Normalized solutions for a fractional Schr\"{o}dinger-Poisson system with critical growth

In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb R}^3,\end{cases} \] having prescribed mass \[\int_{\mathbb R^3} |u|^2dx=a^2,\] where $ s, t \in (0, 1)$ satisfies $2s+2t>3, q\in(2,2^*_s), a>0$ and $\lambda,\mu>0$ parameters and $\alpha\in{\mathbb R}$ is an undetermined parameter. Under the $L^2$-subcritical perturbation $q\in (2, 2+\frac{4s}{3})$, we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. For the $L^2$-supercritical perturbation $q\in (2+\frac{4s}{3}, 2^*_s)$, by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions.


Introduction
In the last decade, the following time-dependent fractional Schrödinger-Poisson system (1.1) has attracted much attention, where Ψ : R × R 3 → C, s, t ∈ (0, 1), λ ∈ R. It is well-known that, the first equation in (1.1) was used by Laskin (see [17,18]) to extend the Feynman path integral, from Brownian-like to Lévy-like quantum mechanical paths.This class of fractional Schrödinger equations with a repulsive nonlocal Coulombic potential can be approximated by the Hartree-Fock equations to describe a quantum mechanical system of many particles; see, for example, [7,20,21], and [26,27] for more applied backgrounds on the fractional Laplacian.
|x − y| 3+2s dy, x ∈ R 3 , s ∈ (0, 1), and P.V. stands for the Cauchy principal value on the integral, and C s is a suitable normalization constant.
We note that, when α ∈ R is a fixed real number, there was a lot of attention in recent years on the system (1.2) for the existence and multiplicity of ground state solutions, bound state solutions and concentrating solutions, see for examples [34,36,37,39] and references therein.Especially, Zhang, do Ó and Squassina [39] considered the existence and asymptotical behaviors of positive solutions as λ → 0 + , for the fractional Schrödinger-Poisson system (−∆) s u + λφu = g(u), x ∈ R 3 , where λ > 0 and g may be subcritical or critical growth satisfying the Berestycki-Lions conditions.
In [31], Teng studied the existence of a nontrivial ground state solution for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent where µ ∈ R + is a parameter, 1 < q < 2 * s − 1, s, t ∈ (0, 1) with 2s + 2t > 3. The potential V satisfies some suitable hypotheses.By the monotonicity trick, concentration-compactness principe and a global compactness Lemma, the author establishes the existence of ground state solutions.Formally, system (1.1) with s = t = 1 can be regarded as the following classical Schrödinger-Poisson system −∆u which appears in semiconductor theory [26] and also describes the interaction of a charged particle with the electrostatic field in quantum mechanics.The literature on the Schrödinger-Poisson system in presence of a pure power nonlinearity is very rich, we refer to [34,36,38] and references therein.Alternatively, from a physical point of view, it is interesting to find solutions of (1.2) with prescribed L 2 -norms, α appearing as Lagrange multiplier.Solutions of this type are often referred to as normalized solutions.The occurrence of the L 2 -constraint renders several methods developed to deal with variational problems without constraints useless, and the L 2 -constraint induces a new critical exponent, the L 2 -critical exponent given by q := 2 + 4s 3 , and the number q can keep the mass invariant by the law of conservation of mass.Precisely for this reason, 2 + 4s 3 is called L 2 -critical exponent or mass critical exponent, which is the threshold exponent for many dynamical properties such as global existence, blow-up, stability or instability of ground states.In particular, it strongly influences the geometrical structure of the corresponding functional.Meanwhile, the appearance of the L 2 -constraint makes some classical methods, used to prove the boundedness of any Palais-Smale sequence for the unconstrained problem, difficult to implement.In [22], Li and Teng proved the existence of normalized solutions to the following fractional Schrödinger-Poisson system: where s ∈ (0, 1), 2s + 2t > 3, λ ∈ R and f ∈ C 1 (R, R) satisfies some general conditions which contain the case f (u) ∼ |u| q−2 u with q ∈ ( 4s+2t s+t , 2 + 4s 3 ) ∪ (2 + 4s 3 , 2 * s ), i.e., the nonlinearity f is L 2mass subcritical or L 2 -mass supercritical growth, but is Sobolev subcritical growth.In [37], Yang, Zhao, and Zhao showed the existence of infinitely many solutions (u, λ) to (1.3) with subcritical nonlinearity µ|u| q−2 u, by using the cohomological index theory.
