Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth

We are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth in the sense of the Hardy–Littlewood–Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti–Rabinowitz-type conditions, still using variational methods.


Introduction and main results
This paper deals with the following class of nonlinear and nonlocal Schrödinger equations: where ε > 0 is the dimensionalized Planck constant, N ≥ 3, α ∈ (0, N), F is the primitive function of f , I α is the Riesz potential defined for every x ∈ ℝ N \ {0} by , Γ is the Gamma function, and the external potential V satisfies: (V1) V ∈ C(ℝ N , ℝ) and inf x∈ℝ N V(x) > 0. When ε = 1, V(x) = a > 0, equation (1.1) reduces to the following nonlocal elliptic equation: which is variational, in the sense that solutions of (1.2) turn out to be critical points of the energy functional L a (u) = 1 2 ∫ ℝ N |∇u| 2 + au 2 − (I α * F(u))F(u), u ∈ H 1 (ℝ N ).
In particular, in the relevant physical case of dimension N = 3, α = 2 and F(s) = s 2 2 , (1.2) turns into the socalled Choquard equation − ∆u + au = (I 2 * u 2 )u, x ∈ ℝ 3 , (1. 3) which goes back to the seminal work of Fröhlich [24] and Pekar [50], modeling the quantum Polaron and then used by Choquard [35] to study steady states of the one component plasma approximation in the Hartree-Fock theory [38]. Equation (1.3) appears also in quantum gravity in the form of Schrödinger-Newton systems [51][52][53] in which a single particle is moving in its own gravitational field (self-gravitating matter), see also [30]. Lieb in [35] proved the existence and uniqueness of positive solutions to (1.3) by using rearrangements techniques. Multiplicity results for (1.3) were then obtained by Lions [39,40] by means of a variational approach. A class of solutions which turn out to be of great interest in Physics as well as Mathematics are minimal energy solutions, which were predicted by Pekar to have a stochastic characterization in terms of Brownian motion, a conjecture proved just thirty years later by Donsker and Varadhan [21,22]. We refer to [47] and references therein for an extensive survey on the topic. which is a prototype in semilinear equations and in particular it is well known since the work of Gidas, Ni and Nirenberg [28] that positive solutions with finite energy are radially symmetric, unique and non-degenerate (in the sense that the kernel of the linearized operator at the solution u is generated by ∇u), see [28,49]. In contrast with the local problem (1.5), moving planes methods are somehow difficult to be used and is difficult to be used and the classification of positive solutions to (1.4) (even for p = 2) has remained open for a long time. By using a suitable version of the moving planes method developed by Chen, Li and Ou [15], Ma and Zhao [42] gave a breakthrough to this open problem by considering equivalent Bessel-Riesz integral systems. By requiring some involved assumptions on α, p and N, they proved that positive solutions of (1.4) are, up to translations, radially symmetric and unique. In [44], Moroz and Van Schaftingen established the existence of ground state solutions to (1.4) in the optimal range (1.6) The endpoints in the above range of p are extremal values for the Hardy-Littlewood-Sobolev inequality [36] and sometimes called lower and upper H-L-S critical exponents. From the PDE point of view, a Pohozaev-type identity prevents the existence of finite energy solutions. In the upper critical case, as in the local Sobolev case, the appearance of a group invariance which yields explicit extremal functions to the H-L-S inequality is responsible for the lack of compactness. The lack of compactness can not be recovered by the presence of an external potential. In the lower critical case, equivalent variational characterizations of the ground state level still allow the H-L-S extremal functions to preventing compactness: this casts the problem within the class of Brezis-Nirenberg-type problems [8].
