Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions

In the present paper, we consider the following singularly perturbed problem: \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^2\triangle u+V(x)u=\varepsilon^{-\alpha}(I_{\alpha}*F(u))f(u),&x\in \R^N; u\in H^1(\R^N), \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $N\ge 3$, $\alpha\in (0, N)$, $F(t)=\int_{0}^{t}f(s)\mathrm{d}s$ and $I_{\alpha}: \R^N\rightarrow \R$ is the Riesz potential. By introducing some new tricks, we prove that the above problem admits a semiclassical ground state solution ($\varepsilon\in (0,\varepsilon_0)$) and a ground state solution ($\varepsilon=1$) under the general"Berestycki-Lions assumptions"on the nonlinearity $f$ which are almost necessary, as well as some weak assumptions on the potential $V$. When $\varepsilon=1$, our results generalize and improve the ones in [V. Moroz, J. Van Schaftingen, T. Am. Math. Soc. 367 (2015) 6557-6579] and [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, our approach is useful for many similar problems.

Note that (F1)-(F3) were almost necessary and su cient conditions and regarded as the Berestycki-Lions type conditions to Choquard equations, which were introduced by Moroz and Van Schaftingen in [22] for the study of (1.1) with ε = .
In recent years, semiclassical problems like (1.1), i.e. the parameter ε goes to zero, have received attention from the mathematical community. For small ε > , bound states are called semiclassical states, which describe a kind of transition from Quantum Mechanics to Newtonian Mechanics. There are some nice work on semiclassical states for (1.1). For example, for a special form of (1.1) with N = , α = and F(s) = s / , by proving the uniqueness and non-degeneracy, of the ground states for the limit problem, Wei and Winter [37] constructed a family of solutions by a Lyapunov-Schmidt type reduction; Cingolani et al. [9] proved the existence of solutions concentrating around several minimum points of V by a global penalization method. Moroz and Van Schaftingen [24] developed a nonlocal penalization technique to show that problem (1.1) with F(s) = |s| p /p and p ≥ has a family of solutions concentrating at the local minimum of V provided V satis es some additional assumptions at in nity. However, for (1.1) with general nonlinearity F which only satis es (F1)-(F3), there seem to be no results in the existing literature. One of main purpose of this paper is to deal with this case.
If the potential V(x) ≡ V∞, then (1.2) reduces to the following autonomous form: its energy functional is as follows: introduced by Pekar [27] at least in 1954, describing the quantum mechanics of a polaron at rest. In 1976, Choquard [16] used (1.7) to describe an electron trapped in its own hole. In 1996, Penrose [19] proposed (1.7) as a model of self-gravitating matter. In this context (1.7) is usually called the nonlinear Schrödinger-Newton equation, see Moroz-Schaftingen [22]. If we let α → in (1.5), then we can get the following limiting problem: where g = Ff . In the fundamental paper [4], Berestycki-Lions proved that (1.8) has a radially symmetric positive solution provided that g satis es the following assumptions: (G1) g ∈ C(R, R) is odd and there exists a constant C > such that To prove the above result, Berestycki-Lions [4] considered the following constrained minimization problem min ∇u : (1.10) they rst showed that by the Pólya-Szegö inequality for the Schwarz symmetrization, the minimum can be taken on radial and radially nonincreasing functions. Then they showed the existence of a minimum w ∈ H (R N ) by the direct method of the calculus of variations. With the Lagrange multiplier Theorem, they concluded thatū(x) :=ŵ(x/tŵ) with tŵ = N− N ∇ŵ is a least energy solution of (1.8). By noting the oneto-one correspondence between S and Jeanjean-Tanaka [13] proved thatū minimizes the value of the energy functional on the Pohozaev manifold for (1.8). However, the approach of Berestycki-Lions [4] fails for nonlocal problem (1.5) due to the appearance of the nonlocal term. In [22], Moroz-Van Schaftingen proved rstly the existence of a least energy solution to (1.5) under (F1)-(F3). To do that, they employed a scaling technique introduced by Jeanjean [11] to construct a Palais-Smale sequence ((PS)-sequence in short) that satis es asymptotically the Pohozaev identity (a Pohozaev-Palais-Smale sequence in short), where the information related to the Pohozaev identity helps to ensure the boundedness of (PS)-sequences, and then used a concentration compactness argument to overcome the di culty caused by lack of Sobolev embeddings. Such an approach could be useful for the study of other problems where radial symmetry of solutions either fails or is not readily available. For more related results on nonlocal problems, we refer to [6,17,18,26,31,38].
We would like to point out that the approach used in [22] is only valid for autonomous equations, it does not work any more for nonautonomous equation (1.2) with V ≠ constant, since one could not construct a Pohozaev-Palais-Smale sequence as Moroz-Van Schaftingen did in [22]. Thus new techniques are required for the study of the nonautonomous equation (1.2) with f satisfying (F1)-(F3), which is another focus of this paper.
In view of [22,Theorem 3], every solution u of (1.5) satis es the following Pohozaev type identity: (1.11) Therefore, the following set is a natural constraint for the functional I ∞ . Moreover, the least energy solution u obtained in [22] satis es A natural question is whether there exists a solutionū ∈ M ∞ such that In the rst part of this paper, motivated by [4,7,13,22,33,35], we shall develop a more direct approach to obtain a ground state solution for (1.2) which has minimal "energy" I in the set of all nontrivial solutions, moreover, this solution also minimizes the value of I on the Pohozaev manifold associated with (1.2), under (F1)-(F3), (V1) and the following two additional conditions on V: To state our rst result, we de ne a functional on H (R N ) as follows: 13) which is associated with the Pohozaev identity P(u) = of (1. (1.14) Our rst result is as follows.