We note that, when s = t = 1, problem (1.3), are related to the the following equation Recently, Jeanjean and Trung Le in [15] studied the existence of normalized solutions for (1.4) when γ > 0 and a > 0, both in the Sobolev subcritical case p ∈ (10/3, 6) and in the Sobolev critical case p = 6, they showed that there exists a c 1 > 0 such that, for any c ∈ (0, c 1 ), (1.4) admits two solutions u + c and u − c which can be characterized respectively as a local minima and as a mountain pass critical point of the associated energy functional restricted to the norm constraint.While in the case γ < 0, a > 0 and p = 6 the authors showed that (1.4) does not admit positive solutions.Bellazzini, Jeanjean and Luo [4] proved that for c > 0 sufficiently small, there exists a critical point which minimizes with prescribed L 2 -norms.In [14], Jeanjean and Luo studied the existence of minimizers for with L 2 -norm for (1.4), and they expressed a threshold value of c > 0 separating existence and nonexistence of minimizers.In [32], Wang and Qian established the existence of ground state and infinitely many radial solutions to (1.4) with a|u| p−2 u replaced by a general subcritical nonlinearity af (u), by constructing a particular bounded Palais-Smale sequence when γ < 0, a > 0. In [23], Li and Zhang studied the existence of positive normalized ground state solutions for a class of Schrödinger-Popp-Podolsky system.For more results on the existence and no-existence of normalized solutions of Schrödinger-Poisson systems, we refer to [2,3,5,6,12,14,15,24,35,37] and references therein.
After the above bibliography review we have found only two papers [22,37] considering the normalized solutions for the fractional Schrödinger-Poisson system by the prescribed mass approaches with the nonlinearity f (u), being Sobolev subcritical growth.
A natural question arises: How to obtain solutions to system (1.3) in presence of the nonlinear term f (u) = µ|u| q−2 u+|u| 2 * s −2 u, combining the Sobolev critical term with a subcritical perturbation?The main contribution of this paper is to give an affirmative answer to this question and fill this gap.To be specific, in the present paper we aim to study the following fractional Schrödinger-Poisson system where s, t ∈ (0, 1) satisfies 2s + 2t > 3, q ∈ (2, 2 * s ) and α ∈ R is an undetermined parameter, µ, λ > 0 are parameters.For this purpose, applying the reduction argument introduced in [39], system (1.5) is equivalent to the following single equation where .
We shall look for solutions to (1.5)-(1.6),as a critical points of the action functional restricted on the set with α being the Lagrange multipliers, Clearly, each critical point u a ∈ S a of I µ | Sa , corresponds a Lagrange multiplier α ∈ R such that (u a , α) solves (1.7).In particular, if u a ∈ S a is a minimizer of problem then there exists α ∈ R as a Lagrange multiplier and then (u a , α) is a weak solution of (1.7).As far as we know, there is no result about the existence of normalized solutions for Schrödinger-Poisson system with a critical term in the current literature.For this aim, we shall focus our attention on the existence, asymptotic and multiplicity of normalized solutions for problem (1.5)-(1.6).
Finally, we present an existence result of normalized solutions under the L 2 -supercritical perturbation, when parameter µ > 0 is large.
3 improve and complement the main results in [31,39] in the sense that, we are concerned with the normalized solutions.
(ii) Our studies improve and fill in gaps of the main works of [22,30,37], since we consider the existence of normalized solutions to (1.5)-(1.6)with Sobolev critical growth.
