Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation

: In this article, we study the fractional critical Choquard equation with a nonlocal perturbation:

having prescribed mass

Introduction and main results
In this article, we consider the fractional critical Choquard equation with a nonlocal perturbation: is a positive parameter, and { } → I : \ 0 μ N is the Riesz potential, which is defined by: , and the fractional Laplacian ( ) −Δ s is defined for where ( ) S N denotes the Schwartz space of rapidly decreasing smooth functions, P.V. stands for the principle value of the integral, and C N s , is the positive normalization constant.The nature function space associated with ( ) −Δ s in N dimension is ( ) H s N , which is a Hilbert space equipped with the inner product and norm, respectively, given by: The working space is the homogeneous fractional Sobolev space ( ) defined by: Problems (1.1) and (1.2) arise in seeking standing waves for the following nonlinear fractional Schrödinger equation: A standing wave of (1.4) is a solution having the form ( ) ( ) = − ψ t x e u x , iλt for some ∈ t and u satisfying (1.1).So (1.1) is the stationary equation of the time-dependent equation (1.4).
We say that a function ( ) ∈ u H s N is a weak solution to (1.1) provided ( ) For fixed > a 0, we aim at finding a real number ∈ λ and a function ( ) ∈ u H s N weakly solving (1.1) with ‖ ‖ = u c.

2
To study the normalized solutions to (1.1), we need to study the following energy functional and the constraint: N with λ being the Lagrange multipliers.We shall prove the existence of normalized solutions to equation (1.1) in different cases by comparing q and the L 2 -critical exponent.Equation (1.1) is a special standing wave problem from the following fractional Schrödinger equation: It is a fundamental equation of the space-fractional quantum mechanics, where the fractional power of the Laplacian ( ) ∈ s 0, 1 was introduced by Laskin in [28], which comes from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths [2].The operator has a wide range of applications and appears in various areas of pure and applied mathematics such as charge transport in biopolymers, water waves, crystal dislocations, neural systems, and Bose-Einstein.For more details about the application background on fractional Laplacian, we refer to [20,31,46] and references therein.
Condition (1.2) is called as the normalization condition, which imposes a normalization on the L 2 -masses of u.In order to obtain the solution to (1.1)-(1.2),one needs to consider the critical point on ( ) S c ; in this case, λ appears as Lagrange multipliers with respect to the mass constraint, which cannot be determined a priori, but are part of the unknown.The prescribed mass approach that we shall follow here, has seen an increasing interest in these last few years and has been applied to various related problems (see, for example, [3,5,25,55] and references within).This approach is particularly relevant from a physical point of view.Indeed, the L 2 -norm is a preserved quantity of the evolution, and the variational characterization of such solutions is often a strong help to analyze their orbital stability/instability (see, for example, [7,8,24,53,54], for future references).
Recently, the normalized solution of fractional Schrödinger equations with the Choquard term as follows:  [14,17] studied the existence of stable standing waves of (1.8).In [58], Yang studied the existence and asymptotic properties of normalized solutions for (1.5)  for μ a , small, by using a refined version of the minmax principle, the author obtained a mountain-pass-type solution.Furthermore, he gave some asymptotic properties of the solutions.In [34], Li et  .The authors mainly extended the results of [6] and [14] concerning the aforementioned problem from L 2 -subcritical and L 2 -critical setting to L 2 -supercritical setting with respect to q, involving Sobolev critical case especially.
It is well known that when = s 1, equation (1.8) is related to the nonlinear Choquard equation: . When = p 2, the problem goes back to the description of quantum theory of a polaron at rest by Pekar [47] in 1954.And it is a certain approximation to Hartree-Fock theory of onecomponent plasma (see [18,19,29]) in the work [29] of Choquard on the modeling of an electron trapped in its own hole in 1976.In 1996, Penrose also derived the same equation in discussing the self-gravitational collapse of a quantum mechanical wave function in [48][49][50].The equation was also called Schrödinger-Newton system [32].For more results without prescribed mass obtained by variational methods, we refer to [39][40][41][42][43] Normalized solutions for a critical fractional Choquard equation  3 and the references therein.As far as we know, there are a few studies on normalized solutions of Choquard equations in the literature.Li and Ye [33] studied the following equation: is the Riesz potential.Using the minimax method and the concentration compactness principle, it is shown that equation (1.10) has at least a pair of weak solution ( ) u λ , a a that satisfies . Bartsch et al. [4] first proved the existence of the minimum energy solution in all dimensions for (1.10) and the authors used the fountain theorem to prove the existence of infinitely many solutions for (1.10) when ( ) f u is odd.Recently, Li [36] considered the normalized solutions to the Choquard equation with a local perturbation: The author proved the existence, multiplicity, qualitative properties, and orbital stability of the ground states of (1.11) with the upper critical exponent = = [59], and Shang and Ma [52] studied the effect of lower-order nonlocal perturbations in the existence of positive ground state solutions to (1.11).Bellazzini et al. [7] verified the existence of standing wave solutions with L 2 -norm for the following Schrödinger-Poisson equations: (1.12) where . They first showed that the energy functional has a mountain path geometry under constraint conditions and then proved the boundedness of the Palais-Smale sequence in a special case.Furthermore, they proved that critical points exist when > a 0 is sufficient small; on the contrary, a non-exist result is expected.For more results on the normalized solution for the Schrödinger equations or Choquard equations, we refer to [26,27] and references therein.
Motivated by the aforementioned works, especially by [52][53][54]59], in this article, we aim to prove several existence and non-existence results for problems (1.1)-(1.2) by distinguishing the three cases: (i) L 2 -subcritical case: , respectively.In the sequel, we give some preliminary materials that will be useful in our approach.To begin with, we recall that the key point in applying variational method for Problem (1.1)-(1.2) is the following standard estimates for the Riesz potential (Theorem 4.3 [30]).Proposition 1.1.(Hardy-Littlewood-Sobolev inequality [30])

