Semi-classical solutions of perturbed elliptic system with general superlinear nonlinearity
© Liao et al.; licensee Springer 2014
Received: 1 June 2014
Accepted: 21 August 2014
Published: 25 September 2014
This paper is concerned with the following perturbed elliptic system: , , , , , where and . Under some mild conditions on the potential V and nonlinearity W, we establish the existence of nontrivial semi-classical solutions via variational methods, provided that , where the bound is formulated in terms of N, V, and W.
MSC: 35J10, 35J20.
where , is a small parameter, and and satisfy the following basic assumptions, respectively:
(V0) and there exists a such that the set has finite measure;
here and in the sequel if and if or 2;
(W2), as , uniformly in .
For the case , the interest in the study of various qualitative properties of the solutions has steadily increased in recent years. In a bounded smooth domain , similar systems have been extensively studied; see, for instance, – and the references therein. The problem set on the whole space was considered recently in some works. The first difficulty of such a type of problem is the lack of the compactness of the Sobolev embedding. A usual way to recover this difficulty is choosing suitable working space which has compact embedding property, for example, the radially symmetric functions space; see –. The second difficulty is that the energy functional is strongly indefinite different from the single equation case, and so the dual variational methods are involved to avoid this difficulty; see , . Recently, with the aid of the linking arguments in –, the existence of solutions or multiple solutions were obtained with periodic potential and nonlinearity, see – and the references therein.
under various hypotheses on the potential V and the nonlinearity g; for example, see – and references therein. In a very recent paper , Lin and Tang developed a direct and simple approach to show the existence of semi-classical solutions for the single particle equation (1.2) with V satisfying (V0) and (V1). It is well known that the extension of these results of single equation to a system of equations presents some difficulties. One of the main difficulties is that the energy functional associated with (1.1) is strongly indefinite, so the approach used in the single equation is not applicable to system (1.1).
where Ω is a domain of , f and g are power functions, superlinear but subcritical at infinity. The authors established the existence of positive solutions which concentrate, as , on a prescribed finite number of local minimum points of the potential V; also see .
Since Kryszewski and Szulkin  proposed the generalized linking theorem for the strongly indefinite functionals in 1998, Li and Szulkin , Bartsch and Ding  gave several weaker versions, which provide another effective way to deal with such problems. With the aid of the generalized linking theorem, Xiao et al. studied the asymptotically quadratic case and obtained the existence of multiple solutions. For the superquadratic case with magnetic potential, we refer the reader to  and references therein.
In the aforementioned references, it always was assumed that W satisfies a condition of the type of Ambrosetti-Rabinowitz, that is,
and a condition of the type of Ding-Lee ;
As is well known, conditions (1.3) and (1.4) have been successfully applied to Hamiltonian systems, to periodic Schrödinger systems, and to diffusion systems; see ,  and so on. We refer the reader to – and the references therein where the condition (AR) was weakened by more general superlinear conditions. Condition (DL) was firstly given for a single Schrödinger equation by Ding and Lee . Soon after, this condition was generalized by Zhang et al..
Motivated by these works, in the present paper, we shall establish the existence of semi-classical solutions of system (1.1) with a weaker superlinear condition via the generalized linking theorem. To state our results, in addition to the basic hypotheses, we make the following assumptions:
In the present paper, we make use of the techniques developed in ,  to obtain the existence of semi-classical solutions for system (1.1) when , where the bound is formulated in terms of N, V, and W.
We are now in a position to state the main results of this paper.
Before proceeding to the proofs of these theorems, we give two examples to illustrate the assumptions.
satisfies (W1)-(W4), where , with .
satisfies (W1)-(W4), where with .
The rest of the paper is organized as follows. In Section 2, we provide a variational setting. In Section 3, we give the proofs of our theorems.
2 Variational setting
here and in the sequel, by we denote the usual norm in space .
The following generalized linking theorem provides a convenient approach to get a sequence.
Let X be a Hilbert space with and . For a functional , φ is said to be weakly sequentially lower semi-continuous if for any in X one has , and is said to be weakly sequentially continuous if for each .
Suppose that the following assumptions are satisfied:
(I1)is bounded from below and weakly sequentially lower semi-continuous;
(I2)is weakly sequentially continuous;
Such a sequence is called a Cerami sequence on the level c, or asequence.
3 Proofs of the theorems
In this section, we give the proofs of Theorems 1.1 and 1.2.
Let . Then we can prove the following lemma which is very important and crucial.
Now the conclusion of Lemma 3.1 follows by (3.8). □
Applying Lemma 2.1, by standard arguments (see, e.g., ), we can prove the following lemma.
Suppose that (V0), (V1), (W1), (W2), (W3), and (W4) are satisfied. Then any sequencesatisfying (3.9) is bounded in E.
This contradiction shows that .
This contradiction shows that is bounded. □
Proof of Theorem 1.2
Going if necessary to a subsequence, we can assume that in , and . Next, we prove that .