2.1.Remarks on the proofs.We give some comments on the proof for the main results above.Since the critical terms |u| 2 * s −2 u is L 2 -supercritical, the functional I µ is always unbounded from below on S a , and this makes it difficulty to deal with existence of normalized solutions on the L 2 -constraint.One of the main difficulties that one has to face in such context is the analysis of the convergence of constrained Palais-Smale sequences: In fact, the critical growth term in the equation makes the bounded (PS) sequences possibly not convergent; moreover, the Sobolev critical term |u| 2 * s −2 u and nonlocal convolution term λφ t u u, makes it more complicated to estimate the critical value of mountain pass, and one has to consider how the interaction between the nonlocal term and the nonlinear term, and the energy balance between these competing terms needs to be controlled through moderate adjustments of parameter λ > 0. Another of difficulty is that sequences of approximated Lagrange multipliers have to be controlled, since α is not prescribed; and moreover, weak limits of Palais-Smale sequences could leave the constraint, since the embeddings ) are not compact.To overcome these difficulties, we employ Jeanjean's theory [13] by showing that the mountain pass geometry of I µ | Sa allows to construct a Palais-Smale sequence of functions satisfying the Pohozaev identity.This gives boundedness, which is the first step in proving strong H s -convergence.As naturally expected, the presence of the Sobolev critical term in (1.5) further complicates the study of the convergence of Palais-Smale sequences.To overcome the loss of compactness caused by the critical growth, we shall employ the concentration-compactness principle, mountain pass theorem and energy estimation to obtain the existence of normalized ground states of (1.5), by showing that, suitably combining some of the main ideas from [28,29], compactness can be restored in the present setting.
Finally, let us sketch the ideas and methods used along this paper to obtain our main results.For the L 2 -subcritical perturbation: q ∈ (2, 2 + 4s 3 ), it is difficult to get the boundedness of the (PS) sequence by the idea of [13].To get over this difficulty, we use the truncation technique; to restore the loss of compactness of the (PS) sequence caused by the critical growth, we apply for the concentration-compactness principle; and to obtain the multiplicity of normalized solutions of (1.5)-(1.6),we employ the genus theory.For the L 2 -supercritical perturbation: q ∈ (2 + 4s 3 , 2 * s ), we use the Pohozaev manifold and mountain pass theorem to prove the existence of positive ground state solutions for system (1.5)-(1.6)when µ > 0 small.While if the parameter µ > 0 is large, we employ a fiber map and the concentration-compactness principle to prove that the (PS) sequence is strongly convergent, to obtain a normalized solution of (1.5)-(1.6).
2.2.Paper outline.This paper is organized as follows.
• Section 2 provides the main results, and Section 3 presents some preliminary results that will be used frequently in the sequel.
Notations.In the sequel of this paper, we denote by C, C i > 0 different positive constants whose values may vary from line to line and are not essential to the problem.We denote by L p = L p (R 3 ) with 1 < p ≤ ∞ the Lebesgue space with the standard norm u p = R 3 |u| p dx 1/p .

Preliminary stuff
In this section, we first give the functional space setting, and sketch the fractional order Sobolev spaces [27].We recall that, for any s ∈ (0, 1), the nature functions space associated with (−∆) s is H := H s (R 3 ) which is a Hilbert space equipped with the inner product and norm, respectively given by The homogeneous fractional Sobolev space D s,2 (R 3 ) is defined by where 2 * s = 6/(3 − 2s) is the critical exponent.From Proposition 3.4 and 3.6 in [27] we have The best fractional Sobolev constant S is defined as The work space H s rad (R 3 ) is defined by The following two inequalities play an important role in the proof of our main results.Proposition 3.1.(Hardy-Littlewood-Sobolev inequality [20]) Let l, r > 1 and 0 < µ < N be such that 1  r Then there exists a constant C(N, µ, r, l) > 0 such that We recall the fractional Gagliardo-Nirenberg inequality.Lemma 3.2.( [11]) Let 0 < s < 1, and p ∈ (2, 2 * s ).Then there exists a constant C(p, s) = S − δp,s where δ p,s = 3(p − 2)/2ps.
Lemma 3.5.Let u ∈ H s (R N ) be a weak solution of (1.7), then we can construct the following Pohozaev manifold P a = {u ∈ S a : P µ (u) = 0}, where Proof.From Proposition 3.4, we know that u satisfies the Phohzaev identity as follows Moreover, since u is the weak solution of system (1.7), we have Combining with (3.4) and (3.5), we get which finishes the proof.
Finally, we state the following well-known embedding result.

Proof of Theorem 2.1
In this section, we aim to show the multiplicity of normalized solutions to (1.5)-(1.6).To begin with, we recall the definition of a genus.Let X be a Banach space and let A be a subset of X.The set A is said to be symmetric if u ∈ A implies that −u ∈ A. We denote the set Σ := {A ⊂ X \ {0} : A is closed and symmetric with respect to the origin}.
