Normalized solutions for Choquard equations with general nonlinearities

In this paper, we prove the existence of positive solutions with prescribed \begin{document}$ L^{2} $\end{document} -norm to the following Choquard equation: \begin{document}$ \begin{equation*} -\Delta u-\lambda u = (I_{\alpha}*F(u))f(u), \ \ \ \ x\in \mathbb{R}^3, \end{equation*} $\end{document} where \begin{document}$ \lambda\in \mathbb{R}, \alpha\in (0,3) $\end{document} and \begin{document}$ I_{\alpha}: \mathbb{R}^3\rightarrow \mathbb{R} $\end{document} is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any \begin{document}$ c>0 $\end{document} , the above equation possesses at least a couple of weak solution \begin{document}$ (\bar{u}_c, \bar{ \lambda}_c)\in \mathcal{S}_{c}\times \mathbb{R}^- $\end{document} such that \begin{document}$ \|\bar{u}_c\|_{2}^{2} = c $\end{document} .


Introduction
This paper is dedicated to deal with the existence of normalized solutions to the generalized Choquard equation as follows: where λ ∈ R, α ∈ (0, 3), I α : R 3 → R is the Riesz potential. Problem (1.1) is a nonlocal one due to the existence of the nonlocal nonlinearity. When λ ∈ R is a fixed and assigned a parameter or even with an additional external, the existence of (1.1) has been studied during the last decade. For example, when λ = −1, α = 2 and F (u) = u 2 , (1.1) comes back to the description of the quantum theory of a polaron at rest by Pekar [22] and the modeling of an electron trapped in its own hole (in the work of Choquard in 1976), in a certain approximation to Hartree-Fock theory of one-component plasma [17]. The equation is also known as the Schrödinger-Newton equation, which was proposed by Penrose [23] in 1996 as a model of self-gravitating matter. Under this condition, the existence of nontrivial solutions was investigated by various variational methods by Lieb and Menzala [17,19] and also by ordinary differential equations methods [11,21,27]. There are also many papers investigating the Choquard equation under the general pure nonlinearity condition, (1.2) − ∆u + u = (I α * |u| p )|u| p−2 u, x ∈ R N , where N ≥ 3 and α ∈ (0, N ), We can refer to [5,15,17,20]. In [20], Moroz and Van Schaftingen obtained that problem (1.2) has a nontrivial solution when N +α Nowadays, since physicist are more and more interested in the normalized solutions, like [1,2,3,13,14,30], mathematical researchers are committed to investigate the solutions with prescribed L 2 -norm, that is, solutions which satisfy u 2 2 = c > 0 for a priori given constant. Such prescribed L 2 -norm solutions of (1.2) can be obtained by looking for critical points of the following functional (1.3) I N (u) = 1 2 R N |∇u| 2 dx − 1 2 R N (I α * F (u))F (u)dx on the constraint In this sense, the parameter λ ∈ R cannot be fixed but regarded as a Lagrange multiplier, and each critical point u c ∈ S c of I N | S c , corresponds a Lagrange multiplier λ c ∈ R such that (u c , λ c ) solves (weakly) (1.2). In particular, if u c ∈ S c is a minimizer of problem (1.5) σ(c) := inf then there exists λ c ∈ R such that I N (u c ) = λ c u c , hence, (u c , λ c ) is a solution of (1.2).
In [12], Jeanjean proved the existence of normalized solutions of the following Schrödinger equation where N ≥ 1, f : R → R satisfies the following cases: (f1) f ∈ C(R, R) and f is odd; (f2) ∃ α, β ∈ R with 2N +4 N < α ≤ β < 2 * such that Jeanjean deduced the existence of normalized solutions by dealing with the minimization problem and the author verified the existence of the mountain pass structure on the constraint defined by S c . Moreover, one of the highlights in the proof is that the auxiliary functionalĨ : By applying the new functionalĨ, Jeanjean proved that for any fixed c > 0, problem (1.6) has a couple of weak solution (u c , λ c ) ∈ H 1 (R N ) × R − such that u c 2 = c under the conditions (f1), (f2) and (f3).
