On Nonlinear Schrödinger-Bopp-Podolsky System with Asymptotically Periodic Potentials

Abstract

In the paper, we study the existence of ground states for a nonlinear Schrödinger-Bopp-Podolsky system with asymptotically periodic potentials. As a consequence, we also prove ground states for the nonlinear Schrödinger-Bopp-Podolsky system with periodic potentials.

1. Introduction and Main Result

In [8], d’Avenia and Siciliano investigated the following nonlinear Schrödinger-Bopp-Podolsky system

where , which has been considered for the first time in the mathematical literature. Such a system appears when we couple a Schrödinger field with its electromagnetic field in the Bopp-Podolsky electromagnetic theory, and, in particular, in the electrostatic case for standing waves , see ([8], Section 2]) for more details. For more physical details, we refer the reader to the recent papers [1, 3, 4, 7], etc. In addition, we point out that the operator appears also in other different interesting mathematical and physical situations (see [2, 10] and their references).

Since [8], the existence of nontrivial solutions for the nonlinear Schrödinger-Bopp-Podolsky system has begun to receive much attention. We refer the reader to [5, 6, 11, 15, 17, 18], etc.

For the subcritical case, we can see from [8] that there is a difference in the result depending on the range where varies. Moreover, major treatment methods are different if , , or .

In the present paper, we are mainly concerned with the positive solutions of the following nonlinear Schrödinger-Bopp-Podolsky system with subcritical exponent where , and the potentials satisfy

, , for all and

, , for all and

Here,

, denotes the Lebesgue measure in , and satisfy

, , and for all

, , and for all

Condition was introduced by Liu et al. in [13] and unifies the following two types of asymptotic conditions:

, , and

, , and

As is well known, condition was given by Rabinowitz in [16] to study the nonlinear Schrödinger equations and has been widely used in the study of various elliptic problems.

Although Chen et al. [5] also considered nonlinear Schrödinger-Bopp-Podolsky system under condition , they required that the nonlinearity is autonomous, and the function satisfies more assumptions. Thus, to the best of our knowledge, under and , there is no result for systems (2).

Before stating our results, few preliminaries are in order. (i) is the usual Sobolev space endowed with the norm(ii) is the Lebesgue space endowed with the norm(iii) is the Sobolev space endowed with the norm

From [8], we know that for every , there is a unique solution such that

Moreover, it turns out that

By using the classical reduction argument, one is led to study, equivalently, the single equation

This means that we just need to find the solutions of Eq. (9).

Now, we introduce the main result of the present paper.

Theorem 1. Let and . Assume that and hold. Then, Eq. (9) has a ground state solution.
If and , then Eq. (9) reduces to the following periodic case From Theorem 1, we have the following corollary.

Corollary 2. Let and . Assume that and hold. Then, Eq. (10) has a ground state solution.

Remark 3. In fact, the ground state solution we find is positive.

2. Proof of Theorem 1

For the sake of simplicity, we assume . We also use to denote different positive constants whose exact value is inessential. From ([13], Lemma 5), it follows that there is such that

i.e., is equivalent to the norm .

The energy functional associated with Eq. (9) is

When and , the functional turns into

Obviously, is of class and for any ,

As everyone knows, the weak solutions of Eq. (9) correspond to the critical points of the functional . For obtaining a ground state solution (also known as least energy solution), we define where . In addition, for the functional , we also define

To prove the theorem, we introduce a series of lemmas in order.

Lemma 4. Let and . Define Then, .

Proof. From [8], we assume that is a nontrivial solution of the following equation Then, one has i.e., .

From now on, we assume that all the conditions of Theorem 1 are always true.

Lemma 5. For each , there exists a unique such that . Moreover, the function is increasing on and decreasing on and .

Proof. If , then By defining , one has Because , we have that for small enough and for large enough. Then, there exists such that . Note that . Thus, .
Suppose that there exists such that and . Then,

It is a contradiction. Thus, is unique. It is obvious that the function is increasing on and decreasing on and

Lemma 6. .

Proof. From Lemma 5, we know that for any , one has , and there exists such that and . Recall that and for all . Thus, we have According to the arbitrariness of , we get that .

Lemma 7. .

Proof. For any , one has Then, there exists such that Thus, for any , we have Therefore, .

Lemma 8. For any , .

Proof. From (26), we know that for any , which implies that for any , .

Lemma 9. There exists a bounded sequence such that and in , where is the dual space of .

Proof. By using the Ekeland variational principle [9], we get that there exists and such that and in . From (27), we have which deduces that is bounded in . Thus,

Coupling with (28), we obtain . Since is bounded in , by using the Hölder inequality, we get

Therefore,

In order to prove that is achieved, we need the following lemmas.

Lemma 10 (see [13], Lemma 8). If is bounded in and , then for any , the following identities hold.

Lemma 11. If in , then the following identities hold.

Proof. Suppose that . Then, from ([13], Lemma 10), we know that for any , there exists such that where are independent on . Thus, According to the arbitrariness of , we obtain that By using the Hölder inequality and the Sobolev inequality, we have also Because and , the conclusions hold.

Lemma 12 (see [8]). If in , then for any , the formula holds.
Now, we prove that is achieved.

Lemma 13. There exists such that .

Proof. It follows from Lemma 9 that there exists a bounded sequence satisfying , , and in . Up to a subsequence, there exists such that in , in for each , and , a.e., in . Since in , for any using Lemma 12, one has i.e., is a solution of Eq. (9). Thus, .
We divide two steps to complete the proof.

Step 1. Suppose that . Then, . By the Fatou lemma and the weakly lower semicontinuity of norm, we have which deduces .

Step 2. Suppose that . Define If , the Lions lemma [12] implies in for each . We have and then . It is a contradiction. Thus, . Up to a subsequence, there exists and such that Define . Because , there exists such that up to a subsequence, in , in , and , a.e., in . Note that Thus, . If is bounded, there exists such that which contradicts with in . Thus, is unbounded. Up to a subsequence, we may assume . Coupling Lemma 10, we get that for any , i.e., is a nontrivial solution of Eq. (10). Using the Fatou lemma, the weakly lower semicontinuity of norm, Lemma 6, and Lemma 11m we imply Thus, . Because is a nontrivial solution of Eq. (10), . By Lemma 5, there exists such that . Then, one has Therefore, is achieved.

Lemma 14. If is achieved by , then is a solution of Eq. (9).

Proof. We borrow the idea in [14]. Suppose by contradiction that is not a solution of Eq. (9). Then there exists such that . Let small enough such that for all and , . We define a smooth cut-off function satisfying for and for . For , we introduce a curve for and for . Obviously, is a continuous curve and when small enough, for . We claim , for all . Indeed, if , . If , owing to is of , there exists such that Recall that , then and . By the continuity of there exists such that . Thus and , which is a contradiction. Therefore, is a solution of Eq. (9).

Proof of Theorem 1. From Lemma 13 we know that is achieved by . Note that the functionals are even. Thus is achieved by a nonnegative function . It follows from Lemma 14 that is a nonnegative solution of Eq. (9). The strong maximum principle implies that .

Data Availability

The findings in this research do not make use of data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Supported by NNSF of China (11861052), Science and Technology Foundation of Guizhou Province ([2019]5672), Foundation of Education of Guizhou Province (KY [2016] 316, KY [2019] 067), and Foundation of QNUN (qnsy2018017).