The equation holds if and only if
for some ∈ ≠ ∈ A γ , 0 and ∈ a N .
Remark 1.1.By a direct calculation from the Hardy-Littlewood-Sobolev inequality, we have where > 0 is a constant defined depending only on N μ , , and s.
Consequently, we can define the best constant S h l , as: and the authors in [22,45] have proved that the best constant can be attained by the function: ,, , satisfies the following equation: for any fixed ∈ y N and > ε 0 with In addition, the function , and achieves the best Sobolev constant S defined as: and from [12,22], we know that the constant κ is given by: Moreover, the constants S S , h l , , and N μ s , , have the relationship: We recall the fractional Gagliardo-Nirenberg inequality.
, , where Normalized solutions for a critical fractional Choquard equation  5 For any and and Feng [17] has proved that the S q is achieved.
In particular, we can set = − p N N μ and = +∞ s .
In addition, it is easy to enumerate that is the L 2 -critical exponent.We also introduce two constants defined as: 1 q s , Now, the main results can be formulated as the following.
, and , where = * { } min , 1 2 , then (i) J α restricted to S c admits a ground state + u α with ( ) for some suitable k small enough.Moreover, any ground state for | J α Sc is a local minimizer of J α on the set M k .
(ii) The solution + u α is real-valued, positive, radially symmetric, and radially decreasing and solves (1.1) for some , then it is natural for us to try to obtain a second critical point of mountain-pass type on S c .To be more precisely, we have the following assertion: , where { } = * min , u α is real-valued, positive, radially symmetric, and radially decreasing and solves (1.1) for some In the case of L 2 -critical perturbation, | J α Sc may change its structure, which will influence the number of critical points of J α .In fact, we have the following existence result. (ii 4 and (iii) u is real-valued, positive, radially symmetric, and radially decreasing and solves (1.1) for suitable < λ 0.
In the case of L 2 -supercritical perturbation, the existence of ground states can be given by the following theorem, which is similar to Theorem 1.3.
, where > 0 is a certain constant, then | J α Sc has a ground state u with ( ) (ii , it is also true that | J α Sc has a ground state u with ( ) (iii) u is real-valued, positive, radially symmetric, and radially decreasing and solves (1.1) for suitable < λ 0.
This article consists of the following sections: in Section 2, some preliminary results that will be used frequently in the sequel.Especially, we study the convergence of the Palais-Smale sequences.In Section 3, we consider the existence of ground state and mountain-pass-type solutions, respectively, for the case of L 2 -sub- critical, and prove Theorems 1.1 and 1.2.In Section 4, we study L 2 -critical and L 2 -supercritical perturbations, and then complete the proof of Theorems 1.3 and 1.4.
Notation.In the sequel of this article, we denote by different positive constants whose values may vary from line to line and are not essential to the problem.We denote by the Lebesgue space with the standard norm Normalized solutions for a critical fractional Choquard equation  7 2 Pohozaev manifold and compactness lemma