Recently, in [45] the more general Choquard equation (1.2) has been studied by requiring Berestycki-Lions-type conditions, and establishing the existence of ground state solutions in the subcritical case (1.6).
The first purpose of the present work is to investigate the existence of ground state solutions to (1.2) involving the upper H-L-S critical exponent. In presence of lower H-L-S critical exponent, a suitable external potential may lower down the groundstate to the compact region. This turns out to be a Lions-type problem and it is considered in a companion paper [26] as it involves quite different techniques.
Our first main result in this paper is the following: Let us point out that assumption (F3) plays a crucial role. Indeed, under the lonely assumptions (F1) and (F2), equation (1.2) has no solutions for any nontrivial external potential V by means of a Pohozaev-type identity (Lemma 3.2, Section 2). This fact rules out any perturbative argument and casts the problem into a Brezis-Nirenberg-type.
The second purpose of this paper is to investigate the profile of positive solutions to (1.1) as ε → 0. Indeed, in quantum physics one expects that as the Planck constant ε → 0, the dynamic is governed by the external potential V and an interesting class of solutions show up which develop a spike shape around critical points of V. From the physical point of view, these solutions are known as semiclassical states, as they describe the transition from quantum mechanics to classical mechanics. For the detailed physical background, we refer to [49] and references therein. By a Lyapunov-Schmidt reduction approach, based on the non-degeneracy condition, in [23,49] the authors obtained the existence of solutions to the semilinear singularly perturbed Schrödinger equation which exhibit a single peak or multi peaks concentrating, as ε → 0, around any given non-degenerate critical points of V. However, so far, the non-degeneracy condition holds for only a very restricted class of f . In the last decade, a lot of efforts have been devoted to relax or remove the non-degeneracy condition in this family of singularly perturbed problems. By using a variational approach, Rabinowitz [54] obtained the existence of positive solutions to (1.7) for small ε > 0 with the following global potential well condition: Subsequently, by a penalization approach, del Pino and Felmer [18] weakened the above global potential well condition to the local condition (V2) there exists a bounded domain O ⊂ ℝ N such that and proved the existence of a single-peak solution to (1.7). In [18,54], the non-degeneracy condition is not required. Some related results can be found in [3,17,19,20,59] and the references therein. In [10] Byeon and Jeanjean introduced a new penalization approach and constructed a spike layered solution of (1.7) under (V2) and the almost optimal Berestycki-Lions conditions [6], see also [9,11,12] and [65,68]. The second main result of this paper is the following: dist(x ε , M) = 0, and w ε (x) ≡ v ε (εx + x ε ) converges (up to a subsequence) uniformly to a ground state solution of the limit equation We mention that related results under stronger assumptions have been recently obtained in [5]. For the convenience of the reader let us better contextualize our result within the existing literature on the singularly perturbed problem (1.1).
In [60], Wei and Winter considered the nonlocal equation, equivalent to the Schrödinger-Newton system, and by using a Lyapunov-Schmidt reduction method under assumption (V1), proved the existence of multibump solutions concentrating around local minima, local maxima or non-degenerate critical points of V. When the potential is allowed to vanish somewhere, thus avoiding (V1), the problem becomes much more difficult. In [56], Secchi considered (1.8) with a positive decaying potential and by means of a perturbative approach, proved the existence and concentration of bound states near local minima (or maxima) points of V as ε → 0. Recently, by a nonlocal penalization technique, Moroz and Van