Advances in Mathematical Physics
Volume 2015 (2015), Article ID 760157, 10 pages
http://dx.doi.org/10.1155/2015/760157

Research Article

Existence of Multiple Positive Solutions for Choquard Equation with Perturbation

Tao Xie,¹ Lu Xiao,¹ and Jun Wang²

¹School of Management, Jiangsu University, Zhenjiang, Jiangsu 212013, China
²Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received 7 May 2015; Revised 17 July 2015; Accepted 13 September 2015

Academic Editor: Kamil Brádler

Copyright © 2015 Tao Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with the following Choquard equation with perturbation: , , where , , and . This kind of equations is well known as the Choquard or nonlinear Schrödinger-Newton equation. Under some assumptions for the functions , we prove the existence of multiple positive solutions of the equation. Moreover, we also show that these results still hold for more generalized Choquard equation with perturbation.

1. Introduction and Main Results

In present paper, we are concerned with the existence of multiple solutions to the following Choquard equation with perturbation:

where , , , and is nonnegative function.

This kind of (1) arises in various physical contexts, especially in the case where , , , and . Then (1) becomes

It is called the stationary nonlinear Choquard equation or the nonlinear Schrödinger-Newton equation. In general, many mathematicians are concerned with the positive solitary solutions of the following nonlinear generalized Choquard equation:

where the powers and . In order to obtain the solitary solutions of (3), we set ( is a constant) in (3) and get the stationary equation of (1) without perturbation, where .

In 1954, paper [1] proposed model (2) to the description of the quantum theory of a polaron. Later, (2) was proposed by Choquard in 1976 as an approximation to Hartree-Fock theory for one component plasma [2]. In the 1990s the same equation reemerged as a model of self-gravitating matter [3, 4] and is known in that context as the Schrödinger-Newton equation. In recent years, many papers are concerned with the existence of solutions of (3). Lieb [2] proved the existence and uniqueness of the ground state to (2). Lions [5] obtained the existence of a sequence of radially symmetric solutions for (2) by using variational methods. Papers [6, 7] proved the existence of multibump solutions of (2). Paper [8] proved that every positive solution of (2) is radially symmetric and monotone decreasing about some point by using moving plane methods. Furthermore, the authors obtained the uniqueness of positive solutions for (2). Clapp and Salazar [9] proved the existence of positive and sign changing solutions of (2) when and are replaced by an exterior bounded domain and , respectively. Moroz and Van Schaftingen [10] showed the regularity, positivity, and radial symmetry of the ground states for the optimal range of parameters, and they also obtained decay asymptotic at infinity for these ground states. The more general system (3) was considered in [11]. Moroz and Van Schaftingen [12] obtained the nonexistence and optimal decay of supersolutions of (3). Cingolani and Secchi [13] considered the existences of ground states for the pseudorelativistic Hartree equation. For semiclassical cases, the existence of multiple semiclassical solutions was considered in [14]. Paper [15] considered the existence of semiclassical regime of standing wave solutions of a Schrödinger equation in presence of nonconstant electric and magnetic potentials. Cingolani and Secchi [16] studied the semiclassical limit for the pseudorelativistic Hartree equation. Under the assumptions on the decay of , paper [17] proved the existence of positive solutions by using variational methods and nonlocal penalization technique.

Motivated by the works we mentioned above, in this paper we study the existence of multiple solutions to the nonlinear Choquard equation with perturbation. This kind of problems is often referred to as being nonlocal because of the appearance of the term in the energy functional. This leads to the fact that (1) is no longer a pointwise identity. The main difficulties when dealing with this problem lie in the presence of the nonlocal term and the lack of compactness due to the unboundedness of the domain . Under some conditions on , in the present paper we recover the compactness and find two nontrivial solutions of (1) by using variational methods.

In what follows, we assume that and satisfies the following condition:

) is coercive; that is, .

A solution is called a ground state solution (or positive ground state solution) if its energy is minimal among all the nontrivial solutions (or all the nontrivial positive solutions) of (1). A bound state solution refers to limited-energy solution.

Then, we have the following main results.

Theorem 1. Assume that holds. Then there exists such that, for every with , problem (1) admits two nontrivial solutions. Furthermore, if is positive, problem (1) has one positive ground state solution and one positive bound state solution.

Remark 2. As we will see later, condition guarantees that the embedding is compact (see [18]), where . So, similar to [19], condition can be replaced by the following general condition:

()There exists such that, for any ,

 where and is the Lebesgue measure in .

2. Variational Setting

Throughout the paper, we use the following notations:

(i) is the norm in defined by .(ii) is the norm in defined by for .(iii) Let and be some positive numbers.

The main purpose of this section is to establish the variational setting for problem (1). We first recall the following classical Hardy-Littlewood-Sobolev inequality (see [20, Theorem 4.3]).

Lemma 3. Assume that and . Then one has

where , , , and .