In order to overcome the loss of compactness of the (PS) sequences, we need to apply for the following concentration-compactness principle.Lemma 4.1.( [40]) Let {u n } be a bounded sequence in D s,2 (R 3 ) converging weakly and a.e. to some u ∈ D s,2 (R 3 ).We have that |(−∆) s ⇀ ζ in the sense of measures.Then, there exist some at most a countable set J, a family of points {z j } j∈J ⊂ R 3 , and families of positive numbers {ζ j } j∈J and {ω j } j∈J such that Then it follows that For u ∈ S r,a , in view of Lemma 3.2, and the Sobolev inequality, one has that where Recalling that 2 < q < 2 + 4s 3 , we get that qδ q,s < 2, and there exists β > 0 such that, if µa q(1−δq,s) ≤ β, the function g attains its positive local maximum.More precisely, there exist two constants 0 Let τ : R + → [0, 1] be a nonincreasing and C ∞ function satisfying In the sequel, let us consider the truncated functional For u ∈ S r,a , again by Lemma 3.2, and the Sobolev inequality, it is easy to see that where Then, by the definition of τ (•), when a ∈ (0, ( β µ ) 1 q(1−δq,s) ), we have In what follows, we always assume that a ∈ (0, ( β µ ) 1 q(1−δq,s ) ).Without loss of generality, in the sequel, we may assume that Lemma 4.3.The functional I µ,τ has the following characteristics: 1 > 0 large.Proof.We can obtain conclusions (i) and (ii) by a standard argument.To prove item (iii), let {u n } be a (P S) c sequence of I µ,τ restricted to S r,a with c < 0. By (ii), we see that (−∆) for large n, and thus {u n } is a (P S) c sequence of I| Sr,a with c < 0; i.e., I(u n ) → c < 0 and Moreover, we have that u ≡ 0. Indeed, assume by contradiction that, u ≡ 0, then lim n→∞ R 3 |u n | q dx = 0. From (4.8) and the definition of I µ,τ , we infer that which is absurd.On the other hand, setting the function Θ(v) : Then, by Proposition 5.12 in [33], there exists α n ∈ R such that and if we choose ϕ = u n , we get (4.12)(−∆) From (4.12), and the boundedness of {u n } in D s,2 (R 3 ), we can deduce that {α n } is bounded in R.
Then we can assume that, up to a subsequence, α n → α for some α ∈ R.Then, by (4.11), we can derive that u solves the following equation Indeed, for any ϕ ∈ H s rad (R 3 ), it follows by ), and u n (x) → u(x) a.e. on R 3 , we obtain that and so, Thus, we have Therefore, u solves equation (4.13).
In the sequel, by the concentration-compactness principle, we can prove that (4.15) In fact, since (−∆) ) or there exists a (at most countable) set of distinct points {x j } j∈J ⊂ R 3 and positive numbers {ζ j } j∈J such that Moreover, there exist some at most a countable set J ⊆ N, a corresponding set of distinct points {x j } j∈J ⊂ R 3 , and two sets of positive numbers {ζ j } j∈J and {ω j } j∈J such that items (4.1)-( 4.3) holds.Now, assume that J = ∅.We split the proof into three steps.
Step 1.We prove that ω j = ζ j , where ω j , and By the boundedness of {u n } in H s rad (R 3 ), we have that {ϕ ρ u n } is also bounded in H s rad (R 3 ).Thus, one has that It is easy to check that where For T 1 , by (4.16), we obtain From Hölder's inequality, we have .
Analogously to the proof of Lemma 3.4 in [40], we obtain and lim Again by (4.16), we have By the definition of ϕ ρ , and the absolute continuity of the Lebesgue integral, one has that Thus, by Proposition 3.1 and Lemma 3.6, we have Summing up, from (4.17)-(4.19)and (4.21), taking the limit as n → ∞, and then the limit as ρ → 0, we arrive at Using again the boundedness of {u n } and {u n ψ R } in H s rad (R 3 ), we have It is easy to derive that where |x − y| 3+2s dxdy and For T 3 , by (4.16) and Lemma 4.2, we infer to By virtue of Hölder's inequality, we get .
