On a class of mixed Choquard-Schr\"odinger-Poisson system

We study the system $$ \left\{ -\Delta u+u+K(x) \phi |u|^{q-2}u&=(I_\alpha*|u|^p)|u|^{p-2}u&&\mbox{ in }{\mathbb R}^N, -\Delta \phi&=K(x)|u|^q&&\mbox{ in }{\mathbb R}^N, \right. $$ where $N\geq 3$, $\alpha\in (0,N)$, $p,q>1$ and $K\geq 0$. Using a Pohozaev type identity we first derive conditions in terms of $p,q,N,\alpha$ and $K$ for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach.


Introduction
In this paper we are concerned with the following system −∆u + u + K(x)φ|u| q−2 u = (I α * |u| p )|u| p−2 u in R N , where p, q > 1 are real numbers and K ≥ 0 satisfies some more properties as we shall state below. Here I α : R N → R is the Riesz potential of order α ∈ (0, N ), N ≥ 3, given by (1.2) When K ≡ 0, system (1.1) reduces to the single equation which bears the name Choquard or Choquard-Pekar equation. For N = 3, p = α = 2, equation (1.3) was introduced in 1954 by S.I. Pekar [28] as a model in quantum theory of a Polaron at rest (see also [12]). In 1976, P. Choquard used (1.3) in a certain approximation to Hartree-Fock theory of one component plasma (see [16]). In 1996, equation (1.3) appears in a different context, being employed by R. Penrose [29] as a model of self-gravitating matter (see, e.g., [14,22]) and it is known in this context as the Schrödinger-Newton equation.
The Choquard equation (1.3) has been investigated for a few decades by variational methods starting with the pioneering works of E.H. Lieb [16] and P.-L. Lions [17,18]. More recently, new and improved techniques have been devised to deal with various forms of (1.3) (see, e.g., [1,23,25,26,27,31] and the references therein). In [23] existence, regularity, positivity, asymptotic behavior and radial symmetry of solutions to (1.1) is discussed for optimal range of parameters. We also mention here the works [10,11] where the fractional version of (1.3) is considered. For a nonvariational approach to Choquard equation the reader may consult [13,19,24].
Back to (1.1), we should point out that since for all ϕ ∈ C ∞ 0 (R N ), I α * ϕ → ϕ as α → 0, the system −∆u may be seen as a formal limit of (1.1) when α → 0. The nonlocal nonlinear Schrödinger equation is used as an approximation to Hartree-Fock model of a quantum many-body system of electrons under the presence of an external potential V ext (see [15]). In such a setting, (1.4) and its stationary counterpart bear the name of Schrödinger-Poisson-Slater [5], Schrödinger-Poisson-X α [3,20], or Maxwell-Schrödinger-Poisson [2,7] equations. The convolution term in (1.4) represents the Coulombic repulsion between the electrons. The local term |u| 2p−2 u was introduced by Slater [30] as a local approximation of the exchange potential in the Hartree-Fock model [5,20].
Notations. Throughout in this paper we use the following notations.
• H 1 (R N ) denotes the standard Sobolev space endowed with the usual norm We shall denote by ·, · the duality pairing between H 1 (R N ) and its dual H −1 (R N ).
• D 1,2 (R N ) is the Hilbert space and the associated scalar product • L s (R N ) is the usual Lebesgue space in R N of order s ∈ [1, ∞] whose norm will be denoted by · s .