Bellazzini, Jeanjean and Luo [4] verified the existence of standing waves with prescribed L 2 -norm for the following Schrödinger-Poisson equation: where q ∈ ( 10 3 , 6), and which different from [12] is that the function defined by is no more bounded from below on the constraint: To overcome this difficulty, they first investigated the mountain-pass structure ofÎ on the constraint S c , and then they show the existence of special bounded Palais-Smale sequence {u n } at the level γ(c) which surrounds around the constraint set In particular, M c used in [4] acts as a natural restriction and γ(c) equals numerically to As far as we know, there seems to be only one paper [16] dealing with the Choquard equation in the sense of prescribed L 2 -norm, Li and Ye considered the Choquard equation (1.1) in the N −dimension space under the following conditions: (F1 ) f (s) = 0 for s ≤ 0 and there exists r ∈ ( N +α+2 (F6 ) there exists 0 < θ 2 < 1 and t 0 such that for all s ∈ R and |t| ≤ t 0 , In fact, the nonlinearity term in paper [16] needs the assumption f ∈ C 1 (R, R). A natural question is whether the above result in [16] on the existence of normalized solutions to (1.1) can be generalized to more general f ∈ C(R, R). The purpose of the present paper is to address this question. To this end, we introduce the following assumptions: (F1) f (s) = 0 for s ≤ 0 and there exists r ∈ ( t is nondecreasing on (−∞, 0) and (0, +∞). In this paper, we define (1.12) where the definition of S c is given by (1.9). Our main result is as follows: Notice that, we proved the existence of normalized solution of problem (1.1) under the assumptions (F1)-(F5). Compared to [16], case (F5) plays an important role to overcome the difficulty caused by the absence of condition (F5 ), that is, we generalized the problem (1.1) concerning the prescribe L 2 -norm solutions to fit on more general nonlinearity term. But also, the absence of (F5 ) in [16] causes new difficulties. In the proof, we present a new and more general approach to overcome this difficulty. Now, we give our main idea for the proof of Theorem 1.1. By (F1) and (F2), there exists some C > 0 such that (1.13) |F (s)| ≤ C(|s| r + |s| 3+α ).
Then for any s > 0, F (s) is nondecreasing in s > 0. By (F1), we conclude that: Then by (1.14), we see that f (s) ≥ 0 for all s ∈ R and F (s) is nondecreasing in s ∈ R.
As in [4,12], I is no more bounded from below on S c by (F1), similarly we shall seek for a critical point satisfying a minimax characterization, i.e., we try to prove, I possesses a mountain pass geometry on the constrain S c . Let us recall this, to obtain this conclusion, the authors in [4] constructed some sequence of paths {g n } ⊂ Γ c which have nice 'shape' properties, and by Taylor's formula which is relies on I ∈ C 2 (H 1 (R 3 ), R), the author deduced a localization lemma concerning the specific (PS) sequence. Different from his work, in the present paper we shall investigate the following auxiliary functional: and also we shall know the fact thatĨ possesses the same mountain pass structure on S c × R as the functional I Sc . Based on this fact, in Lemma 2.3, we find a (PS) γ(c) sequence {u n } with the additional property J(u n ) → 0, and then prove the convergence of {u n }, this idea comes from [12] in which the classical Schrödinger equation (1.6) was studied.
Since we have obtained the boundness of {u n }, next we using scaling tramsform to verify the convergence of {u n }. Because the nonlocal term and the gradient term in I scale differently, we overcome this difficulty by verifying whether γ(c) is nonincreasing. As in [8], we first prove that γ(c) is nonincreasing and then combining with the fact γ(c) = m(c) which is verified in Lemma 2.10, then we can prove the convergence of {u n }.
In [4] the fact I may be not C 2 prevents us using the Implicit Function Theorem which influence above approach, then there needs new techniques and more subtle analyses to apply to more general f ∈ C 1 . To deduce the convergence of (PS) γ(c) sequence {u n }, we shall establish a new key inequality with the help of (F5), see Lemma 2.4, which is also inspired by [6,7,9,10,24,26,28]. In particular, we present a new and more general approach to recover the compactness of minimizing sequence.
Throughout the paper we use the following notations: • H 1 (R 3 ) denotes the usual Sobolev space equipped with the inner product and norm positive constants possibly different in different places.