Combining the above proof, we conclude that lim By Lemma 4.2, we have Analogous the proof of Lemma 3.3 in [40], we infer to Moreover, we can obtain Summing up, from (4.24)-(4.27),taking the limit as n → ∞, and then the limit as R → ∞, we have Step 3. We claim that ζ j = 0 for any j ∈ J and ζ ∞ = 0. Suppose by contradiction that, there exists j 0 ∈ J such that Step 1, Step 2, and Lemmas 4.1, 4.2 imply that (4.28) and Consequently, we get . If the former case occurs, we have (4.30) Taking the limit ρ → 0 in the last inequality, we get ) which contradicts (4.9).If the last case happens, we have Taking the limits n → ∞ and R → ∞ in (4.31), we infer to ) which also contradicts (4.9).Therefore, ζ j = 0 for any j ∈ J and ζ ∞ = 0.As a result, by Lemma 4.1, we obtain that u n → u in L 2 * s loc (R 3 ); while by Lemma 4.2, we know that u n → u in L 2 * s (R 3 ).Now, we prove there exists µ * 1 > 0 independently on n ∈ N such that if µ > µ * 1 , the Lagrange multiplier α < 0 in (4.13).Indeed, note that {u n } ⊂ S r,s and (−∆) s 2 u n 2 ≤ R 1 , as can be seen from the previous proof of this lemma, and (3.2)-(3.3)that, there exists Q 1 > 0 independently on n, such that where By (4.32)-(4.34)we have Recall by (4.13) and its Pohozaev identity P µ (u) = 0, we infer to Thus, from (4.36), we infer to lim n→+∞ α n = α < 0. Hence, taking into account (4.12), we derive (4.37) Since α < 0 for µ > µ * 1 large, we obtain by Fatou's Lemma, that is, Thus, by (4.37) we have Theerfore, u n → u in H s rad (R 3 ) and u 2 = a.The proof is complete.
For ε > 0, we introduce the set By the fact that I µ,τ (u) is continuous and even on H s rad (R 3 ), I −ε µ,τ is closed and symmetric.
Lemma 4.4.For any fixed k ∈ N , there exists The proof of Lemma 4.4 is similar to Lemma 3.2 in [1], so we omit it here.
In the sequel, we define the set and by Lemma 4.3-(ii), we know that for all k ∈ N. To prove Theorem 2.1, we introduce the critical value, we define Then, we can derive the following conclusion: Especially, I µ,τ (u) admits at least ℓ + 1 nontrivial critical points.
Proof.For ε > 0, it is easy to check that I −ε µ,τ ∈ Σ.For any fixed k ∈ N, by Lemma 4.4, there exists Then, by Lemma 4.3-(iii), I µ,τ (u) satisfies the (P S) c -condition at the level c < 0. So, K c is a compact set.By Theorem 2.1 in [1], or Theorem 2.1 in [16], we know that the restricted functional I µ,τ | Sa possesses at least ℓ + 1 nontrivial critical points.
Proof of Theorem 2.1.
, we see that the critical points of I µ,τ (u) found in Lemma 4.5 are the critical points of I µ , which completes the proof.

Proof of Theorem 2.2
From Lemma 3.5, we see that any critical point of I µ | Sa belongs to P a .Consequently, the properties of the manifold P a have relation to the mini-max structure of I µ | Sa .For u ∈ S a and t ∈ R, we introduce the transformation (e.g.[29]): It is easy to check that the dilations preserve the L 2 -norm such that θ ⋆ u ∈ S a , by direct calculation, one has Notice that by q > 2, we infer to Hence, item (i) follows.Using 2s + 2t > 3, it is easy to obtain that 3(2 * s −2) 2 > 3 − 2t, and conclusion (ii) holds.Lemma 5.2.There exist K = K a > 0 and a > 0 such that for all 0 < a < a, where Proof.By Lemma 3.2, we have for any q ∈ (2, 2 * s ), that (5.6) .
By Lemma 5.2, we can deduce the following Corollary 5.1.Let K a , a be given in Lemma 5.2, and u ∈ S r,a with u 2 ≤ K a , then I µ (u) > 0. Furthermore, we have Proof.As in the proof of Lemma 5.2, we have that
Next, we show the existence of the (P S) cµ(a) -sequence for I(u, θ) on S r,a × R ⊂ H.It is obtained by a standard argument using Ekeland's variational principle and constructing pseudo-gradient flow, see Proposition 2.2 [13].