Main Results
Our first result provides sufficient conditions for the nonexistence of solutions to (1.1).
then, the only solution (u, φ) of (1.1) that satisfies and  Let us now discuss the existence of a solution to (1.1). Crucial to our approach will be the Hardy-Littlewood-Sobolev inequality It is more convenient to reduce our system (1.1) to a single equation. More exactly, for any u ∈ H 1 (R N ) define If K ∈ L r (R N ), with 1 r + q + 1 2 * = 1 and 1 < q < then, by Hölder and Sobolev inequality one gets that T u is linear and continuous. By Lax-Milgram theorem, there exists a unique φ u ∈ D 1,2 (R N ) such that Hence φ u = I 2 * (K|u| q ). (2.9) More properties of φ u are given in Lemma 3.1 below. We should finally note that with φ u given by (2.9), system (1.1) reduces implicitly to the single equation Let us remark that (2.10) has a variational structure. If N +α N < p < N +α N −2 and q, r satisfy (2.7) then functional is well defined for all u ∈ H 1 (R N ) and any critical point u of J is a weak solution to (2.10). Our existence result is the following.
Moreover, u is a ground state of (2.10).
In order to deal with the lack of compactness of H 1 (R N ) into the Lebesgue spaces L s (R N ), 2 ≤ s ≤ 2 * , we rely on a careful analysis of the Palais-Smale (in short (P S)) sequences for J restricted to its Nehari manifold N . Roughly speaking, we have that any (P S) sequence of J | N either converges strongly to its weak limit or differs from it by a finite number of sequences, which are nothing but translated solutions of (1.3), centered at points whose distances from the origin and whose interdistances go to infinity (see Proposition 5.2). Then, a further evaluation of the energy levels of J allows us to locate some ranges for which the compactness is still preserved. Such an approach was successfully applied for the Schrödinger-Poisson system (1.4) in [8,9] and recently adapted to the study of the non-autonomous fractional Choquard equation in [10]. Unlike the approach in [10] where a direct energy estimation is possible due to the presence of suitable non-autonomous terms, we shall rely essentially on several nonlocal Brezis-Lieb type results as we describe in Section 3.2.
The remaining part of the paper is organised as follows. Section 3 contains some preliminary results which we will use in the study of the existence of a ground state to (1.1). Sections 4 and 5 contain the proofs of our main results.

Preliminary results
Proof. (i) and (ii) follow from the definition of φ u .
(iii) For a proof of this part in dimension N = 3 the reader may consult [8, Proposition 2.2(a)]. Here we provide a different argument.
Let us note first that from the definition of φ u in (2.8) we deduce Using the continuous embedding of

Some nonlocal versions of Brezis-Lieb lemma
In this part we collect some useful results in dealing with the existence of a ground state solution to (1.3). We first recall the concentration-compactness lemma of P.-L. Lions formulated in an inequality setting.
Using a similar proof to that in the original Brezis-Lieb lemma [6, Theorem 2] (see also [32,Proposition 4.7.30]) we have A first nonlocal version of Bezis-Lieb lemma in the literature appeared in [23] (see also [21]) and reads as follows.
N+α (R N ) that converges almost everywhere to some u : R N → R. Then Below we state and prove another nonlocal version of Brezis-Lieb lemma.
Proof. Using h = h + − h − , it is enough to prove our lemma for h ≥ 0. Denote v n = u n − u and observe that Apply Lemma 3.4 with q = p, s = 2N p N +α by taking respectively (w n , w) = (u n , u) and then (w n , w) = (u n h 1/p , uh 1/p ). We find Using now the Hardy-Littlewood-Sobolev inequality (2.5) we obtain Also, by Lemma 3.3 we have By Hölder's inequality and Hardy-Littlewood-Sobolev inequality (2.6) with

A Pohozaev identity
The main tool in proving Theorem 2.1 is the following Pohozaev type identity.

Proposition 4.1. Let (u, φ) be a solution of (1.1) that satisfies (2.3)-(2.4). Then
Then N+α (R N ) and from the first equation of (1.1) we have Let us next analyse term by term the above equation. Since u ∈ W 2,2 loc (R N ) we have By Lebesgue dominated convergence theorem, we find Next, Again by Lebesgue dominated convergence theorem we deduce

Further we have
Note that the regularity of K, u, φ and the second equation of (1.1) allow us to derive Thus, we have We obtain Passing now to the limit in (4.1) we obtain the conclusion.

Proof of Theorem 2.1 completed
Let (u, φ) be a solution of (1.1) which satisfies (2.3)-(2.4). It is enough to show that u ≡ 0 as the second equation of (1.1) together with φ ∈ H 1 (R N ) will imply φ ≡ 0. Suppose by contradiction that the solution (u, φ) satisfies u ≡ 0. For convenience, let us denote Since u is a solution of (2.10) we also have But this last inequality is impossible since u > 0, A(u) ≥ 0 and p, q, N, α, γ satisfy (2.1).
(ii) Assume x · ∇K(x) + γK(x) ≤ 0 in R N for some γ ∈ R and that (2.2) holds. It follows that so that (4.3) together with (4.4) yield Note that the above inequality is impossible since u > 0, A(u) ≥ 0 and p, q, N, α, γ satisfy (2.2). This concludes our proof. Define the Nehari manifold associated with J as Remark that for u ∈ H 1 (R N ) \ {0} and t > 0 we have Since p > q > 1, the equation J ′ (tu), tu = 0 has a unique positive solution t = t(u) and the corresponding element t(u)u ∈ N is called the projection of u on N . The main properties of the Nehari manifold which we use in this paper are stated below.
Hence, there exists C 0 > 0 such that Using this fact we have Assume now that u ∈ N is a critical point of J in N . By the Lagrange multiplier theorem, there exists λ ∈ R such that J ′ (u) = λG ′ (u). In particular J ′ (u), u = λ G ′ (u), u . Since G ′ (u), u < 0, it follows that λ = 0 so J ′ (u) = 0.

