Existence of Multispike Positive Solutions for a Nonlocal Problem in R 3

In this paper, we study the following nonlinear Choquard equation −ε2Δu + KðxÞu = ðð1Þ/ð8πε2ÞÞðÐR3ððu2ðyÞÞ/ðjx − yjÞÞdyÞu, x ∈R3, where ε > 0 and KðxÞ is a positive bounded continuous potential on R3. By applying the reduction method, we proved that for any positive integer k, the above equation has a positive solution with k spikes near the local maximum point of KðxÞ if ε > 0 is sufficiently small under some suitable conditions on KðxÞ.


Introduction and Main Results
In this paper, we consider the following nonlinear Choquard equations where ϵ > 0 and KðxÞ 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 selfgravitating 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 φ 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 ψðx, tÞ = e −iht/ϵ uðxÞ, where i is the imaginary unit, h ∈ ℝ and ϵ is the Planck constant. The above Hartree equations also appear in quantum mechanics models (see [4][5][6]) and in the semiconductor theory (see [7][8][9]).
Also, the Choquard equation (2) is a special type of the following generalized Choquard equation where α ∈ ð0, nÞ and p > 1. 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 p when ϵ = 1. 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 n = 3, p = 2, and α = 2 in (4), we get (2). In [14], Lions derived the existence of ground state solutions of (2) under some suitable conditions on KðxÞ if ϵ > 0 is small enough. For any positive integer k > 0, Wei and Winter [15] proved that there exist a solution of (2) concentrating at k points which are all local minimums or local maximums or non-degenerate critical points of KðxÞ under the conditions that inf ℝ 3 K > 0, K ∈ C 2 ðℝ 3 Þ provided ϵ is sufficiently small. Recently, Luo, Peng and Wang [16] showed the uniqueness of positive solutions for (2) concentrating at the nondegenerate critical points of KðxÞ by using a local Pohozaev type identity for ϵ > 0 small enough.
But, when ϵ = 1 and KðxÞ = 1, (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 KðxÞ as follows: (K 1 ) K has a strict local maximum at some point for all x, y ∈ B 2δ ðy 0 Þ. We state our main result as follows: Theorem 1. Assume that ðK 1 Þ, ðK 2 Þ hold, then for any positive integer k, problem (2) has a k-spike solution for sufficiently small ϵ > 0.
As in [21][22][23], we mainly use the finite-dimensional reduction to prove our result. Here, our purpose is to verify that if ϵ is small enough, then for any positive integer k, (2) has a solution with k-spikes concentrating near y 0 corresponding to any strict local maximum y 0 of KðxÞ, namely, a solution with k maximum points converging to y 0 as ϵ ⟶ 0.
In the end of this part, let us outline the sketch of our proof of Theorem 1. Denoted by wðxÞ, the unique radially positive solution of the following problem It follows from [2,15] that wðxÞ is strictly decreasing and satisfies for some constant c 0 > 0. Also, wðxÞ is nondegenerate, namely, if ψðxÞ ∈ H 1 ðℝ 3 Þ solves the lineared equation then ψðxÞ is a linear combination of ð∂w/∂x j Þ, j = 1,2,3.
We will use the unique solution w of (7) to establish the solutions of (2). In what follows, without loss of generality, we assume that y 0 = 0 and Kð0Þ = 1. Let B r ð0Þ = fx ∈ ℝ 3 : jxj < rg and denote its closure by B r ð0Þ. For any positive integer k and large R, we define Furthermore, since inf ℝ 3 K > 0, we can define the following Soblev space with the corresponding norm kuk 2 ϵ = hu, ui ϵ , where 2 Advances in Mathematical Physics and, in this sequel, we denote by j⋅j p the usual norm of L p ðℝ 3 Þ and let kuk D ≔ ð Ð ℝ 3 j∇uj 2 dxÞ 1/2 , kuk H 1 ≔ ð Ð ℝ 3 ðj∇uj 2 + u 2 Þ dxÞ 1/2 be the norms of D 1,2 ðℝ 3 Þ and H 1 ðℝ 3 Þ, respectively. Now fixing y ∈ D ϵ,δ k , set for j = 1, ⋯, k, s = 1,2,3. We will prove Theorem 1 by verifying the following result.

Theorem 2.
Let ðK 1 Þ, ðK 2 Þ hold. Then, for any positive integer k, there is ϵ 0 = ϵðkÞ such that for ϵ ∈ ð0, ϵ 0 Þ, problem (2) has a solution u ϵ of the form for some points y j ϵ ∈ ℝ 3 , j = 1, ⋯, k and ϕ ϵ ∈ H 1 ðℝ 3 Þ satisfying as ϵ ⟶ 0, We want to point out that compared with [15], we introduce a little stronger conditions imposed on KðxÞ than that of [15] and the reduction procedure has been modified here to allow for the degenerate of the critical point of KðxÞ. 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.

