Existence of solutions for a general quasilinear elliptic system via perturbation method
© Jiao et al.; licensee Springer. 2013
Received: 28 May 2013
Accepted: 28 August 2013
Published: 7 November 2013
In this paper, we consider the following quasilinear elliptic 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.
which is referred to as the so-called modified nonlinear Schrödinger system.
Our assumptions on the functions and are as follows.
(A1) The functions , , , , .
(A2) There exist constants , , , satisfying , , and such that
for , , .
for , , .
under suitable assumptions.
It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.2) 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 [16, 17], Floer and Weinstein , Oh [19, 20] and the references therein.
Motivated by the single equation (1.4), the purpose of this paper is to study the existence of both positive and negative solutions for the coupled quasilinear system (1.1). 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 system (1.1) 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 (Theorem 2.1 and Theorem 2.2). In Section 3, we verify the Palais-Smale condition for the perturbed functional. Section 4 is devoted to some asymptotic behavior of the sequences and satisfying some conditions. Finally, our main results will be proved in Section 5.
Throughout this paper, we will use the same C to denote various generic positive constants, and we will use to denote quantities that tend to 0.
2 Perturbation of the functional and main results
In order to obtain the desired existence of solutions for system (1.1), in this section, we introduce a perturbation of the functional and give our main results.
We call a critical point of if , , and for all . That is, is a weak solution for system (1.1).
for all . The idea of this paper is to obtain the existence of the critical points of for small and establish suitable estimates for the critical points as so that we may pass to the limit to get the solutions for the original system (1.1).
Our main results are as follows.
as , and is a critical point of .
Theorem 2.2 Assume that (A1)-(A3) hold, , and . Then has a positive critical point and a negative critical point , and (resp.,) converges to a positive (resp., negative) solution for system (1.1) 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.
has a strongly convergent subsequence in , where is the dual space of .
To give the proof of Proposition 3.1, we need the following lemma firstly.
This completes the proof of Lemma 3.2. □
Now we give the proof of Proposition 3.1.
which implies that , i.e., in . This completes the proof of Proposition 3.1. □
4 Some asymptotic behavior
The following proposition is the key of this section.
as . This completes the proof of Proposition 4.1. □
5 Proof of main results
In this section, we give the proof of our main results. Firstly, we prove Theorem 2.1.
for all , .
as . This completes the proof of Theorem 2.1. □
for some m independent of .
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.
Finally, we give the proof of Theorem 2.2.
Proof of Theorem 2.2 For a positive solution of system (1.1), the proof follows from Proposition 5.1 and Theorem 2.1. A similar argument gives a negative solution of system (1.1). This completes the proof of Theorem 2.2. □
This paper was finished while the first author was a visiting fellow at the School of Mathematical Sciences of Beijing Normal University, and the first author would like to express her gratitude for their hospitality during her visit. This work is supported by the National Science Foundation of China (11061031), Fundamental Research Funds for the Central Universities (31920130004) and Fundamental Research Funds for the Gansu University.
- Poppenberg M, Schmitt K, Wang Z: On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 2002, 14: 329-344. 10.1007/s005260100105MathSciNetView ArticleMATHGoogle Scholar
- Liu J, Wang Y, Wang Z: Soliton solutions for quasilinear Schrödinger equation, II. J. Differ. Equ. 2003, 187: 473-493. 10.1016/S0022-0396(02)00064-5View ArticleMATHGoogle Scholar
- Colin M, Jeanjean L: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 2004, 56: 213-226. 10.1016/j.na.2003.09.008MathSciNetView ArticleMATHGoogle Scholar
- Berestycki H, Lions PL: Nonlinear scalar field equations, I. Arch. Ration. Mech. Anal. 1983, 82: 313-346.MathSciNetMATHGoogle Scholar
- Liu J, Wang Y, Wang Z: Solutions for quasilinear Schrödinger equations via the Nehari method. Commun. Partial Differ. Equ. 2004, 29: 879-901. 10.1081/PDE-120037335View ArticleMATHGoogle Scholar
- Lions PL: The concentration-compactness principle in the calculus of variations. The locally compact case. Part I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 109-145.MATHGoogle Scholar
- Guo Y, Tang Z: Ground state solutions for the quasilinear Schrödinger systems. J. Math. Anal. Appl. 2012, 389: 322-339. 10.1016/j.jmaa.2011.11.064MathSciNetView ArticleMATHGoogle Scholar
- Guo Y, Tang Z: Ground state solutions for the quasilinear Schrödinger equation. Nonlinear Anal. 2012, 75: 3235-3248. 10.1016/j.na.2011.12.024MathSciNetView ArticleMATHGoogle Scholar
- Liu X, Liu J, Wang Z: Quasilinear elliptic equations via perturbation method. Proc. Am. Math. Soc. 2013, 141: 253-263.View ArticleMathSciNetMATHGoogle Scholar
- Bartsch T, Wang Z: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 2000, 51: 366-384.MathSciNetView ArticleMATHGoogle Scholar
- Ambrosetti A, Badiale M, Cingolani S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 1997, 140: 285-300. 10.1007/s002050050067MathSciNetView ArticleMATHGoogle Scholar
- Ambrosetti A, Malchiodi A, Secchi S: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 2001, 159: 253-271. 10.1007/s002050100152MathSciNetView ArticleMATHGoogle Scholar
- Byeon J, Wang Z: Standing waves with a critical frequency for nonlinear Schrödinger equations, II. Calc. Var. Partial Differ. Equ. 2003, 18: 207-219. 10.1007/s00526-002-0191-8MathSciNetView ArticleMATHGoogle Scholar
- Cingolani S, Lazzo M: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 2000, 160: 118-138. 10.1006/jdeq.1999.3662MathSciNetView ArticleMATHGoogle Scholar
- Cingolani S, Nolasco M: Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc. R. Soc. Edinb. 1998, 128: 1249-1260. 10.1017/S030821050002730XMathSciNetView ArticleMATHGoogle Scholar
- Del Pino M, Felmer P: Semi-classical states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré 1998, 15: 127-149. 10.1016/S0294-1449(97)89296-7MathSciNetView ArticleMATHGoogle Scholar
- Del Pino M, Felmer P: Multi-peak bound states for nonlinear Schrödinger equations. J. Funct. Anal. 1997, 149: 245-265. 10.1006/jfan.1996.3085MathSciNetView ArticleMATHGoogle Scholar
- Floer A, Weinstein A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 1986, 69: 397-408. 10.1016/0022-1236(86)90096-0MathSciNetView ArticleMATHGoogle Scholar
- Oh YG: On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 1990, 131: 223-253. 10.1007/BF02161413View ArticleMATHGoogle Scholar
- Oh YG:Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View ArticleMATHGoogle Scholar
- Liu J, Wang Z: Soliton solutions for quasilinear Schrödinger equations, I. Proc. Am. Math. Soc. 2003, 131: 441-448. 10.1090/S0002-9939-02-06783-7View ArticleMATHGoogle Scholar
- Canino A, Degiovanni M: Nonsmooth critical point theory and quasilinear elliptic equations. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 472. Topological Methods in Differential Equations and Inclusions 1995, 1-50. Montreal, PQ, 1994View ArticleGoogle Scholar
- Liu J, Wang Z: Bifurcations for quasilinear Schrödinger equations, II. Commun. Contemp. Math. 2008, 10: 723-743.View ArticleGoogle Scholar
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