Infinitely many solutions for nonhomogeneous Choquard equations

In this paper, we study the following nonhomogeneous Choquard equation −∆u + V(x)u = (Iα ∗ |u|p)|u|p−2u + f (x), x ∈ RN , where N ≥ 3, α ∈ (0, N), p ∈ [N+α N , N+α N−2 ) , Iα denotes the Riesz potential and f 6= 0. By using a critical point theorem for non-even functionals, we prove the existence of infinitely many virtual critical points for two classes of potential V. To the best of our knowledge, this result seems to be the first one for nonhomogeneous Choquard equation on the existence of infinity many solutions.


Introduction
In this paper, we are concerned with the following nonhomogeneous nonlocal problem where N ≥ 3, α ∈ (0, N), p ∈ N+α N , N+α N−2 , and Riesz potential I α is given by where Γ denotes the Gamma function.This equation arises in the study of nonlinear Choquard equations describing an electron trapped in its own hole, in a certain approximation to Hartree Fock theory of one component plasma [4].When f = 0, the existence and qualitative properties of solutions for Choquard type equations (1.1) have been studied widely and intensively in literatures.See [1,3,6,9,13] and Corresponding author.Email: huiguo_math@163.comreferences therein for the existence of ground states, nodal solutions and multiple solutions to (1.1).For the results about qualitative properties such as regularity, symmetry, uniqueness and decay, one can refer to [8][9][10][11]14], for instance.
When f = 0, the authors in [16] (or [18]) proved that (1.1) has a ground state and bound state for f small enough via fibering mapping method.However, as we know, there is no result on the existence of infinitely many solutions of (1.1) with f = 0. Motivated by this, the main purpose of this paper is to consider the existence of infinitely many solutions.
For the potential V, we make the following assumptions that either endowed with the inner product (u, v) = R N (∇u∇v + V(x)uv)dx and norm Let H * be the duality space of H with norm • H * , and •, • denotes duality pairing between H and H * .As usual, the corresponding energy functional of (1.1) is E : It is easy to check that E ∈ C 1 (H, R), and the critical points of E are solutions of (1.1) in the weak sense.To state main results clearly, we consider the following equation which is related to (1.1) Due to the difference of the action space H considered, the results and methods for cases (V1) and (V2) may be different.In view of Theorem 1.1, the range of p in part (ii) is smaller than in part (i).Indeed, for the case (V2), the embedding N−2 while for the case (V1), the embedding N in (ii), we can not guarantee the compactness of nonlocal term, that is, we can not deduce that up to a subsequence, as n → ∞ when u n weakly converge to u in H . Furthermore, the compactness of nonlocal term is critical in the proof of Theorem 1.1 (ii).Therefore, p = N+α N in part (ii).In addition, the proof of (i) makes use of the property of eigenvalues in H tending to infinity as in [17].But this method does not work for the case (V2) because H does not have such a property.So we develop a new technique to overcome this problem by delicate asymptotic analysis of nonlocal term.
The remainder of this paper is organized as follows.In Section 2, some notations and preliminary results are presented.In Section 3, we are devoted to the proofs of our main results.

Preliminaries
In this section, some notations and elementary results are collected as follows.
• N is the set of all the positive integers.
• Denote D(u) = R N (I α * |u| p )|u| p dx, and then for any v ∈ H, • C denotes different positive constants line by line.
First let us recall the Hardy-Littlewood-Sobolev inequality.
Here C(N, α, s) is a positive constant which depend only on N, α, s.
As a consequence of Lemma 2.1 and [7, Proposition 4.3], the following lemma holds true.
Next we give the property of the space H which plays a critical role in recovering the compactness.
(i) Under the assumption (V1), the embedding H → H 1 (R N ) is continuous and H is a Hilbert space.Furthermore, the embedding H → L s (R N ) is compact for s ∈ 2, 2N N−2 ) and the spectrum of the self-adjoint operator of −∆ + V in L 2 (R N ) is discrete, i.e. it consists of an increasing sequence {λ n } n≥1 of eigenvalues with finite multiplicity such that λ n → ∞ as n → ∞ and L 2 (R N ) = ∑ n M n , M n ⊥ M n for n = n , where M n is the eigenspace corresponding to λ n .
(ii) Under the assumption (V2), the embedding H → L s (R N ) is compact for s ∈ (2, 2N N−2 ).In the sequel, we list some definitions from the critical point theory.
Definition 2.4 ((P.S.) c condition).A sequence {u n } n≥1 is a Palais-Smale sequence of the functional E at level c ((P. S.) c sequence for short): if E(u n ) → c and E (u n ) → 0. E is said to satisfy the (P.S.) c condition if any (P.S.) c sequence {u n } n≥1 has a convergent subsequence.Definition 2.5 ((sP.S.) c condition, see [2]).The functional E is said to satisfy the symmetrized Palais-Smale condition at level c ((sP.S.) c condition for short): if E satisfies (P.S.) c condition and any sequence {u n } n≥1 is relatively compact in H whenever it satisfies the following conditions lim Denote the set of critical points at level c by

Theorem 2.7 ([2, Proposition 2.1])
. Let E be a C 1 functional satisfying (sP.S.) c condition on a Hilbert space H = X Y with dim(X) < ∞.Assume that E(0) = 0 as well as the following conditions: (i) there is ρ > 0 and α ≥ 0 such that inf E(S ρ (Y)) ≥ α, where S ρ (Y) = {u ∈ Y : u = ρ}; (ii) there exists an increasing sequence {X n } n≥1 of finite dimensional subspace of H, all containing X such that lim n→∞ dim(X n ) = ∞ and for each n, sup E(S R n (X n )) ≤ 0 for some R n > ρ.
Then E has an unbounded sequence of virtual critical values.
Throughout the paper, we are devoted to the proof of our main result by verifying Theorem 2.7.Therefore, Theorem 1.1 can be restated as follows.