Pohozaev manifold
In this section, we introduce the Pohozaev mainfold and the decomposition of it.To begin with, we introduce the following Pohozaev identity, which can be derived from [11,37].
N be a weak solution of (1.1), then u satisfies the equality: Lemma 2.1.Let ( ) ∈ u H s N be a weak solution of (1.1), then we can construct the following Pohozaev manifold: Proof.From Proposition 2.1, we know that u satisfies the Pohozaev identity as follows: Moreover, since u is the weak solution of Problem (1.1), we have Combining with (2.3) and (2.4), we obtain (2.5) It is easy to check that the dilations preserve the L 2 -norm such that ⋆ ∈ t u S c , hence, we introduce the fiber map naturally as follows: A direct calculus shows that ( )( ) ( ) ; hence, we obtain the proposition immediately. where

Palais-Smale sequence
In this subsection, we shall study the convergence of the Palais-Smale sequence of the energy function ( ) 2 , and Proof.We prove the lemma in three different cases.
Case 1: . In this case, we have , then from the condition ( ) → P u 0 α n and the Inequality (1.12), we can derive that . In this case, we have which implies that for a constant > C 0. By Inequality (1.22), we have Combining this with ( ) → P u 0 α n

, we infer that
Normalized solutions for a critical fractional Choquard equation  9 In this case, we obtain > qγ 1 q s , ; using ( ) → P u 0 α n again, we have which implies that both Then, the conclusion follows immediately.□ Lemma 2.3.Assume that the conditions of Lemma 2.2 are satisfied, and 2 then the sequence has a nontrivial weak limit, where ( ) H r s N , consisting of radially symmetric functions, is the subspace of ( ) Proof.From Lemma 2.2, we know that { } u n is bounded in ( ) H s N , then there is a subsequence, still denoted by itself, such that . We show that the weak limit ≠ u 0. Suppose by contra- diction that ≡ u 0. Since the Sobolev embedding ( ) 2 .Hence, from the the Hardy-Littlewood-Sobolev inequality, we can derive that . Therefore, from ( ) → P u 0 α n and (2.8), we obtain as → ∞ n .Moreover, by Inequality (1.15), we derive and so, , which implies that ( ) , we arrive that Then, one of the following alternatives occurs: weakly in ( ) H s N but not strongly, up to a subsequence.Moreover, u solves equation (1.1) for some (ii) Up to a subsequence , and u is a solution of (1.1), satisfying N is bounded, and we may assume that ⇀ u u n weakly in ( ) We first prove the following claim: It is obviously that For every ∈ p q , and > r 2, there is a simple inequality that Then, by the Hardy-Littlewood-Sobolev Inequality (1.13), the Hölder inequality, and the last inequality, we infer to Normalized solutions for a critical fractional Choquard equation  11 , the compactness of Sobolev embedding ( ) s , and the claim is proved.
Since { } u n is a bounded Palais-Smale sequence restricted on S c for J α , then there exists { } ⊂ λ n that satisfies that by the Lagrange multipliers rule.Taking the text function = φ u n , then we derive { } λ n is bounded, and we obtain that Then, we can derive that < λ 0 by virtue of > ≠ α u 0, 0, and as → ∞ n .Based on the aforementioned fact and using Lemma 1.2, we have as → ∞ n .Combining this with the following fact that as → ∞ n , we derive that as → ∞ n .Then, by the definition of weak convergence, we obtain that for any Therefore, passing to the limit on both sides of (2.10), we obtain that and then ( ) = P u 0 α by the Pohozaev identity.Since ⇀ v 0 n weakly in ( ) H s N , then by the Brezis-Lieb lemma and [22], we have (2.12) and Consequently, from the fact that ( ) → P u 0 α n , (2.9), (2.12), and (2.13), we infer that Combining this with ( ) = P u 0 α , we deduce that and by Inequality (1.15), we infer to , then by (2.12) and (2.13), we obtain Combining with (2.9) and (2.13), and Lemma 1.2, we see that the right-hand side of (2.14) tends to zero, which implies that In this section, we consider that > > N s c α 2 , , 0, and , where . First, we analyze the structure and properties of the Pohozaev manifold c α , .
and c α , is a smooth manifold of codimension 2 in ( ) Normalized solutions for a critical fractional Choquard equation  13 Proof.We first show that . Otherwise, there exists a ∈ ∼ u c α , 0 , which means that Thus by ( ) = ∼ P u 0 μ , we obtain Moreover, using ( ) = ∼ P u 0 μ again, we have and Combining with (3.1) and (3.2), we have qγ q s μ s μ s Next, we show that the right-hand of (3.3) is greater than or equal to . 1 To show we only need to prove that 1, for every 2 2*.
Define the function , and monotone decreasing for (( . Moreover, and Inequality (3.5) holds.Combining (3.3)-(3.4),we obtain a contradiction to . Next, we intend to show that c α , is a smooth manifold of codimension 2 in ( ) H s N .We can rewrite the manifold c α , as follows: N , and . Thus, we have to show that the map ( ( ) ( )) 2 is surjective.For every ∈ η T S u c , which denotes the tangent space to S c in u, we have In addition, we claim that for every ∈ u , c α , there exists η such that , which implies that u is a constrained critical point for ( ) P u α on the set S c ; then, there exists ∈ τ that satisfies by the Lagrange multiplies rule.By the Pohozaev identity of the aforementioned equation, we infer that u satisfies the following equality: Hence, ∈ u c α , 0 , which contradicts with the claim that = ∅ c α , 0 . Therefore, for any ( ) ∈ x y , 2 , the following system and c α , is a smooth manifold of codimension 2 in ( ) , is a critical point for J α constrained on c α , , then for every N by the Lagrange multipliers rule.Hence, we can derive that By the Pohozaev identity, we derive that Combining this with the fact that ( ) = P u 0 α , we infer to , and the conclusion is proved.