Corollary 1.2. Assume that f satis es
With the help of the Pohozaev type identity (1.11) established in [22], we easily prove that the solutionū obtained in Corollary (1.2) is also the least energy solution for (1.5). More precisely, we have the following theorem: There are indeed many functions which satisfy (V1)-(V3). For example Remark 1.5. We point out that, as a consequence of Theorem 1.1, the least energy value m := inf M I has a minimax characterization m = inf u∈Λ max t> I(u t ) which is much simpler than the usual characterizations related to the Mountain Pass level.
In the second part of this paper, we are interested in the existence of the least energy solutions for (1.2) under (F1)-(F3). In this case, we can replace (V3) by the following weaker decay assumption on ∇V: and there exist θ ∈ ( , ) andR ≥ such that In this direction, we have the following theorem. Applying Theorem 1.6 to the following perturbed problem: where V∞ is a positive constant and the function h ∈ C (R N , R) veri es: Then we have the following corollary. In the last part of the present paper, we consider the singularly perturbed nonlinear Choquard equation (1.1), and prove the existence of semiclassical ground state solutions for (1.1) under weaker assumptions on V : V ∈ C (R N , R) and there exists θ ∈ ( , ) such that Condition (V5) was introduced by Rabinowitz in [28]. Our last result is as follows. Remark 1.11. Our approach could be applied to deal with many equations, such as Schrödinger equations, see [7]. In the existing literature, Schrödinger equations were considered by many authors (for example [4,5,10,11,13]).
To prove Theorem 1.1, we shall divide our arguments into three steps: i). Step ii) is the most di cult due to lack of global compactness and adequate information on I (un). To avoid relying radial compactness, we establish a crucial inequality related to I(u), I(u t ) and P(u) (Lemma 2.2), it plays a crucial role in our arguments, see Lemmas 2.7, 2.11, 2.13, 3.5, 4.2. With the help of this inequality, we then can complete Step ii) by using Lions' concentration compactness, the least energy squeeze approach and some subtle analysis. Moreover, such an approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available.
Classically, in order to show the existence of solutions for (1.2), one compares the critical level with the one of (1.5) (i.e. the problem at in nity). To this end, it is necessary to establish a strict inequality similar to for some path γ ∈ C([ , ], H (R N )). Clearly, γ (t) > is a natural requirement under (V1), which usually involves an additional assumption on f besides (F1)-(F3), such as f (t) is odd and f (t)t ≥ , see [22,Theorem 1.4]. We would like to point out that the above strict inequality is not used in our arguments, see Section 2. Our approach could be useful for the study of other problems where paths or the ground state solutions of the problem at in nity are not sign de nite.
To prove Theorem 1.6, as in Jeanjean-Tanaka [13], for λ ∈ [ / , ] we consider the family of functionals These functionals have a Mountain Pass geometry, and denoting c λ the corresponding Mountain Pass levels.
Corresponding to (1.17), we also let is not sign de nite, it prevents us from employing Jeanjean's monotonicity trick [12]. More trouble, it is di cult to show the following key inequality due to the minimizer u ∞ λ being not positive de nite. Thanks to the work of Jeanjean-Toland [15], I λ still has a bounded (PS)-sequence {un(λ)} ⊂ H (R N ) at level c λ for almost every λ ∈ [ / , ]. Di erent from the arguments in the existing literature, by means of u ∞ and the key inequality established in Lemma 2.2, we can nd a constantλ ∈ [ / , ) and then prove directly the following inequality Throughout the paper we make use of the following notations: ♠ H (R N ) denotes the usual Sobolev space equipped with the inner product and norm ♠ For any x ∈ R N and r > , Br(x) := {y ∈ R N : |y − x| < r}; ♠ C , C , · · · denote positive constants possibly di erent in di erent places.
The rest of the paper is organized as follows. In Section 2, we give some preliminaries, and give the proofs of Theorems 1.1 and 1.3. Section 3 is devoted to nding a least energy solution for (1.2) and Theorem 1.6 will be proved in this section. In the last section, we show the existence of semiclassical ground state solutions for (1.1) and prove Theorem 1.9.