Theorem 1.1 is a direct consequence of Theorem 1.2.
This work is partially supported by the NNSF (No. 11471278) of China, Scientific Research Fund of Hunan Provincial Education Department (12C0895), and the Construct Program of the Key Discipline in Hunan Province.
- Clément P, van der Vorst RCAM: On a semilinear elliptic system. Differ. Integral Equ. 1995, 8: 1317-1329.Google Scholar
- Clément P, de Figueiredo DG, Mitidieri E: Positive solutions of semilinear elliptic systems. Commun. Partial Differ. Equ. 1992, 17: 923-940. 10.1080/03605309208820869View ArticleGoogle Scholar
- de Figueiredo DG, Felmer PL: On superquadratic elliptic systems. Trans. Am. Math. Soc. 1994, 343: 97-116. 10.1090/S0002-9947-1994-1214781-2MathSciNetView ArticleGoogle Scholar
- de Figueiredo DG, Ding YH: Strongly indefinite functionals and multiple solutions of elliptic systems. Trans. Am. Math. Soc. 2003, 355: 2973-2989. 10.1090/S0002-9947-03-03257-4MathSciNetView ArticleGoogle Scholar
- Hulshof J, van der Vorst RCAM: Differential systems with strongly variational structure. J. Funct. Anal. 1993, 114: 32-58. 10.1006/jfan.1993.1062MathSciNetView ArticleGoogle Scholar
- Kryszewski W, Szulkin A: An infinite dimensional Morse theory with applications. Trans. Am. Math. Soc. 1997, 349: 3181-3234. 10.1090/S0002-9947-97-01963-6MathSciNetView ArticleGoogle Scholar
- Bartsch T, de Figueiredo DG: Infinitely many solutions of nonlinear elliptic systems. In Topics in Nonlinear Analysis. Birkhäuser, Basel; 1999:51-67. 10.1007/978-3-0348-8765-6_4View ArticleGoogle Scholar
- de Figueiredo DG, Yang J: Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal. 1998, 33: 211-234. 10.1016/S0362-546X(97)00548-8MathSciNetView ArticleGoogle Scholar
- Li G, Yang J: Asymptotically linear elliptic systems. Commun. Partial Differ. Equ. 2004, 29: 925-954. 10.1081/PDE-120037337View ArticleGoogle Scholar
- Ávila AI, Yang J: Multiple solutions of nonlinear elliptic systems. NoDEA Nonlinear Differ. Equ. Appl. 2005, 12: 459-479. 10.1007/s00030-005-0022-7View ArticleGoogle Scholar
- Ávila AI, Yang J: On the existence and shape of least energy solutions for some elliptic systems. J. Differ. Equ. 2003, 191: 348-376. 10.1016/S0022-0396(03)00017-2View ArticleGoogle Scholar
- Bartsch T, Ding YH: Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math. Nachr. 2006, 279: 1267-1288. 10.1002/mana.200410420MathSciNetView ArticleGoogle Scholar
- Li GB, Szulkin A: An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math. 2002, 4: 763-776. 10.1142/S0219199702000853MathSciNetView ArticleGoogle Scholar
- Kryszewki W, Szulkin A: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Differ. Equ. 1998, 3: 441-472.Google Scholar
- Sirakov B:On the existence of solutions of Hamiltonian elliptic systems in . Adv. Differ. Equ. 2000, 5: 1445-1464.MathSciNetGoogle Scholar
- Zhang J, Tang XH, Zhang W: Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms. Nonlinear Anal. 2014, 95: 1-10. 10.1016/j.na.2013.07.027MathSciNetView ArticleGoogle Scholar
- Zhang J, Qin WP, Zhao FK: Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system. J. Math. Anal. Appl. 2013, 399: 433-441. 10.1016/j.jmaa.2012.10.030MathSciNetView ArticleGoogle Scholar
- Zhao F, Zhao L, Ding Y: Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems. ESAIM Control Optim. Calc. Var. 2010, 16: 77-91. 10.1051/cocv:2008064MathSciNetView ArticleGoogle Scholar
- Zhao F, Zhao L, Ding Y:Multiple solution for a superlinear and periodic elliptic system on . Z. Angew. Math. Phys. 2011, 62: 495-511. 10.1007/s00033-010-0105-0MathSciNetView ArticleGoogle Scholar
- Ambrosetti A, Badiale M, Cingolani S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 1997, 140: 285-300. 10.1007/s002050050067MathSciNetView ArticleGoogle Scholar
- Floer A, Weinstein A: Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential. J. Funct. Anal. 1986, 69: 397-408. 10.1016/0022-1236(86)90096-0MathSciNetView ArticleGoogle Scholar
- Oh YG:Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View ArticleGoogle Scholar
- Oh YG: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 1990, 131: 223-253. 10.1007/BF02161413View ArticleGoogle Scholar
- del Pino M, Felmer P: Multipeak bound states of nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1998, 15: 127-149. 10.1016/S0294-1449(97)89296-7MathSciNetView ArticleGoogle Scholar
