Abstract
In this paper, we study the following Schrödinger-Poisson equations where the parameter and . When the parameter is small and the weight function fulfills some appropriate conditions, we admit the Schrödinger-Poisson equations possess infinitely many negative energy solutions by using a truncation technology and applying the usual Krasnoselskii genus theory. In addition, a byproduct is that the set of solutions is compact.
1. Introduction and Main Results
In the present paper, we are interested in the existence of infinitely many negative energy solutions of the following Schrödinger-Poisson equations: where the parameter , , and is a positive continuous weight function satisfying .
In recent decades, the Schrödinger-Poisson system has been studied widely by many authors, because it has strong physical background and interesting meaning. It arises in nonlinear quantum mechanics models and semiconductor theory. From a physical viewpoint, the system describes the interaction between identical charged particles, when the magnetic effects could be ignored in the interaction with each other and its solution is a standing wave for such a stationary system. The nonlinearity models the mutual interaction between many charged particles. The system consists of a Schrödinger equation coupled with a Poisson equation, which implies that the potential is determined by the charge of the wave function. The nonlocal term means that the particles interact with its own electric field. For more information about the mathematical and physical background of the system, we refer the readers to see papers [1–4] and the references therein.
The studies of the Schrödinger-Poisson system have been focused on the existence of positive solutions, ground state solutions, multiplicity of solutions, radial solutions and the semiclassical limit solutions, concentration behavior of solutions, and sign-changing solutions. See references [5–17] and the references therein.
When the nonlinear term is presented as a subcritical growth, there are many results in the literature. Ruiz [18] studied the following system: where the parameter and . When is small, the author showed that there exists at least one positive radial solutions for , and at least two positive radial solutions for . In particular, if , the author proved that is a threshold of existence and nonexistence of positive radial solutions. When in system (2), Azzollini and Pomponio [19] established the existence of ground state solution for . For related system and more results, please refer readers to see [20–28].
In the paper, we are concerned with a critical growth of nonlinearity term and perturbation of low order terms. In this case, there are some results in the references. As regards the following relevant system, where the parameter and , under some suitable conditions, existence of a nontrivial solution was proved in [19] for and in [29] for . Here, we would like to mention some other papers [30–34] for related results. We note that the existence of solutions is very seriously depending on the range of the . As far as we know, there is no result of Schrödinger-Poisson system involving the combination with a critical nonlinearity and sublinear terms.
The compactness of the imbedding into does not hold, and the nonlocal term and the critical nonlinear term appear in the system, which cause many difficulties for us using the variational methods in a standard way to solve the Schrödinger-Poisson system.
Motivated by works mentioned above, particularly, by the results in [16, 24, 29, 35], we overcome these difficulties mentioned above and obtain the existence of infinitely many negative energy solutions to system (1) for and small .
We denote by the best constant for the Sobolev space imbedding into the Lebesgue space , namely, ,
Now, we give our main result as follows.
Theorem 1. Assume . Then, there exists a positive constant such that system (1) possesses infinitely many negative energy solutions for any . Moreover, the set of solutions obtained above is compact.
Remark 2. When , the system (1) has no solution, which follows from Pohožaev’s identity (see [36]). To some extent, we extend the results in [16, 24, 29, 35].
Remark 3. The key ingredient in the proof of Theorem 1 is the genus theory, which plays an important role in obtaining infinitely many solutions of Schrödinger-Poisson equations (1). We followed the methods of Yao and Mu in [37], where the authors studied nonlocal problem of Kirchhoff-type in high dimension ().
The remainder of this paper is organized as follows. In Section 2, we present the abstract framework of the problem as well as some preliminary results. Theorem 1 shall be proved in Section 3.
2. Preliminaries and Functional Setting
In this section, we will define some notations and establish the variational setting for Schrödinger-Poisson equations (1) and list some fundamental results. (i)Let be the usual Sobolev space endowed with the standard inner product and induced norm (ii) is the usual Lebesgue space equipped with the norm (iii) is the completion of with respect to the norm (iv)The letters and denote various positive constants which may vary from line to line and whose exact values are irrelevant(v)The notations and mean strong convergence and weak convergence in corresponding to functional setting, respectively(vi)We use to denote any infinitely small quantity that tends to zero as
For any , the Lax-Milgram theorem implies that there exists a unique such that, for any , that is, is the weak solution of . Furthermore,
We then can rewrite Schrödinger-Poisson equations (1) as the following: and energy functional associated with equation (10) is
It is readily seen that the energy functional belongs to and that for any . Hence, if is a critical point of functional , then is a solution of equation (10) and is a solution of system (1). We denote for simple expressions.
In what follows, we start to state our preliminary results.
Lemma 4. satisfies the following results: (1) is continuous in and (2)If in , then in (3) for any (4). Furthermore,
The proof is omitted here and refers to [18, 29].
Lemma 5. Assume is a for functional in . Then, is bounded in .
Proof. Arguing by contradiction, assume as . According to the Sobolev and Hölder inequality and , we find that This is a contradiction, and is bounded in .
Lemma 6. Suppose is a for functional in . Then, has a convergent subsequence in provided that .
