Ground State Solution for an Autonomous Nonlinear Schrödinger System

Abstract

In this paper, we study the following autonomous nonlinear Schrödinger system (discussed in the paper), where , and are positive parameters; is the critical Sobolev exponent; and satisfies general subcritical growth conditions. With the help of the Pohožaev manifold, a ground state solution is obtained.

1. Introduction and Main Result

In this paper, we consider the following autonomous nonlinear Schrödinger system: where , and are positive parameters satisfying ; is the critical Sobolev exponent; and satisfies the following conditions:

(f₁) is an odd function.(f₂).(f₃).(f₄)There exists such that , where .

Systems of above type arise in nonlinear optics (cf. [1]). It is well known that a solution of system (1) is called a ground state solution if and its energy is minimal among the energy of all the nontrivial solutions.

The following nonlinear Schrödinger system has been studied by many authors. When , , , and small enough, Ambrosetti et al. [2] proved that (2) has multibump solitons. When , , , , and and are replaced by and , Ambrosetti et al. [3] proved that system (2) has a positive ground state solution. When , , and satisfy the integral conditions and and are replaced by , and , respectively, Liu and Liu [4] proved that (2) has a positive solution. When , is a smooth bounded domain in , , and , are replaced by , respectively, Noris and Ramos [5] proved that (2) admits an unbounded sequence of solutions with , and for sufficiently large . When , , and , , Chen and Zou [6] proved that (2) has a positive ground state solution under which satisfied certain conditions. When , , and , Li and Tang [7] proved that (2) has a nontrivial solution.

Inspired by the above literatures, especially [6], we investigate the existence of ground state solution of system (1). When with , by using the Nehari manifold, Chen and Zou [6] obtained the existence of ground state solution of system (1). But in our paper, without the assumption of the monotonicity of , we have to adopt a new method to replace the Nehari manifold.

The following single Schrödinger equation has been widely studied by many researchers, and relevant results can been referred to [8–10] and the references therein. By [9], we know that if satisfies (f₁)-(f₄); then, equation (3) has a ground state solution. Define where and define where is the optimal constant of the Sobolev embedding .

The main result of this paper is the following.

Theorem 1. Assume that , and are positive parameters satisfying and . Suppose that satisfies (f₁)-(f₄). Then, system (1) has a ground state solution.

Remark 2. There are some examples of functions that satisfy the assumptions (f₁)-(f₄), for example, with and with .

Remark 3. It is obvious that system (1) has no semitrivial solutions. Indeed, if is a solution of system (1), then and if is a solution of system (1), then .

Remark 4. There are some recent studies on the ground state solutions for other types of Schrödinger equations or systems, for example, [6, 11]. Moreover, in the bounded domain, the existence and the regularity of solutions to differential problems have been widely investigated by using tools of harmonic and real analysis and variational methods, for example, [12–14].

2. Preliminaries

In order to make a precise explanation of the results in this paper, we will give some notations.

denote various positive constants.

is the usual Lebesgue space endowed with the norm

endowed with the norm

endowed with the norm

For any , we set

For any , we denote for all .

The weak solutions of (1) correspond to critical points of the functional

Obviously, and for all and , we have

Similar to [15, 16], in order to obtain a ground state solution, we define the Pohožaev manifold and consider the constraint minimization problem where is defined as

We also require the following subcritical system of system (1): where , , and are positive parameters satisfying and satisfies (f₁)-(f₄). The energy functional of system (15) is

Define where

3. Proof of Theorem 1

The following two lemmas will be used in proof.

Lemma 5 (compactness lemma of Strauss, see [9, 10]). Let be two continuous functions satisfying Let be a sequence of measurable functions: such that and a.e. in , as . Then, for any bounded Borel set , one has If one further assumes that and as , uniformly with respect to , then converges to in as .

Lemma 6 (Strauss inequality, see [17]). If , there exists such that, for every , a.e. on .
Before proving Theorem 1, we need to prove a series of lemmas.

Lemma 7. Suppose that (f₁)-(f₄) hold. Then, the Pohožaev manifold is not empty.

Proof. From [17], we know that for any , is a positive solution of the following equation: Define a cut-off function as where and . Let and define . By [16], we have Take small enough such that . Let be a positive ground state solution of equation (3). Then, we have the following Pohožaev equality: Then, . Thus, we have Define ; we have We can easily know that for small enough and for large . Since , is a concave function. Then, there exists a unique such that . Hence, there exists a unique such that . Then, we have . Then,

Lemma 8. Suppose that (f₁)-(f₄) hold. Then, .

Proof. Since , there exists such that . For any , we have . By using Young’s inequality, we have Therefore, we have By using Sobolev’s inequality, we have which implies . Therefore, we conclude that for any , we have Therefore, we have .

Lemma 9. Suppose that (f₁)-(f₄) hold. Then, .

Proof. Let be a positive ground state solution of equation (3). Then, (28) holds and Moreover, we have also which is a solution of equation Then, . Since , we have

Lemma 10. Suppose that (f₁)-(f₄) hold. For any , if , then is bounded in .

Proof. Since , we have Because , there exists and such that . Therefore, we have Then, we have Hence, is bounded in .

Lemma 11. Suppose that (f₁)-(f₄) hold. Then, .

Proof. For any , there exists such that . Since , for any , we have Define . Through simple calculations, we have . We can easily see that is increasing for and is decreasing for . Then, we have and for any . By calculation, we have for . Take large such that Then, there exists such that for all and . Then, we have for all . Since for small enough. Then, there exists such that . So, . Hence, we have for all .
From [18, 19], we know that system (15) has a positive and radial ground state solution. Then, for any and , there exists a positive and radial sequence such that By Lemmas 10 and 11, we know that is bounded in .

Lemma 12. Suppose that (f₁)-(f₄) and (46) hold. Then, .

Proof. Similar to the proof of Lemma 8, we have Using Young’s inequality implies Then, So there exists such that up to a subsequence, . On the other hand, Then, .

Proof of Theorem 1. Because (46) holds, there exists such that in , in , and a.e. in . For any , we have i.e., is a solution of system (1). Suppose that . Set and . Through Lemma 5 and Lemma 6, we have as . Since , by using Young’s inequality, we have One has So we have (i) or (ii) . If (i) holds, then we have which contradicts with Lemma 12. If (ii) holds, then we have This is a contradiction. So and through Remark 3, we know that . Applying the weak lower-semicontinuity of the norm, we have This implies . We complete the proof.

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

This study was supported by the National Natural Science Foundation of China (11861052), Science and Technology Foundation of Guizhou Province ([2019]1300, [2019]5653), Innovation group project of Department of Education of Guizhou Province ([2019]067), and Foundation of Qiannan Normal University for Nationalities (QNYSKYTD2018012).