□ To complete the proof of Theorem 1.1 in the L 2 -subcritical perturbation, we need to investigate the behaviors of the constrained functional | J α Sc .By the definition of J α and Inequality (1.15) and (1.22), we have In order to learn more better properties of the right-hand side of Inequality (3.6), we introduce the following function → + F : : Lemma 3.2.Suppose that the inequality holds, then the function ( ) F t has only two critical points with a local strict minimum at the negative level and a global maximum at the positive level.Besides, there are two zero points that satisfy ( ) It is easy to check that ( ) φ t has a unique critical point at where t 0 is the strict maximum.Moreover, Therefore, ( ) , and the fact that ( ) → − F t 0 as → + t 0 and ( ) → −∞ F t as → −∞ t .Therefore, ( ) F t attains its global maximum at the positive level in ( ) t t , 1 2 and a local minimum at the negative level at ( ) t 0, 1 .Moreover,  .Moreover, .For Inequality (3.6), we have Hence, the and a global maximum t u,2 at the positive level on ( ( ‖ ‖ ) ( ‖ ‖ )) 1 .Moreover, ( ) t Ψ u α has no other critical points.In fact, ( )( )  ; hence, we can deduce that if . So far, we have checked that items (i)-(iii) are true.To prove Conclusion (iv), we study the , then by the implicit function theorem, we obtained that → ∀ ∈ u t u S , u c ,1 is of class C 1 .By the same way, we can obtain that → Then, by Lemma 3.3, we can obtain the following conclusion.
, and satisfies for ρ small enough.
Proof.By (3.6) and (3.7) for any ∈ u M t1 , we have . According to the Brézis-Lieb Lemma and Sobolev embedding theorem, we have that Furthermore, for every 2 , there exists and u is a solution for (1.1) with some < λ 0. By the Pohozaev identity, we obtain ( ) = P u 0 α , and combined with Inequality (1.22) we have where we have used the lower semi-continuity of the , we have We set , it is easy to check that ( ) h t has a unique global minimum at and Therefore, we have that which is a contradiction with < m 0 c α , . Hence, the second alternative holds that, up to a sequence, → + u u n α strongly in ( ) H s N .In addition, for some , which implies that + u α is a ground state solution for | J α Sc by Lemma 3.4.Finally, we show that any ground state is a local minimizer of | J α Sc .Assume that u is a critical point for Then, the proof is completed.□ Lemma 3.5.There holds Proof.From Lemma 3.2, we know that ( ) F t has a unique global maximum at the positive level we denote by t max .Hence, for every ∈ − u c α , , we can find τ u such that ‖ ‖ ⋆ = τ u t u max and , could be chosen arbitrarily, we infer to Proof.It is well known that the best Sobolev constant S is achieved at We introduce a cut-off function , and let for > ε 0.Then, by [51], we have (3.10)By Lemma 4.5 [23], we have By the analogous arguments as Lemma 4.5 of [23], we have (3.12) (3.13) And by the proof of Lemma 5.3 in [58], we have , where + u α is given in Lemma 3.4, then (3.15) ) with (3.18) Now, we choose a suitable τ that satisfies . By Lemma 3.3, there exists , .Therefore, Next, we show that , and note that we see that there exists > θ 0 0 large enough, such that for Therefore, we only need to estimate ( ) we have the following assertions: Normalized solutions for a critical fractional Choquard equation  21 for < < s N s 2 4, and for ≥ N s 4 .For notational simplicity, denote by with < λ 0. Combining ( ) By a direct calculus, we obtain where We use the fact that , for any , 0, and we can deduce that By direct calculation, we have Normalized solutions for a critical fractional Choquard equation  23 .
N s 2 2 Using (3.37), arguing as we have done for (3.36), we can derive that where we have used the fact that From (3.37)-(3.39),we infer that Normalized solutions for a critical fractional Choquard equation  for > q 2. In conclusion, Hence, we complete the proof.
Now, we define the function q γ 2 1 q s , By Lemma 3.3, there exists ( ) , and ± t c exists and Moreover, if > b c and Proof.The proof mainly refers to [56].We define  , and ∈ ± u c α , , using the Taylor expansion, we can deduce that q s t b s μ s q s μ s q s μ q s q s μ s q s μ s q s μ q s q s μ s q s q s μ q q q s μ q q μ s μ , , Therefore, , , there holds for every ∈ u S c , which implies that = ∅ + c,0 . If there exists some ∈ u c,0 0 , then we have , and we have Combining with the aforementioned two inequalities and (3.41), we have Normalized solutions for a critical fractional Choquard equation  27  Then, by (1.6), we have , then there exists ρ 0 such that ⋆ ∈ − ρ u c α 0 0 ¯, .We want to prove ≥ t ρ n 0 , up to a subsequence.Otherwise, there is a subsequence still written as { } t n , which satisfies < t ρ n 0 for all n.For ∈ − u n c α , , by Lemmas 3.3, 3.7, the nonlocal Brezis-Lieb lemma, and the fact , we infer that which is a contradiction with Lemma 3.6.Without loss of generality, we can assume ≥ t ρ n 0 for all n.Further- more, we have In this section, we will prove Theorems 1.3 and 1.4.The method used mainly stems from [53].Note that the change in q reduces the difficulty of the proof.We still have the same thing that c α , is a smooth manifold of codimension 1 in S .
c Compared with the previous section, it is found that the geometry structure of and the most significant is the number of critical points of ( ) s Ψ .