Schaftingen [46] obtained a family of single spike solutions for the Choquard equation around the local minimum of V as ε → 0. In [46] the assumption on the decay of V and the range for p ≥ 2 are optimal. More recently, using the penalization argument introduced in [10], Yang, Zhang and Zhang [64] investigated the existence and concentration of solutions to (1.1) under the local potential well condition (V2) and mild assumptions on f . In particular, the Ambrosetti-Rabinowitz condition and the monotonicity of f(t) t are not required. For related results see [4,7,16,43,48,56,58,63]. All the previous results are subcritical in the sense of the Hardy-Littlewood-Sobolev inequality. In [2], the authors considered the ground state solutions of the Choquard equation (1.1) in ℝ 2 . By variational methods, the authors proved the existence and concentration of ground states to (1.1) involving critical exponential growth in the sense of the Pohozaev-Trudinger-Moser inequality. A natural open problem which has not been settled before is to establish concentration phenomena for (1.1) in the critical growth regime. Here we give a positive answer to this open problem in Theorem 1.2.
Overview. We conclude this section by giving the outline of the paper and pointing out major difficulties. In Section 2 we prove some preliminary results which require some efforts to extend a few well-known results in the local setting, to the nonlocal framework. Section 3 is devoted to proving Theorem 1.1. Here, without the Ambrosetti-Rabinowitz condition, to obtain the boundedness of the Palais-Smale sequence becomes a delicate issue. To overcome this difficulty, a possible strategy is to look for a constraint minimization problem. This goes back to Berestycki-Lions [6], in which the authors established the existence of ground state solutions to the scalar mean field equation −∆u = g(u), u ∈ H 1 (ℝ N ). By using a similar strategy, Zhang and Zou [67] extended the result in [6] to the critical case. Precisely, in [6,67], the existence of ground state solutions is reduced to looking at the constraint minimization problem and eventually to get rid of the Lagrange multiplier thanks to some appropriate scaling. However, this approach fails for the nonlocal problem (1.2), since ∫ ℝ N |∇u| 2 , ∫ ℝ N |u| 2 and ∫ ℝ N (I α * F(u))F(u) scale differently in space and hence one has no hope to remove the Lagrange multiplier. The existence of ground state solutions to the nonlocal problem (1.2), in the subcritical case, has been done by Moroz and Van Schaftingen in [45], where they constructed a bounded Palais-Smale sequence satisfying asymptotically the Pohozaev identity and obtained a ground state solution by virtue of a concentration-compactness-type argument and a scaling technique introduced by Jeanjean [31]. Here, to avoid a Ambrosetti-Rabinowitz-type condition, we use the Struwe monotonicity trick, in the abstract form due to [32], to get a bounded Palais-Smale sequence. Clearly, due to the presence of a critical H-L-S term, the Palais-Smale condition fails. By a decomposition technique, we recover compactness and obtain the existence of ground state solutions to (1.2). In Section 4, we first prove some qualitative properties of the set of ground states such as compactness, regularity, symmetry and positivity. Then we use a truncation argument as key ingredient to prove Theorem 1.2. In [64], the authors considered problem (1.1) in the subcritical case and established concentration phenomena. Here, the presence of critical growth prevents to use directly the argument in [64]. We overcome this difficulty by penalizing the problem which is relaxed to a subcritical case. The penalized problem admits a family of spike shaped solutions which develop a concentrating behavior around the local minima of V. Finally, the analysis carried out in Section 3 enables us to prove the convergence of the penalized solution to a solution of the original problem which preserves the same qualitative properties of the penalized problem.