A compactness result
be the energy functional corresponding to (1.3). Also, consider its Nehari manifold Then, there exists a solution u ∈ H 1 (R N ) of (2.10) such that replacing (u n ) with a subsequence the following alternative holds (1) either u n → u strongly in H 1 (R N ); or (2) u n ⇀ u weakly (but not strongly) in H 1 (R N ) and there exists a positive integer k ≥ 1, k functions u 1 , u 2 , . . . , u k ∈ H 1 (R N ) which are nontrivial weak solutions to (1.3) and k sequence of points (y n,1 ), (y n,2 ), . . . , (y n,k ) ⊂ R N such that: (i) |y n,j | → ∞ and |y n,j − y n,i | → ∞ if i = j, n → ∞; u n → u a.e. in R N .

(5.4)
We also need the following result: Proof. We shall prove only (ii) as the (i) part is similar.
We now return to the proof of Proposition 5.2. By (5.4), Lemma 3.6 and Lemma 5.1(ii) it follows that J ′ (u) = 0 so u ∈ H 1 (R N ) is a solution of (2.10).
If u n → u strongly in H 1 (R N ) then the first alternative in the statement of Proposition 5.2 holds and we are done. Assume in the following that (u n ) does not converge strongly in H 1 (R N ) to u and define z n,1 = u n − u. Then (z n,1 ) converges weakly and not strongly to zero in H 1 (R N ) and u n 2 = u 2 + z n,1 2 + o(1). We claim that δ > 0. Indeed, if δ = 0, by Lemma 3.2 we deduce z n,1 → 0 strongly in L 2Np N+α (R N ). Then, by Hardy-Littlewood-Sobolev inequality (2.6) we find This fact combined with (5.12) yields z n,1 → 0 strongly in H 1 (R N ) in contradiction to our assumption. Hence, δ > 0 so that we may find y n,1 ∈ R N with B 1 (y n,1 ) Considering the sequence (z n,1 (· + y n,1 )), there exists u 1 ∈ H 1 (R N ) such that, up to a subsequence, we have z n,1 (· + y n,1 ) ⇀ u 1 weakly in H 1 (R N ), z n,1 (· + y n,1 ) → u 1 strongly in L 2Np N+α loc (R N ), z n,1 (· + y n,1 ) → u 1 a.e. in R N .
Passing to the limit in (5.13) we find so u 1 ≡ 0. Also, since (z n,1 ) converges weakly to zero in H 1 (R N ) it follows that (y n,1 ) is unbounded. Passing to a subsequence we may assume |y n,1 | → ∞. From (5.12) we also obtain E ′ (u 1 ) = 0, so u 1 is a nontrivial solution of (1.3).
If (z n,2 ) converges strongly to zero, the proof finishes (and take k = 1 in the statement of Proposition 5.2). Assuming that z n,2 ⇀ 0 weakly and not strongly in H 1 (R N ), we iterate the process. In k number of steps we find a set of sequences (y n,j ) ⊂ R N , 1 ≤ j ≤ k with |y n,j | → ∞ and |y n,i − y n,j | → ∞ as i = j, n → ∞ and k nontrivial solutions u 1 , u 2 , . . . , u k ∈ H 1 (R N ) of (1.3) such that, denoting z n,j (x) := z n,j−1 (x) − u j−1 (x − y n,j−1 ) , 2 ≤ j ≤ k, we have z n,j (x + y n,j ) ⇀ u j weakly in H 1 (R N ) and J (u n ) = J (u) + k j=1 E(u j ) + E(z n,k ) + o(1).
Since E(u j ) ≥ m E and (J (u n )) is bounded, the process can be iterated only a finite number of times. This concludes our proof.
Corollary 3. Let c ∈ (0, m E ). Then, any (P S) c sequence of J | N is relatively compact.
Proof. Let (u n ) be a (P S) c sequence of J | N . Since E(u j ) ≥ m E in Proposition 5.2, it follows that up to a subsequence u n → u strongly in H 1 (R N ) and u is a solution of (2.10).