Proof. By Lemma 5, we know that is bounded in , and up to a subsequence, there exists a such that
In light of Lions’ second concentration compactness lemma [38], there exist an at most countable index set , a sequence of points , and values , such that
in the measure sense, where is the Dirac mass at . We next shall prove that the index set is empty. By the reduction to absurdity, let us suppose that there exists a such that . Consider that some cut-off function such that
Obviously, the sequence is bounded in , and because is a sequence of functional , then , i.e., for large , there is
Let us start to estimate each terms in the equation above. Using the Hölder inequality and (15), we compute simply
Together with formulas above, we note
By combining this inequality with the third formula in (15), we show that
Take a cut-off function such that
For simplicity of computation, denote
By Hölder’s inequality, we easily know that
since and , there exists some positive constant such that for . Using (15), (21), and (22), we obtain
This is a contradiction with the hypothesis, so the index set is empty.
Take and define
In line with lemma 1.40 in [39], and fulfill the following formulas
Take a cut-off function such that
As above, the sequence is bounded in and , namely,
We need again to estimate every terms in the equation above. By the Hölder inequality and the definition of , it follows that
According to the estimates above, we easily show that . Combination with (28), we know that or . If holds, then we have
This is a contradiction with the assumption, thus holds. Together (15) and (28) with the empty index set , we obtain that
By Fatou’s lemma, we prove that
Therefore, in . Set , and in . Suppose . Since is sequentially weakly continuous in , we have that and
Thus for small , this formula forces and as , and in . The proof is finished.
3. The Proof of Theorem 1
In this section, we shall utilize the Krasnoselskii genus theory to establish a minimax class of critical points for proving that Schrödinger-Poisson equations (1) possess infinitely many negative energy solutions. And we next need to introduce the classical concept and some properties of the genus.
Let be a real Banach space and denote the family of sets such that is closed in and symmetric with respect to , namely, implies For any , we define the genus of as follows:
If there is no finite , then define , and by definition of the genus. For and , we denote by a uniform δ-neighborhood of , that is,
In what follows, we shall list some properties of the genus that prepare for showing our results. More detail content about the genus may be seen in the references ([40], Propositions 7.5-7.8).
Proposition 7. Assume . Then, following several results hold. (1)Normalization: if , then (2)Mapping property: if there is an odd map , then (3)Monotonicity property: if , then (4)Subadditivity: (5)Continuity property: if is compact, then and there exists a such that and (6)If there is an odd homeomorphism between and , then (7)If is the sphere in , then (8)If , then (9)If is a subspace of with codimension and , then .
Proposition 8. Assume . If , then set includes infinitely many points.
We define presently an auxiliary function, which essentially follows the idea and method in [41]. Denote where is a positive constant and determined below. By the Hölder and Sobolev inequality, it follows that
For , we observe that gets its positive unique maximum (see Figure 1).
To find the critical points of the energy functional , we truncate the functional as the following: where and is a decreasing function and satisfies where and are two roots of the function (see Figure 1) and . As above, we have that with and see Figure 2. Obviously, and if , then we observe that and .
Therefore, we only need to find some negative critical values for the truncating functional . To do it, we next shall give some lemmas for constructing the minimax sequence of negative critical values of the truncating functional
Before to state the following results, we give some definitions. Denote and define
Lemma 9. For any positive integer , there is a satisfying .
Proof. To show this lemma, we adopt to the argument used in [41], which handles with the local problem. Fix any and suppose is an -dimensional subspace of . Taking with and , by Figures 1 and 2, we observe then, we can choose and such that for and . Denote for , it is easy to know and note Because and are isomorphic and and are homeomorphic, it follows from Proposition 7 that
Lemma 10. Assume . If for some integer , then
Proof. We first claim that each is negative. Indeed, by Lemma 4, for every integer , there exists such that . Because functional is continuous in and even . In virtue of the definition of and low boundedness of , we know Then, we easily show that is compact by Lemma 6. In the following argument, we will prove the expected result by contradiction. Assume , according to the fifth in Proposition 7, then there exists a closed and symmetric set with such that Due to , we can choose such that . Owing to the deformation lemma [42], there is an odd homeomorphism such that for some with . So and then by the definition of , there exists such that , that is, and It means On the other hand, because of and , then , which implies This is a contradiction with (54). Therefore, the proof is finished.
Proof of Theorem 1. By the analysis above, if , then we know that . Thus, it is sufficient to show that the truncating functional possesses infinitely many negative critical values. Noting that is nondecreasing with respect to , there are only the following two cases:
Case I. There are satisfying . In this case, obviously, the functional has infinitely many negative critical values . It is done.
Case II. There is a positive such that for all In the case, by Lemma 10, we know , which implies that the set has infinitely many points, which means that the functional possesses infinitely many negative critical values. Finally, we shall see that the set of solutions is compact.
Let be a sequence of solutions to the equations (1). In Case I, we know that and ; by Lemma 5, we then easily get that there is a satisfying as Then, it follows conclusion. In Case II, it is trivial to see that the set is compact. Thereupon, the proof is completed.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
There are no competing interests regarding this research work.
Authors’ Contributions
All authors equally have made contributions. All authors read and approved the final manuscript.
Acknowledgments
The authors would like to thank manager editor’s help for our Tex file. The first author is supported by the NSFC Grant (No. 11801335). The second author is supported by the Science Foundation of North University of China (No. 2017028). The third author is supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJQN201800818) and the Research Fund of Chongqing Technology and Business University (Grant Nos: 1952012, 1952026, and 1951075).