Abstract
In this paper, by virtue of the critical point theory, we are pleased to investigate the existence of infinitely many high energy solutions for the generalized Chern-Simons-Schrödinger system with a perturbation.
1. Introduction
In this work, we are concerned with the following generalized Chern-Simons-Schrödinger system with a perturbation: where are positive parameters and satisfy the following conditions: (V1), , and on ;(V2)There exists such that is nonempty and has finite measure;(V3)There exists such that and , where denotes the ball of radius centered at ;(H1), and as ;(H2)There exists such that and for ;(H3) as ;(H4)There exist and such that ;(H5) for ;(g), and , for , where .
Recently, the Chern-Simons-Schrödinger system has been paid more attention by many researchers (for example, see [1–10]), where denotes the imaginary unit, for is the complex scalar field, is the gauge field, and is the covariant derivative for
In [1], the authors studied the nonlocal semilinear Schrödinger equation with the gauge field where the potential satisfies (V1)-(V3) (this potential can also be found in [11, 12]). When satisfies more general 6 superlinear conditions at infinity, they obtained some existence theorems of nontrivial solutions for (3). Some similar results can also be found in [2, 5] with a constant external potential. In [4], the authors improved the results in [1, 2, 5] and used the concentration-compactness principle to obtain two bound state solutions for the generalized Chern-Simons-Schrödinger system where . In [3], the authors used some new techniques joined with the manifold of Pohožaev-Nehari type to study the existence of a semiclassical ground state solution for the generalized Chern-Simons-Schrödinger system where their results are available to the nonlinearity for .
There also are some papers in the literature which consider perturbation terms (see [12–21]) and the references therein (also refer to [22–27]). For example, in [13, 14], the authors used the famous Ambrosetti-Rabinowitz condition (see also [12, 15]) to study the existence of solutions for the following Schrödinger equations: where is the fractional -Laplace operator. It is generally known that our conditions (H2) and (H4) are weaker than the corresponding (AR) condition: there exists such that
So, our results here can be viewed as an extension to the ones in [12–15].
In [16], the authors studied the following nonhomogeneous Schrödinger-Kirchhoff-type fourth-order Elliptic equations in :
They obtained the existence of infinitely many solutions for this system by means of the symmetry mountain pass theorem and the fountain theorem.
Now, we state the main result:
Theorem 1. Suppose that (V1)-(V3), (H1)-(H5), and (g) hold. Then, for arbitrarily small , there exists such that system (1) possesses infinitely many high energy solutions when .
Remark 2. From (H1), (H2), and (H4), we can get a growth condition for . Using (H2) and (H4), for , we have and . Let . Then, from (H4), we have and On the other hand, using (H1) for all , we have . Therefore, we obtain and thus,
2. Preliminaries
Let be the usual -norm for , and stand for different positive constants. We use to denote a Sobolev space with the norm
Define the space with the inner product and norm
Note the large parameter in Theorem 1, so we need the following inner product and norm:
Define ; then, we have which is a Hilbert space. Using (V1)-(V3), there exist positive constants (independent of ) such that
Moreover, by [28], the embedding is continuous for , and is compact for , i.e., there exists such that
For convenience, let .
Now, on , we define the following energy functional:
By (V1)-(V3), (9), (10), and [8], is of class , and
Note that (16) in [3], we have
Consequently, we have
Lemma 3 (see [4, 7, 8]). Suppose that converges to a.e. in and converges weakly to in Let Then, and converge to for and converge to .
We say that satisfies -condition if any sequence such that has a convergent subsequence.
Lemma 4 (see [29]). Suppose that is an infinite dimensional Banach space, and are two subspaces of with where is finite dimensional. If satisfies the -condition for all and
(C1). for all ;
(C2). there exist constants such that ;
(C3). for any finite dimensional subspaces there exists such that on ,
then possesses an unbounded sequence of critical values.
3. Main Results
In order to prove Theorem 1, we provide some lemmas.
Lemma 5. Under assumptions (V1)-(V3), (H1)-(H5), and (g), any sequence satisfying is bounded in .
Proof. To prove the boundedness of , argument by contrary, assume that . Let . Then, , and . Note that by (g); for large , from (16), we have
for the fact that and is an arbitrarily small parameter.
In view of (20), we have
Recall that , and there exists a function such that weakly in , strongly in with and for a.e. . Define a set with , and we consider the following two possible cases.
Case 1. , and weakly in , for a.e. . From (9), we have
On the other hand, by the Hölder inequality, (24), and (H4), we obtain
Combining (26) and (27), we have which contradicts (25).
Case 2. . Hence, let and then, . For , we have , and hence, for large . Note that from Proposition 2.1 in [8] and (2.15) in [3], there exists a constant such that Therefore, by (H3), (29), and (9), note the nonnegativity of and , Fatou’s lemma enables us to obtain
This is also a contradiction.
From Cases 1 and 2, we have that is bounded in .
Lemma 6. Under assumptions (V1)-(V3), (H1)-(H5), and (g), any sequence satisfying (22) has a convergent subsequence in .
Proof. From Lemma 5 and the compactness of for , we have By a standard argument, we can check that From Lemma 3, we easily have Note that from Lemma 1 in [30], there exists such that . Therefore, from (g) and the Hölder inequality, we have By a simple calculation, we have It is clear that . As a result, from (32)-(34), we have Let . Then, we have Note in and from Lemma A.1 of [28], there exists with such that From this and (2.12), for and with we have Recall From (2.11), (2.12), and (29), we have where
Lemma 7. Under assumptions (V1)-(V3), (H1)-(H5), and (g), for any finite dimensional subspace , there hold
Proof. Arguing indirectly, assume that for some sequence with , such that , . Let . Then, , and there is a function such that in . Since , we have in , for a.e., , and . Let . Then, , and for a.e., . From , (29), and (g), we have Note that from the L’Hospital rule and (H3), we have Fatou’s lemma implies that This contradicts ((43)), and thus, ((41)) holds.
Proof of Theorem 1. Note that is a Hilbert space, and let be a total orthonormal basis of , and define . From the compact embedding with and Lemma 3.8 in [28], we have This, together with (10), implies that Note that , , and can be arbitrarily small. So, there exists ; we can choose , , and let with ; by ((47)), we find
In Lemma 4, let , and From Lemma 5-7 and (48), we have that (C2)-(C3) of Lemma 4 hold, and satisfies the -condition for all . Moreover, (H5) implies that (C1) of Lemma 4 is also satisfied. Thus, possesses an unbounded sequence of critical values and then (1) possesses infinitely many high energy solutions, i.e.,
4. Conclusions
In this paper, we use a variational method and critical point theory to study the existence of infinitely many high energy solutions for the generalized Chern-Simons-Schrödinger system (1) with a perturbation. The conditions used in this paper are weaker than the famous Ambrosetti-Rabinowitz condition. Moreover, we consider the effect of the parameters on the existence of solutions.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Authors’ Contributions
The study was carried out in collaboration with all authors. All authors read and approved the final manuscript.
Acknowledgments
This work is supported by Talent Project of Chongqing Normal University (Grant No. 02030307-0040), the China Postdoctoral Science Foundation (Grant No. 2019M652348), Natural Science Foundation of Chongqing (Grant No. cstc2020jcyj-msxmX0123), and Technology Research Foundation of Chongqing Educational Committee (Grant Nos. KJQN2019 00539 and KJQN202000528).