Preliminaries
In this section, we are concerned with the existence of ground state solutions to (1.2). Let a > 0 and denote the least energy of (1.2) by In what follows, let H 1 (ℝ N ) be endowed with the norm Before proving Theorem 1.1, we prove first some preliminary results. First of all, let us recall the following Hardy-Littlewood-Sobolev inequality which will be frequently used throughout the paper.
In particular, if s = r = 2N N+α , the best possible constant is given by

Brezis-Lieb lemma
In this subsection, we prove a nonlocal version of the Brezis-Lieb lemma.
Let {u n } ⊂ H 1 (ℝ N ) be such that u n → u weakly in H 1 (ℝ N ) and a.e. in ℝ N as n → ∞. Then where o n (1) → 0 as n → ∞.
In order to prove Lemma 2.2, we recall the following lemma, which states that pointwise convergence of a bounded sequence implies weak convergence.

Lemma 2.3 ([62, Theorem 4.2.7]).
Let Ω ⊆ ℝ N be a domain and let {u n } be bounded in L q (Ω) for some q > 1. If u n → u a.e. in Ω as n → ∞, then u n → u weakly in L q (Ω) as n → ∞

Proof of Lemma 2.2. Observe that
and there exists C > 0 such that which implies F(u) ∈ L 2N/(N+α) (ℝ N ). For any δ > 0 sufficiently small, by the Hardy-Littlewood-Sobolev inequality there exists K 1 > 0 such that , Again by the Hardy-Littlewood-Sobolev inequality we have where we have used the fact that {u n } is bounded in H 1 (ℝ N ). It is easy to see there exists c > 0 such that Then, by Hölder's inequality, So for δ given above and K 1 fixed but large enough, we get for any n, Similarly, let Ω 2 := {x ∈ ℝ N : |x| ≥ R} \ Ω 1 with R > 0 large enough, we have and for any n, If Ω 3 (n) ̸ = 0, then we know that |u(x)| < K 1 and |x| < R for any x ∈ Ω 3 (n). By noting that u n → u a.e. in Ω as n → ∞, it follows from the Severini-Egoroff theorem that u n converges to u in measure in B R (0), which implies that |Ω 3 (n)| → 0 as n → ∞. Similarly we have, for n large enough, Finally, let us estimate where which implies by the Hardy-Littlewood-Sobolev inequality Noting that H n is bounded in L 2N/(N+α) (ℝ N ) and H n → 0 a. e. in ℝ N as n → ∞, by Lemma 2.3, H n → 0 weakly in L 2N/(N+α) (ℝ N ) as n → ∞. By Remark 2.1, and the arbitrary choice of δ concludes the proof.

Splitting lemma
Next we prove a splitting property for the nonlocal energy.
In order to prove Lemma 2.4, we need first to prove Lemma 2.5 and Lemma 2.6 below.
Finally, combining the previous estimate with (2.2), we conclude the proof.
be bounded and such that, up to a subsequence, for any bounded domain Ω ⊂ ℝ N , g n → 0 strongly in L s (Ω) as n → ∞. Then, up to a subsequence if necessary, Proof. Let us prove that for any fixed positive k ∈ ℕ, passing to a subsequence if necessary, (I α * g n )(x) → 0 a.e. in B k (0) as n → ∞. Let k ∈ ℕ + be fixed and for any δ > 0, there exists K = K(δ) > k such that Obviously, B K (x) ⊂ B 2K (0) for any x ∈ B K (0). Noting that g n χ B 2K (0) ∈ L s (ℝ N ), by Remark 2.1, where the constant C depends only on N, α. It follows that, up to a subsequence, I α * (|g n |χ B 2K (0) ) → 0 strongly in L Ns/(N−αs) (ℝ N ) and a.e. in B k (0) as n → ∞. Then, for almost every x ∈ B k (0), one has Since δ is arbitrary, the proof is completed. Now we are set to prove Lemma 2.4.

Proof of Lemma 2.4. Set
Step 1. We claim where o n (1) → 0 uniformly for any ϕ ∈ C ∞ 0 (ℝ N ) as n → ∞. On the other hand, by virtue of (i) of Lemma 2.5 with q = 2+α N−2 and r = 2N For w n = F(u n ), as well as w n = F(u n − u) and also w n = F(u), one easily gets {w n } bounded in L 2N/(N+α) (ℝ N ).