u α
We first prove the following lemma.Proof.Assume, by contradiction, that there exists a ∈ u ¯c α , 0 , which implies that and by ( ) = P u ¯0 α , we obtain . Hence, from (4.1), we have , and so = u ¯0, which is a contradiction with ∈ u S c .
which implies that the left-hand side of (4.
Normalized solutions for a critical fractional Choquard equation  29 From Inequality (1.22), if By Proposition 2.2, monotonicity, and convexity ( ) t Ψ u α , we can obtain the existence and uniqueness of t u,2 which is a strict maximum point at the positive level.Therefore, Ψ u α has a global maximum point t u,2 at the positive level.In addition, since , then for every ∈ u c α , , by Inequalities (1.15) and (1.22), we have 2 , then for every ∈ u c α , , using Inequalities (1.15) and (1.22), we infer to . Moreover, by the fact that ( ) = P u 0 α , we can derive Therefore, by ( ) = P u 0 α and the aforementioned inequality, we obtain , by Inequalities (1.15) and (1.22), we have that both for k small enough.
2 , then for all ∈ u c α , and ∈ u M k with k small enough, we have * * * * q s q s μ s μ s q s q s μ s μ s .15)and (1.22).Similarly, we can choose k small enough such that .  .α α v q s qγ st μ q q st μ q s qγ st μ q q st μ α , Combining the aforementioned two cases, we infer that = m m c α r c α , , , .□ To prove Theorems 1.3 and 1.4, we need to introduce some preliminary results from [21].
from which, we infer that .

