Ground States for the Schrödinger Systems with Harmonic Potential and Combined Power-Type Nonlinearities

Abstract

We consider a class of coupled nonlinear Schrödinger systems with potential terms and combined power-type nonlinearities. We establish the existence of ground states, by using a variational method. As an application, some symmetry results for ground states of Schrödinger systems with harmonic potential terms are obtained.

1. Introduction

In this paper, we are interested in the steady state of the coupled nonlinear Schrödinger system in , where , , , , and are real constants, , , and , ( when and when ). System (1) has applications in many physical problems, especially in the Hartree-Fock theory for a double Bose-Einstein condensate with interparticle interactions under the magnetic trap. In physic, is the trapping potential for the th species, whose role is to confine the movement of particles. We remark that the harmonic potential is a widely used trapping potential in current experiments [1].

We call solutions of forms , standing wave solutions to (1), where solves the following elliptic system: Here .

The energy functional of (2) is and the work space . Since compactly (for ) and is a Hilbert space with norm (see [2, 3]), the energy functional makes sense in the work space.

We say that is a nontrivial ( or ) bound state of (2) if is a nontrivial critical point of . The ground state is usually defined as the positive minimizer of the following minimization problem:

The existence and structure of ground states for Schrödinger equation or systems have been investigated by many authors (see [4–19] and the references therein).

In the case of a single nonlinear Schrödinger equation , , under some appropriate conditions on , the ground state exists and is radially symmetric [5, 8, 19]. For the single equation with potential terms Rabinowitz [14] used variational methods based on variants of mountain pass theorem to prove that (5) has a positive ground state, for those satisfying and . Similar results were also obtained in [6].

In the case of nonlinear Schrödinger systems, Lin and Wei [11] considered two-component systems of nonlinear Schrödinger equations with trap potentials: where , . Among other things, they showed that there is some such that if , , , , and , , then the ground state solution to (6) always exists.

Sirakov [17] studied a system of two equations, and found that there are always ranges of positive parameters in (7), for which it has a ground state, and ranges of positive parameters for which it does not have ground state.

Maia et al. [13] considered the weakly couple nonlinear elliptic system where for and for . They showed that if is sufficiently large, then there exists a nontrivial positive ground state of (8).

Ma and Zhao [12] considered in which . Under assumptions , , , and , or , then the ground state of (9) exists and is unique up to translations.

Song [18] obtained the existence of ground states for a system of Schrödinger equations with combined power-type nonlinearities and with no trap potentials. It is natural to consider similar results for Schrödinger system with potential terms and the combined power-type nonlinearities.

Motived by the above work, in this paper we focus on the existence and symmetric properties for the ground states of Schrödinger system (2). For simplicity, we only prove the existence result for ground state of where , , and , , , , and are real constants satisfying

To be precise, our first result reads as follows.

Theorem 1. Consider (11) and (12), where Suppose the potential term satisfies Then the ground state of system (10) exists.

Remark 2. Condition (13) is the same as (1.10) and (1.11) in [18], in which Schrödinger systems (2) with were considered. The result in [18] is suitable for the case with all coefficients being positive, while our result can be applied to system (2) with some negative and .

As an application, we give a symmetric result for system with harmonic potential in : where and , , , , and are real constants which satisfy (11), (12), (13), and (14). In this case the work space

We first show that each solution of (16) is classic and decays at infinity.

Theorem 3. Suppose solves (16) with , , , , , and being real constants satisfying , , and . Then (a), for every ;(b) and ;(c).

Using the above regularity and decay result, one can easily show that the ground state is radial symmetric.

Corollary 4. Assume , , , , , (12), (13), and (14). Then there exists a radial symmetric ground state of system (16).

An outline of this paper is as follows. We devote Section 2 to some preparations and the proof of Theorem 1. The proofs of Theorem 3 and Corollary 4 will be given in Section 3.

2. Existence of Ground State

In this section, we always assume (11), (12), (13), (14), and (15). Also, we denote by the norm for . Recall that the work space is a Hilbert space with norm .

We now introduce some lemmas, which will be needed in the proof of Theorem 1.

Lemma 5. All critical points of energy function in are weak solutions of system (10).

Proof. Suppose is a critical point of ; that is, Direct computation shows that, for all , we have Using (12), we have Hence, for all Therefore, is a weak solution of system (10).