Define the Sobolev space

with the norm

This is a Hilbert space and its inner product is denoted by . It is easy to check that the embedding is continuous. Moreover, under condition , we infer that the embedding is compact (see, e.g., [18]).

By Lemma 3, the functional

is well defined if for defined by

Therefore, since , it follows that . Here we only consider the subcritical case. So we have

The energy functional corresponding to system (1) is defined by

for . From (8)–(10) and , we infer that is well defined in . Furthermore, , and a critical point of is a solution of (1).

In order to prove Theorem 1 we will constrain the functional on the set

Usually, this set is called Nehari manifold. It is well-known that critical points of lie in the Nehari manifold. Denote . Thus, we know that

In order to prove the existence of multiple nontrivial solutions for (1), we will divide the Nehari manifold into the following three parts:

Obviously, only contains the element . Furthermore, it is easy to see that and are both closed subsets of .

Next we will give some explanation for the partition of Nehari manifolds . Set

We define the fibering map

Thus,

Obviously, with if and only if . It is well-known that if the function has unique global maximum point, then the set is homotopic to unit ball of . Moreover, the set is a natural constraint for the functional . This means that if the infimum is attained by , then is a solution of (1). However, in our situation, the global maximum point of is not unique. This leads us to partition the set according to the critical points of . This kind of idea was first introduced by Tarantello in [21]. Later, many mathematicians apply this idea to study other problems; for instance, see [22–24] and the references therein.

If we know that

So if small, we infer that has two positive solutions . Moreover, according to the sign of second derivative of the fibering map at these points, one knows that , , and are local minimum, local maximum, and turning point, respectively. Hence, the sets and contain the local minimum, local maximum, and turning point, respectively. Furthermore, one can check that

Now we are ready to study the properties of sets and .

Lemma 4. Assume that the nonzero function . Then the following results hold.

(i)There exists positive constant such that when . Furthermore, .(ii) is closed.

Proof. (i) In order to prove that , we only need to show that, for , has no critical point that is a turning point. Without loss of generality we can assume that . Define

Then . Let us consider the graph of . Since for , is strictly concave. Moreover, , , and for small. Thus, we know that has unique global maximum point and . So we infer from (16) and (17) that if , the equation has exactly two roots satisfying . If , the equation has exactly one root . Since , it follows that , , and . Hence, if , we know that , . If , we know that . Also, since the sets and are nonempty, we infer that . This also implies that .
Next we still need to find such that for

Since and lies on the unit sphere of , we infer from Lemma 3 that has upper bound. So there exists such that

Furthermore, one deduces from Sobolev inequality and Hölder inequality that

where denotes the best constant satisfying . Combining (22) and (23) we know that if , then (21) holds.
(ii) Since , it suffices to show that , where denotes the closure of . Equivalently, we only need to prove that . For any , we denote . Then . From the proof of part (i) we know that if , the equation has exactly one root such that . Thus, , and so . If , the equation has exactly two roots satisfying , which satisfy and . Hence, and so . Anyway, we infer from that . Since and is bounded from above, it follows from (22) that there exists such that . That is,

This ends the proof of the lemma.

Now we define the minimization problems:

We will prove that if the infimum in (25) is attained by , then is a solution of (1). That is, the following results hold.

Lemma 5. If or is attained by or , then is a nontrivial solution of (1). Moreover, for , where is a small positive number.

Proof. Assume that such that . We define . Thus,

Hence, by using [25, Theorem 4.1.1] we infer that there exists Lagrangian multiplier such that

Moreover, it follows from and that . Thus, is a nontrivial solution of (1). Similarly, if is attained by we can also prove that is a nontrivial solution of (1).
Next we prove the latter part of the lemma. As in the proof of Lemma 4, we can take such that and

The equation has exactly two roots and such that , , and . Since , we know that and . Furthermore, it follows from that . Hence, we infer from that . Finally, we will prove that there exists such that for . Since and has upper bound, it follows from (18) that

Moreover, we infer from (23) and (29) that if

then

where is independent of . Hence, as in the proof of Lemma 4 we know that

3. Proof of Theorem 1

In this section we are going to give the proof of the main results. Before doing this we should study the properties for the minimizing sequences for the functional . In the whole paper, we say means that .

Lemma 6. Under the assumptions of Theorem 1, there exists a sequence such that and as .

Proof. It is easy to check that is coercive and bounded from below on or . So, by applying Ekeland’s variational principle (see [26]) on , one obtains a sequence satisfying

From Lemma 5 we know that . Moreover, . So we can assume that satisfying (33) and (34). Hence, in the following it suffices to prove that as .
In order to prove this result, we first claim that is bounded. Since , it follows that

Furthermore, we infer from (33) that

where is given in (10). So we know that is bounded. Next we claim that , where is a constant. In fact, if not, then would converge to zero. So we have that

For any with , we define

Clearly, and . Moreover, one infers from that

By the implicit function theorem, there exists a function defined on some interval , where , such that and

Differentiating (40) in at , we obtain

We infer from and (37) that

Let . Then we know that for

Moreover, it follows from the boundedness of and that

Hence we infer from (41)–(44) that

Let

Then it follows from (40) that for . Moreover, since and is a closed set, it follows that . Thus, there exists such that for

So we know that for . We deduce from (34) that

From Taylor theorem, we have

where as . Furthermore, it follows from (45) and (46) that

where is independent of . Thus, as . One can deduce from (48)–(50) that for any

Hence, one has that

We finish the proof of the lemma.

