Abstract
In this paper, we study the following nonlinear Choquard equation , where and is a positive bounded continuous potential on . By applying the reduction method, we proved that for any positive integer , the above equation has a positive solution with spikes near the local maximum point of if is sufficiently small under some suitable conditions on .
1. Introduction and Main Results
In this paper, we consider the following nonlinear Choquard equations where and is a positive bounded continuous potential. The Choquard equation first appeared as early as in 1954, in a work by Pekar describing the quantum mechanics of a polaron at rest [1]. In 1976, Choquard used it to describe an electron trapped in its own hole in a certain approximation to the Hartree-Fock theory of one component plasma in [2]. Penrose [3] also proposed it as a model of self-gravitating matter, in a programme in which quantum state reduction is understood as a gravitational phenomenon. Moreover, the Choquard equation is also known as the Schrödinger-Newton equation in models coupling the Schrödinger equation of quantum physics together with nonrelativistic Newtonian gravity.
Note that the second equation of (1) can be explicitly solved with respect to and then (1) reduces to the following single nonlocal equation
Equation (2) has attracted considerable attention in recent period and part of the motivation is due to looking for standing waves for the following nonlinear Hartree equations with the form , where is the imaginary unit, and is the Planck constant. The above Hartree equations also appear in quantum mechanics models (see [4–6]) and in the semiconductor theory (see [7–9]).
Also, the Choquard equation (2) is a special type of the following generalized Choquard equation where and . The symmetry and the regularity of solutions of (4) have been established by Ma and Zhao [10] and by Cingolani et al. [11], respectively, under the suitable assumptions on when . Later, in [12] Moroz and Van Schaftingen derived the regularity, positivity, radial symmetry, and sharp asymptotics of ground state solutions of (4) for the optimal range of parameters (see also [13]).
In particular, taking , , and in (4), we get (2). In [14], Lions derived the existence of ground state solutions of (2) under some suitable conditions on if is small enough. For any positive integer , Wei and Winter [15] proved that there exist a solution of (2) concentrating at points which are all local minimums or local maximums or non-degenerate critical points of under the conditions that provided is sufficiently small. Recently, Luo, Peng and Wang [16] showed the uniqueness of positive solutions for (2) concentrating at the non-degenerate critical points of by using a local Pohozaev type identity for small enough.
But, when and , (2) is written as
In [17], Lieb obtained the existence and uniqueness of ground state solutions of (5) by using variational method (see also [18, 19]). Later, Tod and Moroz and Tod [20] and Wei and Winter [15] proved the nondegeneracy of the ground state solutions of (5).
Applying the existence and the nondegeneracy of ground state solutions for (5), inspired by [21, 22], we want to exploit the finite dimensional reduction method to investigate the existence of positive multi-spikes solutions for (2) under the conditions imposed on as follows:
(K1) has a strict local maximum at some point , that is, there is such that for all .
(K2) and there exist constants with such that for all .
We state our main result as follows:
Theorem 1. Assume that , hold, then for any positive integer , problem (2) has a -spike solution for sufficiently small .
As in [21–23], we mainly use the finite-dimensional reduction to prove our result. Here, our purpose is to verify that if is small enough, then for any positive integer , (2) has a solution with -spikes concentrating near corresponding to any strict local maximum of , namely, a solution with maximum points converging to as .
In the end of this part, let us outline the sketch of our proof of Theorem 1. Denoted by , the unique radially positive solution of the following problem
It follows from [2, 15] that is strictly decreasing and satisfies for some constant . Also, is nondegenerate, namely, if solves the lineared equation then is a linear combination of .
We will use the unique solution of (7) to establish the solutions of (2). In what follows, without loss of generality, we assume that and . Let and denote its closure by . For any positive integer and large , we define
Furthermore, since , we can define the following Soblev space with the corresponding norm , where and, in this sequel, we denote by the usual norm of and let , be the norms of and , respectively.
Now fixing , set for . We will prove Theorem 1 by verifying the following result.
Theorem 2. Let , hold. Then, for any positive integer , there is such that for , problem (2) has a solution of the form for some points and satisfying as ,
We want to point out that compared with [15], we introduce a little stronger conditions imposed on than that of [15] and the reduction procedure has been modified here to allow for the degenerate of the critical point of . Also, the appearance of nonlocal term forces us to face much difficulties in the reduction process which involves some more delicate estimates.
The rest of the paper is organized as follows. In Section 2, we will carry out a reduction procedure. We prove our main result in Section 3. Finally, in Appendix, some technical estimates and an energy expansion for the functional corresponding to problem (2) will be established.
