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Semi-classical solutions of perturbed elliptic system with general superlinear nonlinearity
Boundary Value Problems volume 2014, Article number: 208 (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.
The goal of this paper is to establish the existence of semi-classical solutions to the following perturbed elliptic system of Hamiltonian form:
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;
(W1), and there exist constants and such that
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.
For the problem with a small parameter , it is called the semi-classical problem, which describes the transition between of quantum mechanics and classical mechanics with the parameter ε goes to zero. There is much literature dealing with the existence of semi-classical solutions to the single particle equation
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).
Inspired by the single equation, there are a few works considering the perturbed elliptic systems; see , – and references therein. To the best of our knowledge, the first approach to the singular perturbed system in a bounded domain, with Neumann boundary, and appeared in  by means of a dual variational formulation of the problem. Moreover, in , Sirakov and Soares considered the superquadratic case by using dual variational methods. In , Ramos and Tavares considered the following problem:
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,
(AR) there is a such that
together with a technical assumption that there is such that
or the superquadratic condition
and a condition of the type of Ding-Lee ;
(DL) for and there exist and such that
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..
Observe that conditions (1.5) and , in (AR) or , , in (DL) play an important role in showing that any Palais-Smale sequence or Cerami sequence is bounded in the aforementioned works. However, there are many functions which do not satisfy these conditions, for example,
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:
(W3) there exist and such that
(W4), , and there exist , , and such that
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.
Since , we can choose a such that
Let be such that . Then we can choose such that
Let E be a Hilbert space as defined in Section 2 and . Under assumptions (V0), (V1), (W1), and (W2), the functional
is well defined. Moreover, and for all
We are now in a position to state the main results of this paper.
Assume that V and W satisfy (V0), (V1), (W1), (W2), (W3), and (W4). Then for, (1.1) has a solutionsuch that, and
Assume that V and W satisfy (V0), (V1), (W1), (W2), (W3), and (W4). Then for, (2.1) has a solutionsuch that, and
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
Letting , (1.1) is rewritten as
Analogous to the proof of , Lemma 1], by using (V0), (V1), and the Sobolev inequality, one can demonstrate that there exists a constant independent of λ such that
This shows that is a Hilbert space for . Furthermore, by virtue of the Sobolev embedding theorem, we have
here and in the sequel, by we denote the usual norm in space .
Set , then E is a Hilbert space with the inner product
the corresponding norm is denoted by . Then we have
For any , set
It is obvious that , and are orthogonal with respect to the inner products and . Thus we have . By a simple calculation, one gets
Therefore, the functional defined in (1.9) can be rewritten in a standard way
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 .
Let X be a Hilbert space withand, and letbe of the form
Suppose that the following assumptions are satisfied:
(I1)is bounded from below and weakly sequentially lower semi-continuous;
(I2)is weakly sequentially continuous;
(I3)there existandwithsuch that
Then for some, there exists a sequencesatisfying
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.
Then , moreover,
Let . Then we can prove the following lemma which is very important and crucial.
Suppose that (V0), (V1), (W1), (W2), and (W3) are satisfied. Then
Note that , we have
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), and (W3) are satisfied. Then there exist a constantand a sequencesatisfying
Suppose that (V0), (V1), (W1), (W2), (W3), and (W4) are satisfied. Then any sequencesatisfying (3.9) is bounded in E.
To prove the boundedness of , arguing by contradiction, suppose that . Let , then . If
Let , then . Hence, by virtue of (W4), (3.12), and the Hölder inequality, one gets
This contradiction shows that .
Going if necessary to a subsequence, we may assume the existence of such that . Let . Then
Now we define , then and . Passing to a subsequence, we have in E, in , , and a.e. on . Obviously, (3.15) implies that . For a.e. , we have . Hence, it follows from (2.6), (3.9), (W3), and Fatou’s lemma 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 .
Arguing by contradiction, suppose that , i.e. in E, and so , in , , and , a.e. on . Since is a set of finite measure, we have
Theorem 1.1 is a direct consequence of Theorem 1.2.
Clément P, van der Vorst RCAM: On a semilinear elliptic system. Differ. Integral Equ. 1995, 8: 1317-1329.