- del Pino M, Felmer P: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 2002, 324: 1-32. 10.1007/s002080200327MathSciNetView ArticleGoogle Scholar
- Rabinowitz PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 1992, 43: 270-291. 10.1007/BF00946631MathSciNetView ArticleGoogle Scholar
- Zhang J, Zhao FK: Multiple solutions for a semiclassical Schrödinger equation. Nonlinear Anal. 2012, 75: 1834-1842. 10.1016/j.na.2011.09.032MathSciNetView ArticleGoogle Scholar
- Lin X, Tang XH: Semiclassical solutions of perturbed p -Laplacian equations with critical nonlinearity. J. Math. Anal. Appl. 2014, 413: 439-449. 10.1016/j.jmaa.2013.11.063MathSciNetView ArticleGoogle Scholar
- Ding YH, Lee C, Zhao FK: Semiclassical limits of ground state solutions to Schrödinger systems. Calc. Var. 2013.Google Scholar
- Ramos M: On singular perturbations of superlinear elliptic systems. J. Math. Anal. Appl. 2009, 352: 246-258. 10.1016/j.jmaa.2008.06.019MathSciNetView ArticleGoogle Scholar
- Ramos M, Tavares H: Solutions with multiple spike patterns for an elliptic system. Calc. Var. 2008, 31: 1-25. 10.1007/s00526-007-0103-zMathSciNetView ArticleGoogle Scholar
- Pistoia A, Ramos M: Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions. J. Differ. Equ. 2004, 201: 160-176. 10.1016/j.jde.2004.02.003MathSciNetView ArticleGoogle Scholar
- Sirakov B, Soares SHM: Soliton solutions to systems of coupled Schrödinger equations of Hamiltonian type. Trans. Am. Math. Soc. 2010, 362: 5729-5744. 10.1090/S0002-9947-2010-04982-7MathSciNetView ArticleGoogle Scholar
- Xiao L, Wang J, Fan M, Zhang F: Existence and multiplicity of semiclassical solutions for asymptotically Hamiltonian elliptic systems. J. Math. Anal. Appl. 2013, 399: 340-351. 10.1016/j.jmaa.2012.10.010MathSciNetView ArticleGoogle Scholar
- Zhang J, Tang XH, Zhang W: Semiclassical solutions for a class of Schrödinger system with magnetic potentials. J. Math. Anal. Appl. 2014, 414: 357-371. 10.1016/j.jmaa.2013.12.060MathSciNetView ArticleGoogle Scholar
- Ding YH, Lee C: Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Differ. Equ. 2006, 222: 137-163. 10.1016/j.jde.2005.03.011MathSciNetView ArticleGoogle Scholar
- Ding YH: Variational Methods for Strongly Indefinite Problems. World Scientific, Hackensack; 2008.Google Scholar
- Ding YH, Lee C: Periodic solutions of an infinite dimensional Hamiltonian system. Rocky Mt. J. Math. 2005, 35: 1881-1908. 10.1216/rmjm/1181069621MathSciNetView ArticleGoogle Scholar
- Qin DD, Tang XH, Jian Z: Multiple solutions for semilinear elliptic equations with sign-changing potential and nonlinearity. Electron. J. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-305Google Scholar
- Tang XH: Infinitely many solutions for semilinear Schrödinger equation with sign-changing potential and nonlinearity. J. Math. Anal. Appl. 2013, 401: 407-415. 10.1016/j.jmaa.2012.12.035MathSciNetView ArticleGoogle Scholar
- Tang XH: New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation. Adv. Nonlinear Stud. 2014, 14: 349-361.Google Scholar
- Tang XH: New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum. J. Math. Anal. Appl. 2014, 413: 392-410. 10.1016/j.jmaa.2013.11.062MathSciNetView ArticleGoogle Scholar
- Tang XH: Non-Nehari manifold method for superlinear Schrödinger equation. Taiwan. J. Math. 2014.Google Scholar
- Zhang RM, Chen J, Zhao FK: Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete Contin. Dyn. Syst., Ser. A 2011, 30: 1249-1262. 10.3934/dcds.2011.30.1237MathSciNetView ArticleGoogle Scholar
- Liao FF, Tang XH, Zhang J: Existence of solutions for periodic elliptic system with general superlinear nonlinearity. Z. Angew. Math. Phys. 2014.Google Scholar
- Sirakov B:Standing wave solutions of the nonlinear Schrödinger equations in . Ann. Mat. 2002, 183: 73-83. 10.1007/s102310200029MathSciNetView ArticleGoogle Scholar
- Ding YH, Wei JC: Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials. J. Funct. Anal. 2007, 251: 546-572. 10.1016/j.jfa.2007.07.005MathSciNetView ArticleGoogle Scholar
- Lions PL: The concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 223-283.Google Scholar
- Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.View ArticleGoogle Scholar