Ground state solutions: Proof of Theorem 1.1
Since we are looking for positive ground state solutions to (1.2), we may assume that f is odd in ℝ N . In this section, a key tool is a monotonicity trick, originally due to Struwe [57] and which here we borrow in the abstract form due to Jeanjean and Toland [32,34].
For λ ∈ [ 1 2 , 1], we consider the following family of functionals: Obviously, if f satisfies the assumptions of Theorem 1.1, for λ ∈ [ 1 2 , 1], I λ ∈ C 1 (H 1 (ℝ N ), ℝ) and every critical point of I λ is a weak solution of (3.1) The existence of critical points to I λ is a consequence of the following abstract result Theorem A (see [32]). Let X be a Banach space equipped with a norm ‖ ⋅ ‖ X , let J ⊂ ℝ + be an interval and let a family of C 1 -functionals {I λ } λ∈J be given on X of the form Assume that B(u) ≥ 0 for any u ∈ X, at least one between A and B is coercive on X and there exist two points v 1 , v 2 ∈ X such that for any λ ∈ J, Then, for almost every λ ∈ J, the C 1 -functional I λ admits a bounded Palais-Smale sequence at level c λ . Moreover, c λ is left-continuous with respect to λ ∈ [ 1 2 , 1].
By (3.2), c λ > δ 2 4 for any λ ∈ J. Then, as a consequence of Theorem A, we have: Next, in the spirit of [33,41], we establish a decomposition of such a Palais-Smale sequence {u n }, which will play a crucial role in proving the existence of ground states to (1.2). However, some extra difficulties with respect to the local case are carried over by the presence the nonlocal as well as critical H-L-S nonlinearity.  Before proving Proposition 3.1, we need a few preliminary lemmas.
Let α ∈ (0, N). For any u ∈ D 1,2 (ℝ N ), combining the Hardy-Littlewood-Sobolev inequality with Sobolev's inequality, we have Minimizers for S α are explicitly known from [37,Theorem 4.3] (see also [27,Lemma 1.2]). Actually, and it is achieved by the instanton Now, we use this information to prove an upper estimate for c λ .
Then the following upper bound holds: Proof. Let φ ∈ C ∞ 0 (ℝ N ) be a cut-off function with support B 2 such that φ ≡ 1 on B 1 and 0 ≤ φ ≤ 1 on B 2 , where B r denotes the ball in ℝ N of center at origin and radius r.
Consider first the case, t ε → 0 as ε → 0. Then by (3.5), (3.6) and (3.7), there exist c 1 , c 2 > 0 (independent of ε) such that for ε small, where we used the fact that q < N+α N−2 : hence a contradiction and t ε ≥ t 0 . By (3.9), one has which implies, combining (3.5) and (3.7), that t ε ≤ t 1 for some t 1 > 0 and ε small. By the Claim just proved and (3.8), we have for some K 7 > 0, and hence on the one hand the following: ).
Step 1. We claim I λ (u λ ) = 0 in H −1 (ℝ N ). As a consequence of Lemma 2.4, it is enough to show, up to a subsequence, that for any fixed ϕ ∈ C ∞ 0 (ℝ N ), In fact, by (F1)-(F2), there exists C > 0 such that By virtue of the Hardy-Littlewood-Sobolev inequality and Rellich's theorem, up to a subsequence, for some C (independent of n) we have → 0 as n → ∞.