2 2
is the fractional Hardy-Littlewood-Sobolev critical exponent.Under the L 2 -subcri- tical perturbation ( | | )| | − α obtain the existence of normalized ground states and mountain-pass-type solutions.Meanwhile, for the L 2 -critical and L 2 -supercritical cases

s
is the fractional Sobolev exponent.From Propositions 3.4 and 3.6 in[46], we have that

1 2 ,
then (i) J α restricted to S c has another critical point ∈ The solution −

2 impliesProposition 3 . 1 .
that a and b can be solved easily.Consequently, the surjectivity is proved, and c α , is a smooth manifold of codimension 2 in ( )H s N .□Suppose the conditions of Theorem 1.1 are satisfied, then c α , is a smooth manifold of codimension 1 in S c .Moreover, if ∈ u c α, is a critical point for J α constrained on c α , , then u is a critical point for | J α Sc .Proof.From Lemma 3.1, we know that = ∅ c α , 0

From ( 3 .
21)-(3.23) and (3.25) we deduce that Furthermore, it is easy to check that and (3.40)  holds by applying the implicit function theorem.For (

, * 1
Proof.A direct calculation shows that Ψ u 0 has a unique maximum point at t u,0 such that

4 L 2 -
critical and supercritical case , is a smooth manifold of codimension 1 in ( ) H s N .

□ 31 Lemma
In order to recover the compactness of the Palais-Smale sequence by applying Proposition 2.3, we need to consider is the subset of the radial functions in S c .Normalized solutions for a critical fractional Choquard equation  the symmetric decreasing rearrangement of modulus u, which lies in S c r , .Then, we have

From Lemma 3 . 8 , we know that Ψ v 0 ε has a unique maximum point t v , 2
* 1 qs , which means that ( ) F t has only a global strict maximum at the positive level and a local strict minimum at the negative level, and no other critical points.□Fromthe aforementioned analysis, we can derive the following lemma, which is important to prove Theorem 1.2.
The maps → u t u,1 and → u t u,2 for ∀ ∈ u S c are of class C 1 .
α .(iv) min .Normalized solutions for a critical fractional Choquard equation  17 n t 1 .Obviously, v n are all real-valued, non-negative, radially sym- metric, and decreasing in | | = r x as well as a Palais-Smale sequence for | [56]osition 2.3and the Pohozaev identity to prove the compactness of the Palais-Smale sequence in Theorem 1.1 and the following Theorems 1.3 and 1.4 is no longer applicable.To overcome this difficulty, we shall prove the compactness of the Palais-Smale sequence mainly by Lemma 3.7, and the method mainly due to[56].
[21]nition 4.1.Let B be a closed subset of X .We say that a class of compact subsets of X is a homotopy-stable family with extended boundary B if (i) each set in contains B; (ii) for any set A in and any Lemma 4.6.[21].Let φ be a C 1 -functional on a complete connected C 1 -Finsler manifold X and consider a