Define and the Nehari manifold Note that when (11), (12), (14), and (15) are satisfied. Actually, by using (12) and (14), one can prove that, for all , Therefore, we can choose such that .

The following lemma shows that distance between the Nehari manifold and is positive in or in .

Lemma 6. Assume , , , and (15). Then there are positive constants , such that

Proof. Choose . For each , by using (15) and Young’s inequality, we have Since is embedded into , we know that where we have used the fact that .
Therefore, combining the above inequality with (26), we obtain
Putting (28) into (26), we obtain

For each , define Since , , , and , there holds for small and for large. Therefore, and it is achieved at some . Let Solving , we obtain Recalling (24), we can see .

To conclude, for each , there hold

Define

Proposition 7. Consider .

Proof. Firstly, for all , which implies .
Next, we show that . On one hand, for all , we know that and , which imply That is, . On the other hand, for all , where we have used the fact that . That is, . Therefore, .

Lemma 8. is sequentially weakly lower semicontinuous on with respect to .

Proof. Let weakly in . By the uniform boundedness theorem, is bounded in . Moreover, since the embedding is compact (see [2, 3]), there is a subsequence strongly in . By using Fatou’s lemma, we obtain which ends the proof.

Lemma 9. is achieved on .

Proof. Let be a minimizing sequence for ; that is, . We have , which implies is bounded in . Since compactly, up to a subsequence, there is satisfying By using Lemma 6, , we have . Hence, .
Using Lemma 8, Proposition 7, and the fact that , we obtain By using (36) Let We obtain and it is a minimizer for .

Lemma 10. defined as in (44) is a critical point of .

Proof. Define
From the proof of Lemma 9, we know that is a minimizer for under the constrained condition . Hence, there is a Lagrange multiplier such that
Choosing , we have Choosing , we have Combining (47) with (48) and using , we obtain By the fact that , namely, , that is, +, we obtain Since , , which leads to . Therefore, .

Let us now give a proof of Theorem 1.

Proof of Theorem 1. Lemma 10 shows that is the critical point of energy functional . By Lemma 5, is a solution of system (10). We now claim that and . Hence is a ground state of system (10).
Assume for contradiction that , . On one hand, one can see that is a solution of scalar equation Using Lemma 9 and Proposition 7, we have , where Let . Let be the ground state of (51) (see [14] for the existence). There hold and the least energy
By the fact that and (54), we know that
Notice that . Using (24), (53), (13), and (14), we get and
By using the definition of , (53), (54), and (56), we compute which contradicts (55).
Hence, we obtain and . Therefore is a ground state of (10).

3. Symmetry of Ground State

In this section we first use a bootstrap argument similar to in [20] (see also Theorem  8.1.1 of [21]) to obtain a regularity result (Theorem 3). Then we use this regularity result to prove that the ground state of (16) is symmetric about origin.

Lemma 11. Let (for some ) be a solution of (16). Then .

Proof. Since , we have by using Hölder’s inequality. By the fact that is a maximal accretive operator in (see Theorem  2.5 in [22]), we have which implies .

Proof of Theorem 3. Consider the sequence defined by Set . Since , . Direct computation shows that is to say, is decreasing and . Since , it follows that there exists such that
We claim that . , by Sobolev embedding . If , for some , we have by Lemma 11 Using Sobolev embedding again , for all such that . In particular, . By induction, .
Using Lemma 11 and Sobolev embedding again, we have Therefore Part (a) then follows form Lemma 11.
By Sobolev’s embedding, , which implies are uniformly Lipschitz continuous. Since , there holds that decay to zero at infinity; that is, (b) is valid.
To prove (c), we first take derivative to obtain for every
By (a) and Sobolev’s embedding, . Hence, , , and . Using (a), we know that . By the fact that (see Theorem  2.5 in [22]), we have that is, . Therefore, the right hand side of the first equation in (66) belongs to . Using Lemma 11, . Similarly, .

Proof of Corollary 4. This result is a corollary of Theorem 1 and Corollary  3 in [23]. By using Theorem 1, there exists a ground state of (16). From Theorem 3, the ground state is of class and satisfies , . The maximum principle applied to each single equation in (16) suggests that . Then, by using Theorem  1 in [24] (see also Corollary  3 in [23]), all positive and decay solutions of (16) are radially symmetric about the origin, which proves this corollary.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China no. 11201025 and the Fundamental Research Funds for the Central Universities.