Clément P, de Figueiredo DG, Mitidieri E: Positive solutions of semilinear elliptic systems. Commun. Partial Differ. Equ. 1992, 17: 923-940. 10.1080/03605309208820869
de Figueiredo DG, Felmer PL: On superquadratic elliptic systems. Trans. Am. Math. Soc. 1994, 343: 97-116. 10.1090/S0002-9947-1994-1214781-2
de Figueiredo DG, Ding YH: Strongly indefinite functionals and multiple solutions of elliptic systems. Trans. Am. Math. Soc. 2003, 355: 2973-2989. 10.1090/S0002-9947-03-03257-4
Hulshof J, van der Vorst RCAM: Differential systems with strongly variational structure. J. Funct. Anal. 1993, 114: 32-58. 10.1006/jfan.1993.1062
Kryszewski W, Szulkin A: An infinite dimensional Morse theory with applications. Trans. Am. Math. Soc. 1997, 349: 3181-3234. 10.1090/S0002-9947-97-01963-6
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_4
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)00548-8
Li G, Yang J: Asymptotically linear elliptic systems. Commun. Partial Differ. Equ. 2004, 29: 925-954. 10.1081/PDE-120037337
Ávila AI, Yang J: Multiple solutions of nonlinear elliptic systems. NoDEA Nonlinear Differ. Equ. Appl. 2005, 12: 459-479. 10.1007/s00030-005-0022-7
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)00017-2
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.200410420
Li GB, Szulkin A: An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math. 2002, 4: 763-776. 10.1142/S0219199702000853
Kryszewki W, Szulkin A: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Differ. Equ. 1998, 3: 441-472.
Sirakov B:On the existence of solutions of Hamiltonian elliptic systems in . Adv. Differ. Equ. 2000, 5: 1445-1464.
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.2013.07.027
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.2012.10.030
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:2008064
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/s00033-010-0105-0
Ambrosetti A, Badiale M, Cingolani S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 1997, 140: 285-300. 10.1007/s002050050067
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)90096-0
Oh YG:Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585
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/BF02161413
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)89296-7
del Pino M, Felmer P: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 2002, 324: 1-32. 10.1007/s002080200327
Rabinowitz PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 1992, 43: 270-291. 10.1007/BF00946631
Zhang J, Zhao FK: Multiple solutions for a semiclassical Schrödinger equation. Nonlinear Anal. 2012, 75: 1834-1842. 10.1016/j.na.2011.09.032
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.2013.11.063
Ding YH, Lee C, Zhao FK: Semiclassical limits of ground state solutions to Schrödinger systems. Calc. Var. 2013.
Ramos M: On singular perturbations of superlinear elliptic systems. J. Math. Anal. Appl. 2009, 352: 246-258. 10.1016/j.jmaa.2008.06.019
Ramos M, Tavares H: Solutions with multiple spike patterns for an elliptic system. Calc. Var. 2008, 31: 1-25. 10.1007/s00526-007-0103-z
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.2004.02.003
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/S0002-9947-2010-04982-7
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.2012.10.010
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.2013.12.060
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.2005.03.011
Ding YH: Variational Methods for Strongly Indefinite Problems. World Scientific, Hackensack; 2008.
Ding YH, Lee C: Periodic solutions of an infinite dimensional Hamiltonian system. Rocky Mt. J. Math. 2005, 35: 1881-1908. 10.1216/rmjm/1181069621
Qin DD, Tang XH, Jian Z: Multiple solutions for semilinear elliptic equations with sign-changing potential and nonlinearity. Electron. J. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-305
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.2012.12.035
Tang XH: New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation. Adv. Nonlinear Stud. 2014, 14: 349-361.
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.2013.11.062
Tang XH: Non-Nehari manifold method for superlinear Schrödinger equation. Taiwan. J. Math. 2014.
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.2011.30.1237
Liao FF, Tang XH, Zhang J: Existence of solutions for periodic elliptic system with general superlinear nonlinearity. Z. Angew. Math. Phys. 2014.
Sirakov B:Standing wave solutions of the nonlinear Schrödinger equations in . Ann. Mat. 2002, 183: 73-83. 10.1007/s102310200029
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.2007.07.005
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.
Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.
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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Liao, F., Tang, X., Zhang, J. et al. Semi-classical solutions of perturbed elliptic system with general superlinear nonlinearity. Bound Value Probl 2014, 208 (2014). https://doi.org/10.1186/s13661-014-0208-1
- semi-classical solutions
- perturbed elliptic system
- generalized linking theorems