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Semi-classical solutions of perturbed elliptic system with general superlinear nonlinearity
Boundary Value Problems volume 2014, Article number: 208 (2014)
Abstract
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.
1 Introduction
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;
(V1);
(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, [1]–[6] 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 [7]–[9]. 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 [10], [11]. Recently, with the aid of the linking arguments in [12]–[14], the existence of solutions or multiple solutions were obtained with periodic potential and nonlinearity, see [15]–[19] 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 [20]–[27] and references therein. In a very recent paper [28], 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 [11], [29]–[35] 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 [11] by means of a dual variational formulation of the problem. Moreover, in [33], Sirakov and Soares considered the superquadratic case by using dual variational methods. In [31], 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 [30].
Since Kryszewski and Szulkin [6] proposed the generalized linking theorem for the strongly indefinite functionals in 1998, Li and Szulkin [13], Bartsch and Ding [12] 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.[34] studied the asymptotically quadratic case and obtained the existence of multiple solutions. For the superquadratic case with magnetic potential, we refer the reader to [35] 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 [36];
(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 [37], [38] and so on. We refer the reader to [39]–[43] 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 [36]. Soon after, this condition was generalized by Zhang et al.[44].
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,
or
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
and
In the present paper, we make use of the techniques developed in [28], [45] 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.
Theorem 1.1
Assume that V and W satisfy (V0), (V1), (W1), (W2), (W3), and (W4). Then for, (1.1) has a solutionsuch that, and
Theorem 1.2
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.
Example 1.3
satisfies (W1)-(W4), where , with .
Example 1.4
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
Let
and
Analogous to the proof of [46], 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
and
Let
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
Moreover,
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 .
Lemma 2.1
([37], Theorem 4.5], [13], Theorem 2.1])
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
where
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.
From now on we assume without loss of generality that (see [47]), that is, , then defined by (1.8) satisfies
Let
Then , moreover,
and
Let . Then we can prove the following lemma which is very important and crucial.
Lemma 3.1
Suppose that (V0), (V1), (W1), (W2), and (W3) are satisfied. Then
Proof
Note that , we have
It follows from (W3), (1.7), (1.9), (3.3), (3.4), (3.5), and (3.7) that
Now the conclusion of Lemma 3.1 follows by (3.8). □
Applying Lemma 2.1, by standard arguments (see, e.g., [45]), we can prove the following lemma.
Lemma 3.2
Suppose that (V0), (V1), (W1), (W2), and (W3) are satisfied. Then there exist a constantand a sequencesatisfying
Lemma 3.3
Suppose that (V0), (V1), (W1), (W2), (W3), and (W4) are satisfied. Then any sequencesatisfying (3.9) is bounded in E.
Proof
By virtue of (W3), (2.6), and (3.9), one gets
To prove the boundedness of , arguing by contradiction, suppose that . Let , then . If
then by Lions’ concentration compactness principle [48] or [49], Lemma 1.21], in for . Hence, it follows from (W4), (2.4), (3.10), and the Hölder inequality that
From (2.6), (2.7), and (3.9), one has
Let , then . Hence, by virtue of (W4), (3.12), and the Hölder inequality, one gets
Combining (3.11) with (3.13) and using (1.10), (3.9), and (3.10), we have
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
Applying Lemmas 3.1, 3.2, and 3.3, we deduce that there exists a bounded sequence satisfying (3.9) and (3.10) with
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
and
For , it follows from (2.3), (2.5), (3.17), (3.18), and the Hölder inequality that
and
According to (W4), (3.10), (3.17), and (3.18), one gets
By virtue of (1.9), (1.10), and (3.9), we have
Using (W4), (3.16), (3.19), (3.20) with , and (3.22), we obtain
which, together with (1.10), (3.9), (3.21), and (3.23), yields
resulting in the fact that , which contradicts (3.10). Thus . By a standard argument, we easily verify that and . Then is a nontrivial solution of (2.1), moreover,
□
Theorem 1.1 is a direct consequence of Theorem 1.2.
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Acknowledgments
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.
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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
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DOI: https://doi.org/10.1186/s13661-014-0208-1