Next we will prove the following compactness lemma for the functional .

Lemma 7. Assume that . Then satisfies the -condition. That is, if such that and , as , then has a convergent subsequence.

Proof. Let such that and , as . As in Lemma 6, it is easy to check that is bounded. Without loss of generality we assume that in and in . From the weak continuity of the derivative of , we know that . First, we have

We infer from Hardy-Littlewood-Sobolev inequality (see Lemma 3) that

where . Moreover, one can check that

Hence, it follows from (53)–(55) that

On the other hand, it follows from that

Similar to (54) and (55) one has that

So we infer from (56) and (58) that .

Proof of Theorem 1. We first prove that is attained by some positive . First, from Lemma 6, we know that there exists a sequence such that and as . By Lemma 7 we infer that in . Thus, and . is a nontrivial solution of (1).
Next we will prove that is attained by positive if is positive. As in proof (i) of Lemma 4, we know that there exists a unique such that . We claim that . In fact, since , , and , it follows from and proof (i) of Lemma 4 that , where , and are defined in (26) and (18). Since is the minimum point of for , it follows from that

Hence, we have

By Lemma 5, we know that is nonnegative solution of (1). Furthermore, by maximum principal we infer that is a positive solution of (1).
Finally we will prove that there exists positive such that . Similar to Lemma 6, we can check that there exists a sequence such that

We infer from Lemma 7 that in . Thus, and . is a nontrivial solution of (1). As in proof (i) of Lemma 4, one can check that there exists unique such that . Moreover, we deduce from , , and and that . Thus, one infers from and which is the unique maximum point of that

So by Lemma 5, we know that is nonnegative solution of (1). Furthermore, by maximum principal we infer that is a positive solution of (1).

4. Solutions for the Choquard Equation with General Nonlinearity

In this section we will look for the positive solutions for the following Choquard equation with general nonlinearity:

where . Since we only care about the existence of positive solutions, in what follows, we assume that verifies the following conditions.

()There exist and such that

()There exists such that

() is strictly increasing on .

Then, we have the following main results.

Theorem 8. Assume that and hold. Then there exists small enough such that, for every positive function with , problem (63) admits two positive solutions. Furthermore, we know that one of the two positive solutions is a positive ground state solution and another is positive bound state solution.

Since the idea of the proof of Theorem 8 is the same as Theorem 1, we only state the differences. Set

We define the functional

It is easy to check that , and the critical points of are solutions of (63). Let

Then

where . We claim that has a unique maximum point for . Indeed, since

we infer from condition that for small. Moreover, one deduces from that . So has at least one maximum point. If such that and are maximum points of , then one sees that

Hence we have that

Obviously, one has

Moreover, it follows from that

So we infer from (73) and (74) that

Similar to (73) and (74) one infers from that

This contradicts with (72). So has a unique maximum point. By using this property of , one can establish Lemmas 4 and 5 for sufficiently small.

In the proof of Lemma 6, we need check the following differences: (i) the boundedness of ; (ii) , where . We first check the boundedness of . Similar to (36) one infers from and that

Therefore, is bounded.

On the other hand, we prove that . Similar to (42) one deduces from that

Arguing by contradiction, if as , we infer from that

We conclude from and that

Moreover, since , it follows from that if is sufficiently small,

So substituting (80) and (81) into (79), we obtain a contradiction if is sufficiently small. Hence, the conclusions of Lemma 6 hold. Thus, following the line of proof of Theorem 1, one can prove the results of Theorem 8.

Remark 9. Our arguments in this paper can also apply to the following generalized nonlinear Choquard equation with perturbation:

where the dimension is given, is an external potential, and is the Riesz potential [27] of order defined for every by

. The results of Theorem 1 remain true for (82).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful for the referee’s thoughtful reading of details of the paper and nice suggestions to improve the results. This work was supported by Natural Science Foundation of China (nos. 11201186, 11571140, 11371090, and 71373103), Natural Science Foundation of Jiangsu Province (nos. BK2012282, BK20140525, and BK20150478), Jiangsu University foundation Grant (nos. 11JDG117, 13JDG121, and 14JDG070), Social Science Foundation of China (no. 15CRK010), China Postdoctoral Science Foundation funded project (nos. 2012M511199, 2013T60499, and 2015M571710), and Jiangsu Postdoctoral Science Foundation funded project (1401055C).

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