Ground state solutions for (1.2)
In this section, we give the proofs of Theorems 1.1 and 1.3. To this end, we give some useful lemmas. Since V(x) ≡ V∞ satis es (V1)-(V3), thus all conclusions on I are also true for I ∞ . For (1.5), we always assume that V∞ > . First, by a simple calculation, we can verify Lemma 2.1. Lemma 2.1. The following two inequalities hold: Moreover (V3) implies the following inequality holds: Proof. According to Hardy inequality, we have This shows that (2.4) holds.
From Lemma 2.2, we have the following two corollaries.
Next we claim that tu is unique for any u ∈ Λ. In fact, for any given u ∈ Λ, let t , t > such that u t , u t ∈ M. Then P (ut ) = P (ut ) = . Jointly with (2.4), we have and  The following lemma is a known result which can be proved by a standard argument(see [32]).
Proof of Theorem 1.1. In view of Lemmas 2.9, 2.13 and 2.14, there existsū ∈ M such that This shows thatū is a nontrivial solution of (1.2).

The least energy solutions for (1.2)
In this section, we give the proof of Theorem 1.6. Proposition 3.1. [15] Let X be a Banach space and let J ⊂ R + be an interval, and be a family of C -functional on X such that i) either A(u) → +∞ or B(u) → +∞, as u → ∞; ii) B maps every bounded set of X into a set of R bounded below; iii) there are two points v , v in X such that Then, for almost every λ ∈ J, there exists a sequence {un(λ)} such that Similar to the proof of [22,Theorem 3], we can prove the following lemma.
In both cases, we obtain that c λ < m ∞ λ for λ ∈ (λ, ]. (ii) where we agree that in the case l = the above holds without w k .

Semiclassical states for (1.1)
In this section, we give the proof of Theorem 1.9. From now on we assume without loss of generality that x = , that is V( ) < V∞. Performing the scaling u(x) = v(εx) one easily sees that problem (1.1) is equivalent where Vε(x) = V(εx). The energy functional associated to problem (4.1) is given by As in Section 3, we also de ne, for λ ∈ [ / , ] and ε ≥ , the family of functionals I ε λ : Since V ∈ C(R N , R), V( ) < V∞ and u ∞ ∈ H (R N ) \ { }, then there existr > and R > such that and .
Proof of Theorem 1.9. By a similar argument as the proof of Theorem 1.6, we can prove Theorem 1.9 by using Lemmas 4.2, 4.3 and 4.4 instead of 3.5, 3.8 and 3.9, respectively, so, we omit it.