfying
Finally, we give the proof of Theorem 2.
Proof of Theorem 2. For a positive solution of the system (1), the proof follows from Proposition 6 and Theorem 1. A similar argument gives a negative solution of the system (1). This completes the proof of Theorem 2.
This paper was finished while Y. Jiao visited School of Mathematical Sciences of Beijing Normal University as a visiting fellow, and she would like to express her gratitude for their hospitality during her visit. Y. Jiao is supported by the National Science Foundation of China (11161041 and 31260098) and Fundamental Research Funds for the Central Universities (zyz2012074).
- S. Kurihara, “Large-amplitude quasi-solitons in superfluid films,” Journal of the Physical Society of Japan, vol. 50, no. 10, pp. 3262–3267, 1981.
- 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.
- 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 B, vol. 5, no. 10, pp. 3539–3550, 1993.
- 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.
- 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.
- B. Ritchie, “Relativistic self-focusing and channel formation in laser-plasma interactions,” Physical Review E, vol. 50, no. 2, pp. R687–R689, 1994.
- 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.
- J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, “Soliton solutions for quasilinear Schrödinger equations II,” Journal of Differential Equations, vol. 187, no. 2, pp. 473–493, 2003.
- M. Colin and L. Jeanjean, “Solutions for a quasilinear Schrödinger equation: a dual approach,” Nonlinear Analysis A, vol. 56, no. 2, pp. 213–226, 2004.
- H. Berestycki and P. L. Lions, “Nonlinear scalar field equations. I: existence of a ground state,” Archive for Rational Mechanics and Analysis, vol. 82, no. 4, pp. 313–345, 1983.
- J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, “Solutions for quasilinear Schrödinger equations via the Nehari method,” Communications in Partial Differential Equations, vol. 29, no. 5-6, pp. 879–901, 2004.
- Y. Guo and Z. Tang, “Ground state solutions for quasilinear Schrödinger systems,” Journal of Mathematical Analysis and Applications, vol. 389, no. 1, pp. 322–339, 2012.
- P. L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case I,” Annales de l'Institut Henri Poincaré, vol. 1, no. 2, pp. 109–145, 1984.
- Y. Guo and Z. Tang, “Ground state solutions for the quasilinear Schrödinger equation,” Nonlinear Analysis A, vol. 75, no. 6, pp. 3235–3248, 2012.
- T. Bartsch and Z. Q. Wang, “Multiple positive solutions for a nonlinear Schrödinger equation,” Zeitschrift für Angewandte Mathematik und Physik, vol. 51, no. 3, pp. 366–384, 2000.
- A. Ambrosetti, M. Badiale, and S. Cingolani, “Semiclassical states of nonlinear Schrödinger equations,” Archive for Rational Mechanics and Analysis, vol. 140, no. 3, pp. 285–300, 1997.
- A. Ambrosetti, A. Malchiodi, and S. Secchi, “Multiplicity results for some nonlinear Schrödinger equations with potentials,” Archive for Rational Mechanics and Analysis, vol. 159, no. 3, pp. 253–271, 2001.
- J. Byeon and Z. Q. Wang, “Standing waves with a critical frequency for nonlinear Schrödinger equations II,” Calculus of Variations and Partial Differential Equations, vol. 18, no. 2, pp. 207–219, 2003.
- S. Cingolani and M. Lazzo, “Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,” Journal of Differential Equations, vol. 160, no. 1, pp. 118–138, 2000.
- S. Cingolani and M. Nolasco, “Multi-peak periodic semiclassical states for a class of nonlinear Schrödinger equations,” Proceedings of the Royal Society of Edinburgh A, vol. 128, no. 6, pp. 1249–1260, 1998.
- M. Del Pino and P. L. Felmer, “Multi-peak bound states for nonlinear Schrödinger equations,” Annales de l'Institut Henri Poincaré, vol. 15, no. 2, pp. 127–149, 1998.
- M. del Pino and P. L. Felmer, “Semi-classical states for nonlinear Schrödinger equations,” Journal of Functional Analysis, vol. 149, no. 1, pp. 245–265, 1997.
- A. Floer and A. Weinstein, “Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,” Journal of Functional Analysis, vol. 69, no. 3, pp. 397–408, 1986.
- Y. G. Oh, “On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,” Communications in Mathematical Physics, vol. 131, no. 2, pp. 223–253, 1990.
- Y. G. Oh, “Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class ,” Communications in Partial Differential Equations, vol. 13, no. 12, pp. 1499–1519, 1988.
- X. Q. Liu, J. Q. Liu, and Z. Q. Wang, “Quasilinear elliptic equations via perturbation method,” Proceedings of the American Mathematical Society, vol. 141, no. 1, pp. 253–263, 2013.
- J. Liu and Z. Q. Wang, “Soliton solutions for quasilinear Schrödinger equations I,” Proceedings of the American Mathematical Society, vol. 131, no. 2, pp. 441–448, 2003.
- A. Canino and M. Degiovanni, “Nonsmooth critical point theory and quasilinear elliptic equations,” in Topological Methods in Differential Equations and Inclusions, P. Q. Montreal, Ed., vol. 472 of Nato Advanced Study Institute Series C: Mathematical and Physical Sciences, pp. 1–50, Kluwer Academic, 1995.
- J. Q. Liu and Z. Q. Wang, “Bifurcations for quasilinear elliptic equations II,” Communications in Contemporary Mathematics, vol. 10, no. 5, pp. 721–743, 2008.