It follows from the above proposition, we can obtain a special (P S) cµ(a) -sequence for I µ (u) on S r,a ⊂ H s (R 3 ).Proposition 5.5.Under the assumption 2 + 4s 3 < q < 2 * s , there exists a sequence {u n } ⊂ S r,a such that (1) It follows from Proposition 5.4 that, there exists a sequence {(v n , θ n )} ⊂ S r,a × R such that as n → ∞, one has (5.9) and by (5.9), item (1) holds.To prove conclusion (2), we utilize which implies item (2) by (5.10).To show item (3), we set z n ∈ T un .Then, Denote by z n (x) = e − 3s 2 z n (e −θn x), then we get It is easy to check that Therefore, we see that ( z n , 0) ∈ T (vn,θn) .On the other hand, where the last inequality follows by θ n → 0. Consequently, we conclude item (3).
Remark 5.1 From Propositions 5.4,5.5, we know that u n := θ n ⋆ v n ⊂ S r,a is a (PS) sequence for I µ with the level c µ (a), that is

and
(5.34) a 2 * s −2 , under suitable conditions.To this aim, we distinguish the following three subcases.
In what follows we focus on an upper estimate of max θ∈R Ψ µ vε (θ).We split the argument into two steps.
Lemma 5.8.Let {u n } be the (P S) sequence in S r,a at level c µ (a), with c µ (a) < s 3 S 3 2s , assume that u n ⇀ u, then, u ≡ 0.
Lemma 5.9.Let {u n } be the (P S) sequence in S r,a at level c µ (a), with c µ (a) < s 3 S 3 2s , assume that P µ (u n ) → 0 when n → ∞, and λ < λ * 1 small.Then one of the following alternatives holds: (i) either going to a subsequence u n ⇀ u weakly in H s rad (R 3 ), but not strongly, where u ≡ 0 is a solution to where α n → α < 0, and ; (ii) or passing to a subsequence u n → u strongly in H s rad (R 3 ), I µ (u) = c µ (a) and u is a solution of (1.5)-(1.6)for some α < 0.
Proof.By Lemma 5.6, we have that {u n } ⊂ S r,a is a bounded (P S) sequence for I µ in H s rad (R 3 ), and so u n ⇀ u in H s rad (R 3 ) for some u.By the Lagrange multiplier principle, there exists for any ϕ ∈ H s rad (R 3 ).Moreover, one has lim n→∞ α n = α < 0. Letting n → ∞ in (5.64), we have which implies that u solves the equation and we have the Pohozȃev identity P µ (u) = 0. Let v n = u n − u, then v n ⇀ 0 in H s rad (R 3 ).According to Brezis-Lieb lemma [33] and Lemma 3.3, one has (5.66) (−∆) , and (5.67) .
By P µ (u) = 0, we have Passing to a subsequence, we may assume that (5.69) lim Then, it follows from Sobolev's inequality that ℓ ≥ Sℓ (5.70) which means that item (i) holds.
), it remains only to prove that u n → u in L 2 (R 3 ).Fix ψ = u n − u as a test function in (5.64), and u n − u as a test function of (5.65), we deduce that (5.71) Passing the limit in (5.71) as n → ∞, we have and then u n → u in L 2 (R 3 ).Therefore, item (ii) holds.Now, we are ready to complete the proof of Theorem 2.2.

Proof of Theorem 2.3
In this section, we deal with the L 2 -supercritical case 2 + 4s 3 < q < 2 * s , when parameter µ > 0 large.In view of 3(q−2) 2s > 2, the truncated functional I µ,τ defined in Section 4 is still unbounded from below on S r,a , and the truncation technique can not be applied to study problem (1.5)-(1.6).
Proof.From Lemma 5.6, we know that {u n } is bounded in H s rad (R 3 ), and by Lemma 3.6, up to a subsequence, there exists u ∈ H s rad (R 3 ) such that u n ⇀ u weakly in H s rad (R 3 ), u n → u strongly in L t (R 3 ), for t ∈ (2, 2 * s ), u n → u a.e. on R 3 .In view of 2 + 4s 3 < q < 2 * s , and Lemmas 3.3, 3.6, then (6.9)