2. The Finite-Dimensional Reduction
Observe that the variational functional corresponding to (2) is
Then by the direct computation, we have for any , where we use the fact that solves
In order to find a critical point for , we need to discuss each terms in the expansion (17). First, we have
Lemma 3. For any , there holds
Proof. Taking , then which implies the conclusion holds.
Lemma 4. There exists a positive constant independent of such that where denotes the th derivative of .
Proof. Note that So it is easy to check that First, we estimate . Notice that if we denote then satisfies in . So, which implies that As a result, from Lemma 3, we have Combining the definition of and (27), (28), we find Now, we discuss . Similar to (27) and (28), we get So, by the estimates above, we infer that and then Finally, by the same argument as above, we find The estimates above imply that This completes our proof.
Lemma 5. There holds where and is a constant independent of .
Proof. Recall that First, we can write and we have Then, by the assumption , Lemma 3 and the decay property of , we find where we used the fact that On the other hand, by Hölder inequality, we have which, together with (41), implies that By the same argument as above, we also deduce that Hence, Now, in order to estimate , we recall the Hardy-Littlewood-Sobolev inequality (see [2]): if and , , then Thus, by Hölder inequality and (45), one has Finally, using Lemma A.1 and Hölder inequality, we infer that which, together with (44) and (46), concludes this proof.
Now, associated to the quadratic form , we define to be a bounded linear map from to as
Here, we come to show the invertibility of in .
Proposition 6. There exist such that for ,
Proof. We argue by contradiction. Suppose that there exists , , and such that
Without loss of generality, we can assume that . Fix and let
So,
which implies that is bounded in . Thus, up to a subsequence, there exists such that as ,
Next we will prove . To this end, from (50), we find that satisfies for any ,
where
for and .
But, for , we can decompose as follows
where and Then, by the exponential decay of , we have
for and . On the other hand,
So, up to a subsequence, we can easily check that as for , while for some . Inserting into (54) and letting , we infer that
Since solves
We find that
which implies that
Combining (59) and (62), we have
Considering that is arbitrary and is nondegenerate, there exist such that
Moreover, being , we have
which, together with (64), yields . Finally, by Lemma A.1, we deduce that
where as .
Similarly, the Hardy-Littlewood-Sobolev inequality (45) implies that
As a result, by (50),
which is impossible. So we complete this proof.
Proposition 7. Suppose that and hold. Then, for any given , there exist such that for , there is a map from to , satisfying
Proof. We use the contraction mapping theorem to prove the wanted result. It follows from Lemma 5 that is a bounded linear map in . So applying Reisz representation theorem, there exists an such that Thus, finding a critical point for is equivalent to solving Since is invertible in from Proposition 6, (71) can be rewritten as Define We shall verify that is a contraction mapping from to itself. First, for , by Lemmas 4 and 5, we have which tells that maps to . On the other hand, for any , using Lemma 4, where . Therefore, is a contraction map from to , and then, applying the contraction mapping theorem, we can find a unique satisfying (71). So the conclusion follows.
3. Proof of the Main Results
In this section, we come to prove our main results. Let and be as in Proposition 7. Define and let satisfies
Next, we can show that is an interior point of and thus a critical point of for small .
Lemma 8. Suppose that satisfies (77). Then, as , , and .
Proof. It follows from Lemma A.2 and Proposition 7 that Take for some and some vectors with . Then as , which means that if is small enough. Combining (77) and (78), we have where . Hence, which yields that So, as and for , we find from which, the conclusion follows.
Proof of Theorem 2. By Lemma 8, we can check that (77) can be obtained by some , which is an interior point of for small and satisfies for . Moreover, from Proposition 7, as . Finally, it is well-known that if is a critical point of , then is a solution of (2)Thus, we finish this proof.
Appendix
Energy Expansion
In this section, we give some basic estimates and the energy expansion for the approximate solutions.
Lemma A.1. There exists a positive constant independent of such that
Proof. The proof of this Lemma can be obtained arguing as Lemma B.1 of [24] exactly. We omit the details here.
Lemma A.2. There exists a positive constant independent of such that
Proof. Recall that We have Now, we discuss each terms in the right hand of (A.4). First, by the direct computation, one has In view of (A.5), we have and similarly, Thus, from the estimates above and (A.5), we find Now, we estimate . We have By using the Hardy-Littlewood-Sobolev inequality (45), we have Moreover, it follows from Lemma A.1 that Thus, combining (A.4)-(A.9), we deduce that
Data Availability
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work was partially supported by NSFC (No. 11601194) and the Natural Science Foundation of Jiangsu Province of China (BK20180976).