Proof of the main results
In this section, we prove Theorem 1.1 (i) in Subsection 3.1 and (ii) in Subsection 3.2.

Case
The functional E satisfies the (sP.S.) c condition.
Proof.We first show that E satisfies (P.S.) c condition.Let {u n } n≥1 be a sequence such that which implies that {u n } n≥1 is bounded.Up to a subsequence, u n u 0 in H and u n → u 0 in L 2N p N+α (R N ).By Lemma 2.2, it follows that for any v ∈ H, E (u n ), v → E (u 0 ), v and hence E (u 0 ) = 0. Note that E (u n ), u n → 0 and E (u 0 ), u 0 = 0.By using Lemma 2.2 again, it follows that u n → u 0 and then u n → u 0 in H. Furthermore, this yields that E(u 0 ) = c, E (u 0 ) = 0 and u 0 = 0 due to f = 0.
Next, we prove that if a sequence {v n } n≥1 ⊂ H satisfies (2.1) and (2.2), then {v n } n≥1 is relatively compact.Indeed, we can conclude from (2.1) that f , v n → 0, and from (2.2) that This means that where E 0 (u Then by the definition of operator norm and (3.2), there exists C 1 > 0 independent of n such that Without loss of generality, we assume that, up to a subsequence, v n v 0 in H and then On the other hand, by Lemma 2.2, we deduce from (3.2) that This, together with (3.3) yields that lim n→∞ The proof is completed.
In view of Lemma 2.3, let {e k } k≥1 ⊂ H be an orthonormal basis of eigenvectors of the operator −∆ + V. Let X = span{e 1 , e 2 , • • • , e k 0 } and Y be the orthogonal complement of X.Clearly, dim X = k 0 , where dim X denotes the dimension of the space X.Lemma 3.2.For k 0 = dim X large enough, there exist θ > 0 and ρ > 0 such that E(u) ≥ θ for all u ∈ Y with u = ρ.
Proof.By Lemma 2.3, it follows that for any u ∈ Y, where s ∈ (0, 1) and 2s Then by (3.4) and Sobolev inequality, we have Note that there exists ρ > 0 such that θ : Thus, for any u ∈ Y with u = ρ, we conclude from (3.6) that E(u) ≥ θ.
(ii) When p = N+α N , it follows that 2N p N+α = 2.By Lemmas 2.1 and 2.3, Then by similar arguments as those in (i), there also exist ρ > 0 and θ > 0 such that E(u) ≥ θ for all u ∈ Y with u = ρ.
To sum up, the proof is completed.
Lemma 3.3.Let ρ be defined in Lemma 3.2, and {X n } n≥1 ⊂ H containing X be an increasing sequence of finite dimensional subspace with lim n→∞ dim X n = ∞.Then for each n, there exists R n > ρ such that sup Proof.For each n, define S n = {u ∈ X n : u = 1} and d n = min u∈S n D(u).Since X n is a finite dimensional subspace, the set S n is compact and by Lemma 2.2, d n can be achieved and d n > 0. Then for any R > 0 and u ∈ X n with u = R, it holds that Therefore, there is R n > ρ large enough such that E(u) ≤ 0 for u = R n .The lemma follows.
Proof of Theorem

Case V = const
Let V(x) ≡ V 0 > 0 be a constant function.In order to prove Theorem 2.8, we first show the following lemma.Denote by {v i } i the orthogonal basis of H. Proof.To this end, we denote γ k = sup u∈Y k , u =1 D(u).Then (ii) For each k ≥ 1, there exists u k ∈ Y k with u k = 1 such that γ k = D(u k ).
Clearly, since Y k+1 ⊂ Y k for k ≥ 1, (i) is trivial.In addition, (ii) follows by using minimizing method.In fact, for each k ≥ 1, there exists a sequence {u k j } j≥1 ∈ Y k such that u k j = 1 and D(u k j ) → γ k as j → ∞.Up to a subsequence, u k In view of (ii), we can choose a subsequence {u n k } of {u k } k≥1 with u n k = 1 such that γ n k = D(u n k ) and X n k {u n k } ⊂ X n k+1 .

Definition 2 . 6 (
[2]).Denote the set of Z 2 -resonant points at level c byK f c = {u ∈ H : E(u) = E(−u) = c, E (u) = λE (−u), λ > 0},and the set of virtual critical points at level c by I c = K f c K c .The corresponding value c is called virtual critical values.

Theorem 2 . 8 .
Under the same assumptions of Theorem 1.1, problem (1.1) has an unbounded sequence of virtual critical values.

Lemma 3 . 4 .
Let X k = span{v 1 , v 2 , • • • , v k } ⊂ H with dim X k < ∞ and Y k be the orthogonal complement of X k in H. Then lim