In fi nitely many non-radial solutions for a Choquard equation

is radial and unique up to translations. While Lions [2] showed the existence of a sequence of radially symmetric solutions via variational methods. In [3,4], the authors proved, if u is a ground state of equation (1.2), then u is either positive or negative and there exist ∈ x0 3 and a monotone function ( ) ∈ ∞ ∞ v C 0, such that for every ∈ x 3, ( ) (∣ ∣) = − u x v x x0 . Without loss of generality, we can suppose > U 0 and = x 0 0 , that is, ( ) (∣ ∣) = U x U x . For the non-degeneracy of the ground states, we may see [5–8]. Chen [9] proved that the ground state solution U is non-degenerate, i.e., the kernel of the linearized equation


Introduction and main results
In the past two decades, many authors have devoted to the study of existence, multiplicity, and properties of the solutions of the non-linear Choquard equation (1.1), In a early paper [1], Lieb proved that the ground state U of the equation (1.2) is radial and unique up to translations. While Lions [2] showed the existence of a sequence of radially symmetric solutions via variational methods. In [3,4], the authors proved, if u is a ground state of equation (1.2), then u is either positive or negative and there exist ∈ x 0 3 and a monotone function Without loss of generality, we can suppose > U 0 and = x 0 0 , that is, ( ) (| |) = U x U x . For the non-degeneracy of the ground states, we may see [5][6][7][8]. Chen [9] proved that the ground state solution U is non-degenerate, i.e., the kernel of the linearized equation . Moreover, where τ is an arbitrary number in (0, 1), | | = r x, and C is a positive constant depending on τ. By this fact, we also have for some > C β , 0. For more background and recent literature of the non-linear Choquard equation, we may turn to [1][2][3][4]7,8,[10][11][12][13][14] and references therein.
The aim of the present article is to consider the following non-linear Choquard equation: where potential ( ) V x satisfies the following assumptions: (Without loss of generality, we may assume that = V 1 0 ). To apply variational methods, we introduce the energy functional associated with equation (1.4) by The Hardy-Littlewood-Sobolev inequality implies that J is well defined on ( ) H 1 3 and belongs to 1 . And so u is a weak solution of (1.4) if and only if u is a critical point of the functional J .
The main result of this article is to establish the existence of infinitely many non-radial solution for (1.4) under assumption ( ) V . The result says that To prove the main results, we will adopt the idea introduced by Wei and Yan in [15] to use the unique ground state U of equation (1.2) to build up the approximate solutions for (1.4) with large number of bumps near the infinity. As in [15], let where m is the constant in the expansion for V , > δ 0 is a small constant, and β is given in (1.3). We denote This article is organized as follows. In Section 2, we prove two basic estimates. In Section 3, we carry out the reduction. Then, we study the reduced finite dimensional problem and prove Theorem 1.2 in Section 4.

Preliminaries
Throughout this article we write | | ⋅ q for the ( ) , always assume that condition ( ) V holds, and the norm of ( ) H 1 3 is defined as follows: Then, we have the following basic estimates: Proof. In view of the symmetry, we only estimate the function Thus for any > α 0, using (2.3), we know Therefore, Consequently, (2.2) follows. □

The reduction argument
Applying Lemma 2.2, there exists a bounded linear operator L from E to E such that Thus, we have Lemma 3.1. There is a constant ′ > C 0, independent of k, such that for any ∈ r S k , Next, we show that L is invertible in E.
There is a constant ″ > C 0, independent of k, such that for any ∈ r S k , Proof. Suppose to the contrary that there are → +∞ k , ∈ r S k k , and ∈ v E k , with We may assume that ‖ ‖ = v k k 2 . By symmetry, we see from (3.1), In particular, , we see that ( ) ⊂ B z Ω R 1 1 . As a result, from (3.3), we find that for any > R 0, So, we may assume that there is a ( ) ∈ v H 1 3 , such that as → +∞ k , Since v k is even in x h , = h 2, 3, it is easy to see that v is even in x h , = h 2, 3. On the other hand, from U y z k k U y z Now, we claim that v satisfies x y x y U x u x U y y x y x y : : be any function, satisfying that ψ is even in . With the argument in [15] we find Similarly, Thus, we have

(3.6)
On the other hand, since v is even in x h , = h 2, 3, (3.6) holds for any function ( ) ∈ ∞ ψ C 0 3 , which is odd in x h , = h 2, 3. Therefore, (3.6) holds for any . By the density of ( ) is true for any ( ) ∈ ψ H 1 3 . So, we have proved (3.5). Since U is non- As a result, On the other hand, it follows from Lemma 2.1 that for any small > η 0, there is a constant > C 0, such that Expand ( ) I φ as follows: In order to find a critical point ∈ φ E for ( ) I φ , we need to estimate each term in the expansion.
There is a constant > C 0, independent of k, such that for any ( ) ∈ φ H 1 3 , Proof. Similar to the proof of (3.1), we have that for any Proof. By the symmetry of the problem, Proposition 3.5. There is an integer Moreover, there is a small > σ 0, such that Proof. Since ( ) l φ is a bounded linear functional in E, we know that there is an ∈ l E k , such that k Thus, finding a critical point for ( ) I φ is equivalent to solving By Lemma 3.2, L is invertible. Thus, (3.14) can be rewritten as  Thus, A maps S into S.
By (3.11), we have, which is an interior point of S k , it is easy to check that (4.1) is achieved by some r k , which is in the interior of S k . Thus, r k is a critical point of ( ) F r . As a result, is a solution of (1.4). □ Funding information: Fashun Gao was partially supported by NSFC (11901155). Minbo Yang is the corresponding author who was partially supported by NSFC (11971436, 12011530199) and ZJNSF (LZ22A010001, LD19A010001).

Conflict of interest:
Authors state no conflict of interest.