Step 3. By (3.10) and v 1 n ⇀ 0 weakly in H 1 (ℝ N ) as n → ∞, there exists {z 1 n } ⊂ ℝ N such that |z 1 n | → ∞ as n → ∞ and lim Let u 1 n = v 1 n ( ⋅ + z 1 n ). Then, up to a subsequence, u 1 n → v 1 λ weakly in H 1 (ℝ N ) as n → ∞ for some v 1 λ ̸ = 0. By Lemma 2.2 and Lemma 2.4, we have Similarly as above, If v 2 n → 0, i.e. u 1 n → v 1 λ strongly in H 1 (ℝ N ) as n → ∞, then and we are done. Otherwise, if v 2 n ̸ → 0 strongly in H 1 (ℝ N ) as n → ∞, similarly as above Then there exists {z 2 n } ⊂ ℝ N such that |z 2 n | → ∞ as n → ∞ and Let u 2 n = v 2 n ( ⋅ + z 2 n ). Then, up to a subsequence, u 2 n ⇀ v 2 λ weakly in H 1 (ℝ N ) as n → ∞ for some v 2 λ ̸ = 0. We have I λ (v 2 λ ) = 0 and Let v 3 n = u 2 n − v 2 λ . Then If v 3 n → 0, i.e., u 2 n → v 2 λ strongly in H 1 (ℝ N ) as n → ∞, then and we are done. Otherwise, we can iterate the above procedure and by Lemma 3.2, we will end up in a finite number k of steps. Namely, let x j n = ∑ j i=1 z i n to have Step 4. Clearly, |x Recalling that ‖u n − u λ ‖ ̸ → 0 as n → ∞, we have Λ 2 ̸ = 0. Let x i n ∈ Λ 2 and Then similarly as above, up to a subsequence, for someṽ i Then, as n → ∞, Without loss of generality, we may assume thatṽ i λ ̸ = 0. Noting that u n ( ⋅ + x i n ) →ṽ i λ a.e. in ℝ N as n → ∞, we get I λ (ṽ i λ ) = 0 in H −1 (ℝ N ). Then we redefine v i λ :=ṽ i λ and as n → ∞, By repeating the argument above at most (k − 1) times and redefining {v j λ } if necessary, we end up with Λ ⊂ Λ 2 such that Finally, by Lemma 2.2 one has c λ = I λ (u λ ) + ∑ j∈Λ I λ (v Then by (3.13), up to a sequence, there exists c 0 ∈ [γ, c 1 ] such that where we used the fact that c λ is continuous from the left at λ. Moreover, by (3.13), for any ϕ ∈ C ∞ 0 (ℝ N ), Similarly as above, there exists some C > 0 such that ≤ C‖ϕ‖ uniformly for all ϕ ∈ C ∞ 0 (ℝ N ), n = 1, 2, . . . , and by the Hardy-Littlewood-Sobolev inequality Then N ̸ = 0 and inf u∈N L a (u) = E a ∈ [γ, c 1 ]. We conclude the proof of Theorem 1.1 by showing that E a is achieved. Clearly, there exists {v n } ⊂ N such that as n → ∞, L a (v n ) → E a and L a (v n ) = 0 in H −1 (ℝ N ). Thus {v n } is bounded in . By the definition of E a , v 0 = 0, k = 1 and L a (v 1 ) = E a , which yields v 1 as a ground state solution to (1.2). The proof is now complete. Then by Theorem 1.1, N a ̸ = 0 for any a > 0. Since L a is invariant by translations, N a cannot be compact in H 1 (ℝ N ). However, this turns out to be the only way to loose compactness as we have the following result. Proof. Let {u n } ⊂ N a . Then L a (u n ) = E a and L a (u n ) = 0 in H −1 (ℝ N ). Similarly as above {u n } is bounded in H 1 (ℝ N ). Assume that u n ⇀ u 0 weakly in H 1 (ℝ N ) as n → ∞; then L a (u 0 ) = 0 in H −1 (ℝ N ). If u n → u 0 strongly in H 1 (ℝ N ), we are done. Otherwise, by virtue of Proposition 3.1, up to a subsequence, there exists which implies that u 0 = 0, k = 1, v 1 ∈ N a and ‖u n ( ⋅ + x 1 n ) − v 1 λ ‖ → 0 as n → ∞.