Preliminary results
To prove Theorem 1.1, recalling the Gagliardo-Nirenberg inequality, that is, let p ∈ [2, 6), In the following lemma, we show that I possesses the mountain pass geometry on the constraint S c . Lemma 2.1. Assume that (F1), (F2) and (F4) hold. Then for any c > 0, there exist 0 < k 1 < k 2 and u 1 , u 2 ∈ S c such that u 1 ∈ A k1 and u 2 ∈ A k2 , where Moreover, I has a mountain pass geometry on the constraint S c .
Proof. Given any k > 0, let We have known that F (u) ∈ L 6 3+α (R 3 ), then by the Hardy-Littlewood-Sobolev inequality, Sobolev embedding inequality and the Gagliardo-Nirenberg inequality, we can find (2.5) Hence, we have that Since 3r − 3 − α > 2 and 6 + 2α > 2, it follows from (2.6) that there exist k 2 > 0 small and ρ > 0 such that On the other hand, use (2.5) again, we have Combining (2.7) with (2.9), there exists k 1 ∈ (0, k 2 ) small such that and (2.4) follows. Let Then u t 2 = u 2 , and so u t ∈ S c for any u ∈ S c and t > 0. Note that Using (F1), (2.10) and Fatou's Lemma, which is inspired by [16], we can see that (2.12) Hence, we have 2 > k 2 . This fact indicates that u 1 ∈ A k1 and u 2 ∈ A k2 . We next claim that I possesses a mountain pass geometry on S c . For which, together with the arbitrariness of g ∈ Γ c , implies Indeed, to obtain the desired conclusion, it suffices to check that Γ c = ∅. For any u ∈ S c , set It follows from (2.14) that g 0 ∈ γ(c). Hence, Γ c = ∅ and the proof is completed.
Next, inspired by [6,12], we will show the existence of a (PS) sequence for the functional I on the constraint S c attaching the property J(u n ) → 0, where (2. 16) To achieve this, we define a continuous map β : where H is a Banach space equipped with the product norm (v, t) H := v 2 + |t| 2 1/2 . We introduce the following auxiliary functional: It is easy to see thatĨ ∈ C 1 (H, R), and for any (w, s) ∈ H,   1))} = max g∈Γc max{Ĩ(g(0)),Ĩ(g(1))}.
Following by [29], we recall that for any c > 0, S c is a submanifold of H 1 (R 3 ) with codimension 1 and the tangent space at S c is given The norm of the C 1 restriction functional I| Sc is defined by And the tangent space at (u, t) ∈ S c × R is given as The norm of the derivative of the C 1 restriction functionalĨ| Sc×R is defined by Learning from [12, Proposition 2.2], we have the following proposition.
Proposition 1. Assume thatĨ has a mountain pass geometry on the constraint S c × R. Letg n ∈Γ c be such that Then there exists a sequence (u n , t n ) ∈ S c × R such that Applying proposition 1 toĨ and also by [8], we conclude the following key lemma.
In connection with the additional minimax characterization of γ(c), we have the following Lemma 2.10. To achieve this goal, we have to establish some new inequalities, which is the crucial procedure for our convenience to obtain our final conclusion of this paper.
Proof. For any t ∈ R, we have Now we only need to study q(t, τ 1 , τ 2 ) which is defined by the following form: By (F5) and (1.15), we can easily get the above conclusion, which implies that h(t) ≥ h(1) = 0 for all t > 0, This shows that (2.35) holds.
By the preceding scaling (2.10), we have

It can be easily checked that
, where the definition of J is given in (2.16). Set After basic calculations, we can see Inspired by [7,25], we obtain the following key inequality. 3 Proof This shows that (2.40) holds. Letting t → 0 in (2.40), we derive that (2.41) holds.
Following the Lemma 2.4 naturally, we obtain the following corollary.
Combining the Corollary 1 and Lemma 2.5, we can easily obtain the following facts. Proof. To achieve this purpose, it is sufficient to verify whether in the condition that for any c 1 < c 2 and ε > 0 arbitrary, we have By the definition of m(c 1 ), there exists u ∈ M c1 such that I(u) ≤ m( For any small δ ∈ (0, 1], let It is easy to obtain that u δ → u in H 1 (R 3 ) as δ → 0. Then we have From Lemma 2.5, for any δ > 0, there exists t δ > 0 such that u δ t δ ∈ M c for some c > 0. Next we show that {t δ } is bounded. Actually, if t δ → ∞ as δ → 0, since u δ → u = 0 in H 1 (R 3 ) as δ → 0, in view of (F1), we infer that which is a contradiction. So we may assume that up to a subsequence, t δ →t as δ → 0, and so J(u δ t δ ) → J(ut), which jointly with J(u) = 0 implies thatt = 1. By (2.40), we have which, together with (2.50), implies that there exists δ 0 ∈ (0, 1) small enough such that Let for which we have v 0 following which we can easily obtain Then (2.55), (2.56) and (F2) imply that as λ → 0, and by (2.57), we have (2.58) which lead to (2.59) I(w λ ) → I(u δ0 ) and J(w λ ) → J(u δ0 ).