The Finite-Dimensional Reduction
Observe that the variational functional corresponding to (2) is Then by the direct computation, we have for any ϕ ∈ E ϵ,k , where we use the fact that w ϵ,y j ðj = 1,⋯,kÞ solves In order to find a critical point for J ϵ ðy, ϕÞ, we need to discuss each terms in the expansion (17). First, we have

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Proof. Taking u ϵ ðxÞ = uðϵxÞ, then which implies the conclusion holds.

Lemma 4.
There exists a positive constant C independent of ϵ such that where R i ϵ,y ðϕÞ denotes the ith derivative of R ϵ,y ðϕÞ.
Proof. Note that So it is easy to check that First, we estimate R ϵ,y ðϕÞ. Notice that if we denote which implies that As a result, from Lemma 3, we have Combining the definition of R ϵ,y ðϕÞ and (27), (28), we find Now, we discuss R ϵ,y ′ ðϕÞ. Similar to (27) and (28), we get So, by the estimates above, we infer that and then

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Finally, by the same argument as above, we find The estimates above imply that This completes our proof.

Lemma 5. There holds
where i, j = 1, ⋯, k, i ≠ j and C is a constant independent of ϵ.
Proof. Recall that First, we can write and we have Then, by the assumption ðK 2 Þ, Lemma 3 and the decay property of w, we find where we used the fact that On the other hand, by Hölder inequality, we have which, together with (41), implies that

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By the same argument as above, we also deduce that Hence, Now, in order to estimate ℓ 2 , we recall the Hardy-Littlewood-Sobolev inequality (see [2]): if 1 < p, q < ∞, 0 < t < 3 and ð1/pÞ + ð1/qÞ + ðt/3Þ = 2, f ∈ L p ðℝ 3 Þ, g ∈ L q ðℝ 3 Þ, 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 L ϵ,y ðϕÞ, we define L ϵ,y to be a bounded linear map from E ϵ,k to E ϵ,k as Here, we come to show the invertibility of L ϵ,y in E ϵ,k .
Proof. We argue by contradiction. Suppose that there exists ϵ n ⟶ 0, y n = ðy n,1 ,⋯,y n,k Þ ∈ D ϵ n ,δ k , and ϕ n ∈ E ϵ n ,k such that Without loss of generality, we can assume that kϕ n k 2 ϵ n = ϵ 3 n . Fix i ∈ f1,⋯,kg and let ϕ n,i = ϕ n ϵ n x + y n,i À Á : Advances in Mathematical Physics So, which implies that ϕ n,i is bounded in H 1 ðℝ 3 Þ. Thus, up to a subsequence, there exists ϕ ∈ H 1 ðℝ 3 Þ such that as n ⟶ +∞, Next we will prove ϕ = 0. To this end, from (50), we find that ϕ n,i satisfies for any where for j = 1, ⋯, k and s = 1,2,3:. But, for g ∈ C ∞ 0 ðℝ 3 Þ, we can decompose g as follows where g n ∈ E n and a n,j,s ∈ ℝ: Then, by the exponential decay of ð∂ w ϵ n ,y n,j /∂y K ϵ n x + y n,i À Á∂ w ϵ n ,y n,j ∂y n,i s g n dx = o n 1 ð Þ, for h ≠ j and h, j = 1, ⋯, k. On the other hand, So, up to a subsequence, we can easily check that a n,j,s ⟶ 0 as n ⟶ ∞ for j ≠ i, while a n,i,s ⟶ a i,s for some a i,s ∈ ℝ. Inserting g n ðx − y n,i /ϵ n Þ into (54) and letting n ⟶ + ∞, we infer that Since w solves We find that which implies that Considering that g ∈ C ∞ 0 ðℝ 3 Þ is arbitrary and w is nondegenerate, there exist a s , s = 1,2,3 such that Moreover, being ϕ n ∈ E ϵ n ,k , we have which, together with (64), yields ϕ = 0. Finally, by Lemma A.1, we deduce that where o R ð1Þ ⟶ 0 as R ⟶ +∞.
Similarly, the Hardy-Littlewood-Sobolev inequality (45) implies that W ϵ n ,y n 6/5 ϕ n j j 6/5 dx As a result, by (50), which is impossible. So we complete this proof.

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Thus, finding a critical point for J ϵ ðy, ϕÞ is equivalent to solving Since L ϵ,y is invertible in E ϵ,k from Proposition 6, (71) can be rewritten as Define We shall verify that A is a contraction mapping from S ϵ to itself. First, for ∀ϕ ∈ S ϵ , by Lemmas 4 and 5, we have which tells that A maps S ϵ to S ϵ . On the other hand, for any ϕ 1 , ϕ 2 ∈ S ϵ , using Lemma 4, where ν ∈ ð0, 1Þ. Therefore, A is a contraction map from S ϵ to S ϵ , and then, applying the contraction mapping theorem, we can find a unique ϕ ϵ satisfying (71). So the conclusion follows.