Regularity, positivity and symmetry
Here we borrow some ideas from [4,45] to establish boundedness, decay, positivity and symmetry of ground state solutions to (1.2).

Proposition 4.2.
Let a > 0. The following hold: (ii) For any u ∈ N a , u ∈ C 1,γ loc (ℝ N ) for γ ∈ (0, 1). (iii) For any u ∈ N a , u has constant sign and is radially symmetric about a point. (iv) E a coincides with the mountain pass value.

Proof of Theorem 1.2
Let u(x) = v(εx), V ε (x) = V(εx) and consider the following problem: Let H ε be the completion of C ∞ 0 (ℝ N ) with respect to the norm For any set B ⊂ ℝ N and ε > 0, we define B ε ≡ {x ∈ ℝ N : εx ∈ B} and B δ ≡ {x ∈ ℝ N : dist(x, B) ≤ δ}. Since we are looking for positive solutions of (1.1), from now on, we may assume that f(t) = 0 for t ≤ 0. For u ∈ H ε , let Fix an arbitrary ν > 0 and define as well as Let Γ ε : H ε → ℝ be given by Γ ε (u) = P ε (u) + Q ε (u).
To find solutions of (4.2) which concentrate inside O as ε → 0, we look for critical points u ε of Γ ε satisfying Q ε (u ε ) = 0. The functional Q ε that was first introduced in [13] will act as a penalization to forcing the concentration phenomena inside O. In what follows, we seek the critical points of Γ ε in some neighborhood of ground state solutions to (1.2) with a = m.

The truncated problem
Denote S m by the set of positive ground state solutions of (1.2) with a = m satisfying u(0) = max x∈ℝ N u(x), where m is given in Section 1. Proof. By Proposition 4.2, S m ̸ = 0. For any {u n } ⊂ S m , without loss of generality, we assume that u n ⇀ u 0 weakly in H 1 (ℝ N ) and a.e. in ℝ N as n → ∞. Let us first prove that u 0 ̸ = 0. Indeed, by (v) of Proposition 4.2, there exist c, C > 0 (independent of n) such that |u n (x)| ≤ C exp (−c|x|) for any x ∈ ℝ N . By the Lebesgue dominated convergence theorem, u n → u 0 strongly in L p (ℝ N ) as n → ∞ for any p ∈ [2, 2N N−2 ]. So if u 0 = 0, one has u n → 0 strongly in H 1 (ℝ N ) as n → ∞, which contradicts the fact E m > 0. We claim u n → u 0 strongly in H 1 (ℝ N ) as n → ∞. Indeed, if not, by Proposition 3.1, there exist k ∈ ℕ + and {v j } k j=1 ⊂ H 1 (ℝ N ) such that v j ̸ = 0, L m (v j ) = 0 in H −1 (ℝ N ) for all j and E m = L m (u 0 ) + ∑ k j=1 L m (v j ). Noting that L m (u 0 ) ≥ E m and L m (v j ) ≥ E m , we get a contradiction. Finally, u 0 ∈ S m . Clearly, u 0 ∈ N m is positive and radially symmetric. Recalling that 0 is the same maximum point u n for any n, by the local elliptic estimate, 0 is also a maximum point of u 0 . The proof is complete. For k > max t∈[0,κ] f(t) fixed, let f k (t) := min{f(t), k} and consider the truncated problem − ε 2 ∆v + V(x)v = ε −α (I α * F k (v))f k (v), v ∈ H 1 (ℝ N ), (4.4) whose associated limit problem is where F k (t) = ∫  Proof. Denote by E k m the least energy of (4.5). Notice that any u ∈ S m is also a solution to (4.5). Then E k m ≤ E m . By [45], E k m is a mountain pass value. Combining (iv) of Proposition 4.2 with the fact f k (t) ≤ f(t) for t > 0 and f k (t) = f(t) = 0 for t ≤ 0, we have E k m ≥ E m and so E k m = E m , which yields S m ⊂ S k m .

Proof of Theorem 1.2
In the following, we use the truncation approach to prove Theorem 1.2. First, we consider the truncated problem (4.4). By Lemma 4.2, S m is a compact subset of S k m . Inspired from [10] we show that (4.4) admits a nontrivial positive solution v ε in some neighborhood of S m for small ε. Then we show that there exists ε 0 > 0 such that ‖v ε ‖ ∞ < κ for ε ∈ (0, ε 0 ).
As a consequence, v ε turns out to be a solution to the original problem (1.1). For this purpose, set δ = 1 10 min{dist(M, O c )}.

Conflict of interest.
The authors declare they have no conflict of interest.