Inspired by the above works, we have established the additional minimax characterization of γ(c), which can be summarized as the following lemma. Proof. By (2.14), for any u ∈ M c , there exist t 1 < 0 small and t 2 > 1 large such that u t1 ∈ A k1 and u t2 ∈ A k2 . Set we haveḡ ∈ Γ c . By (2.43), we have and so γ(c) ≤ inf u∈Mc I(u) = m(c) for any c > 0.
On the other hand, by (2.41), we have which implies Moreover, it is easy to verify that there exists u 0 ∈ B k1 such that J(u 0 ) > 0. Hence, any path in Γ c has to go through M c . We deduce that max τ ∈[0,1] and so γ(c) ≥ m(c) for any c > 0. Therefore, γ(c) = m(c) for any c > 0.
Let H be a real Hilbert space, we define its norm and scalar products as · H and (·, ·) H respectively. Let (X, · X ) be a real Banach space, and devoted its dual space by X * satisfying X → H → X * and M = {x ∈ X | x H = 1} be a submanifold of X of codimension 1. Lemma 2.9. Let J : X → R be a C 1 functional and J| M be a C 1 functional restricted to M , assume that {x n } ∈ M is a bounded sequence in X. Then the following are equivalent: x n x n → 0 in X * as n → +∞. Lemma 2.10. Let {v n } ∈ S c be a bounded (P S) γc sequence of I| S (c) . Then there exists a sequence {λ n } ∈ R and λ c ∈ R, v c ∈ H 1 (R 3 ) such that , by preceding Lemma 2.11, we obtain that It means that for any ω ∈ H 1 (R 3 ),

Then
(2.63) and λ n is bounded which is deduced by the boundedness of {v n } and the Hardy-Littewood-Sobolev inequality. Finally, there exists λ c ∈ R such that λ n → λ c .
Lemma 2.11. Assume that f ∈ C(R, R) satisfies the following condition: Next Lemma can also be found in [16], for the sake of completeness and convenience for reading, we show it here again. Proof. Since (v c ,λ c ) ∈ S(c) × R is a weak solution of (1.1), by Lemma 2.13, we infer that where we use that Then, we have i.e.v c ∈ M c . By (2.66), which, combining with v n 2 2 = c, implies {v n } is bounded in H 1 (R 3 ). Then there exists v ∈ H 1 (R 3 ) such that up to a subsequence, v n v in H 1 (R 3 ), v n → v in L s loc (R 3 ) for 2 ≤ s < 6 and v n → v a.e. in R 3 . Since m(c) = γ(c) > 0, by Lions' concentration compactness principle [29, Lemma 1.21] and a standard procedure, we can obtain that {v n } is non-vanishing, and so there exist δ > 0 and {y n } ⊂ R 3 such that B1(yn) |v n | 2 dx > δ. Letv n (x) = v n (x + y n ). Then we have v n = v n and  If there exists a subsequence {w ni } of {w n } such that w ni = 0, by (F4), (3.7), (3.8), the Fatou's lemma and the weak lower semicontinuity of norm, we can deduce that ∇v n − ∇v 2 → 0. Next, we verify that this still be true for w n = 0. Assume that w n = 0. We claim that J(v) ≤ 0. Otherwise, if J(v) > 0, then (3.8) implies J(w n ) < 0 for large n. According to the Lemma 2.5, there exists t n > 0 such that (w n ) tn ∈ Mc n . Then we can know from (1. which implies ∇v n −∇v 2 → 0 for w n = 0. In the end, we prove that v n −v 2 → 0. Using Lemma 2.12, there existsλ c ∈ R such that And by condition (F1) and the strong maximum principle, we conclude that u(x) > 0 for all x ∈ R 3 . This completes the proof.