Existence of Multiple Positive Solutions for Choquard Equation with Perturbation
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) becomesIt 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  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 . 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  proved the existence and uniqueness of the ground state to (2). Lions  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  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  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  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 . Moroz and Van Schaftingen  obtained the nonexistence and optimal decay of supersolutions of (3). Cingolani and Secchi  considered the existences of ground states for the pseudorelativistic Hartree equation. For semiclassical cases, the existence of multiple semiclassical solutions was considered in . Paper  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  studied the semiclassical limit for the pseudorelativistic Hartree equation. Under the assumptions on the decay of , paper  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 ), where . So, similar to , 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 spacewith the normThis 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., ).
By Lemma 3, the functionalis well defined if for defined byTherefore, since , it follows that . Here we only consider the subcritical case. So we haveThe energy functional corresponding to system (1) is defined byfor . 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 setUsually, this set is called Nehari manifold. It is well-known that critical points of lie in the Nehari manifold. Denote . Thus, we know thatIn 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 . SetWe define the fibering mapThus,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 . Later, many mathematicians apply this idea to study other problems; for instance, see [22–24] and the references therein.
If we know thatSo 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 thatNow 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 . DefineThen . 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 thatFurthermore, one deduces from Sobolev inequality and Hölder inequality thatwhere 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 thatMoreover, 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 andThe 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) thatMoreover, we infer from (23) and (29) that ifthenwhere 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 ) on , one obtains a sequence satisfyingFrom 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 thatFurthermore, we infer from (33) thatwhere 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 thatFor any with , we defineClearly, and . Moreover, one infers from thatBy the implicit function theorem, there exists a function defined on some interval , where , such that andDifferentiating (40) in at , we obtainWe infer from and (37) thatLet . Then we know that for Moreover, it follows from the boundedness of and thatHence we infer from (41)–(44) thatLetThen 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) thatFrom Taylor theorem, we havewhere as . Furthermore, it follows from (45) and (46) thatwhere is independent of . Thus, as . One can deduce from (48)–(50) that for any Hence, one has thatWe 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 haveWe infer from Hardy-Littlewood-Sobolev inequality (see Lemma 3) thatwhere . Moreover, one can check thatHence, it follows from (53)–(55) thatOn the other hand, it follows from thatSimilar to (54) and (55) one has thatSo 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 thatHence, we haveBy 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 thatWe 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 thatSo 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. SetWe define the functionalIt is easy to check that , and the critical points of are solutions of (63). LetThenwhere . We claim that