Fractional Laplacian system involving doubly critical nonlinearities in

In this article, we are interested in a fractional Laplacian system in RN , which involves critical Sobolev-type nonlinearities and critical Hardy–Sobolev-type nonlinearities. By using variational methods, we investigate the extremals of the corresponding best fractional Hardy–Sobolev constant and establish the existence of solutions. To our best knowledge, our main results are new in the study of the fractional Laplacian system.


Introduction and main result
In this article, we are concerned with the existence of solutions for the following fractional Laplacian system in R N : where I a (x) = Γ( N−2 2 ) 2 a π N/2 Γ( a where F u denotes the Fourier transform of u.Weak solutions of (1.1) will be found in the space H = Ḣs (R N ) × Ḣs (R N ), where Ḣs (R N ) is defined as the completion of C ∞ 0 (R N ) under the norm Therefore, for s > 0, we have By a (weak) solution (u, v) of problem (1.1), we mean that (u, v) ∈ H satisfies In recent years, much attention has been paid to fractional and non-local operators.More precisely, this type of operators arises in a quite natural way in many different applications, such as, finance, physics, fluid dynamics, population dynamics, image processing, minimal surfaces and game theory, see [4,11] and the references therein.In particular, there are some remarkable mathematical models involving the fractional Laplacian, such as, the fractional Schr ödinger equation (see [18,31]), the fractional Kirchhoff equation (see [1,13,24,25]), the fractional porous medium equation (see [5]) and so on.
Problems with one nonlinearity or two nonlinearities involving the Laplacian and the fractional Laplacian have been studied by many authors.For example, we refer, in bounded domains to [14,20,21,27,28,30], while in the entire space to [12,16,22].In [8], Filippucci, Pucci and Robert proved that there exists a positive solution for a p-Laplacian problem with critical Sobolev and Hardy-Sobolev terms.In [15], Fiscella, Pucci and Saldi dealt with the existence of nontrivial nonnegative solutions of Schr ödinger-Hardy systems driven by two possibly different fractional ℘-Laplacian operators, also including critical nonlinear terms, where the nonlinearities do not necessarily satisfy the Ambrosetti-Rabinowitz condition.It is natural to consider the concentration-compactness principle for critical problems.However, due to the nonlocal feature of the fractional Laplacian, it is difficult to use the concentrationcompactness principle directly, since one needs to estimate commutators of the fractional Laplacian and smooth functions.A natural strategy, which is named by the s-harmonic extension, is to transform the nonlocal problem into a local problem, as Caffarelli and Silvestre performed in [3].Since that, many interesting results in the classical elliptic problems have been extended to the setting of the fractional Laplacian.For example, Ghoussoub and Shakerian in [16] combined the s-harmonic extension with the concentration-compactness principle to investigate the existence of solutions for a doubly critical problem involving the fractional Laplacian.It is worthy pointing out that Yang and Wu in [33] showed the existence of solutions for problem (1.1) with η = 0, by using the elementary approach without the use of the concentration-compactness principle or the extension argument of Caffarelli and Silvestre in [3].
In the doubly critical case, two critical nonlinearities interact to each other.There is an asymptotic competition between the energy carried by the two critical nonlinearities.Obviously, the combination of the two critical exponents induces more difficulties.When one critical exponent is only involved, there are solutions to the corresponding equations: in general, these solutions are radially symmetric with respect to the origin of the domain and are explicit, see for instance [23] for the details.However, very few information have been known in our setting, especially for system, here we just refer the reader to an interesting literature [12].
In this paper, we are interested in the existence of solutions for system (1.1) involving doubly critical exponents, by using a refinement of the Sobolev inequality which is related to the Morrey space.A measurable function u : R N → R belongs to the Morrey space L p,γ (R N ) with p ∈ [1, +∞) and γ ∈ (0, N], if and only if (1.4) By the H ölder inequality, we can verify that Moreover, here holds L p,γ (R N ) → L 1, γ p (R N ) provided that p ∈ (1, +∞) and γ ∈ (0, N).The following refinement of Hardy-Sobolev inequalities were proved in [19] and [32].Proposition 1.1.([32, Theorem 1.1]).For any 0 < b < 2s < N, there exists C > 0 such that for θ and r satisfying max{ for any u ∈ Ḣs (R N ).
In the present paper, we work in the product space H = Ḣs (R N ) × Ḣs (R N ) be the Cartesian product of two Hilbert spaces, which is a reflexive Banach space endowed with the norm where Solutions of (1.1) are equivalent to a nonzero critical points of the functional which is defined on H, and I ∈ C 1 (H, R).We say a pair of functions (u, v) ∈ H is called to be a solution of (1.1) if If (u, v) = (u, 0) or (u, v) = (0, v), we say that they are the semi-nontrivial solution.In this case, system can be seen as a singular equation, that is η = 0, see [33] for the details.
The main result of this paper can be concluded in the following theorem.
, then problem (1.1) possesses at least one nontrivial solution in H.

Remark 1.3.
To the best of our knowledge, Theorem 1.2 is new in the study of the fractional Laplacian system involving doubly critical nonlinearities in the whole space.We mainly follow the idea of [33] to prove our main result.This paper is organized as follows.In Section 2, some preliminary results are presented.In Section 3, the extremals of the corresponding best fractional Hardy-Sobolev constant are achieved.In Section 4, we give the proof of Theorem 1.2.
Throughout this paper, we will use the following notations: 1) is a generic infinitesimal value.We always denote positive constants as C for convenience.

Preliminaries
In this section, we recall the fractional Sobolev inequality.For N > 2s, the fractional Sobolev embedding Ḣs (R N ) → L 2N N−2s (R N ) was considered in [6,7].The continuity of this inclusion corresponds to the inequality The best constant S µ in (2.1) was computed (see Theorem 1.1 in [7]).Using the moving plane method for integral equations, Chen, Li and Ou in [6] classified the solutions of an integral equation, which is related to the problem The positive regular solutions of (2.2), which verify the equality in (2.1), are precisely given by On the other hand, the Hardy-Littlewood-Sobolev inequality yields and the equality in (2.4) holds if and only if u is given by (2.3).Thus, the exponent 2 h,a is critical in the sense that it is the limit exponent for the Sobolev-type inequality (2.4).Taking into account Proposition 1.1 and (2.4), we obtain the following inequality: for u ∈ Ḣs (R N ).

Minimizers of S s,b
In this section, we show that the best constant S s,b in our context can be achieved.Moreover, we investigate the intrinsic relation between S s,b and the best fractional Hardy-Sobolev constant with single equation.For µ > 0, we define Here we remark that µ * in (3.1) was showed in [34] that . Evidently, from (3.1) we have the fractional Hardy-Sobolev inequality and (3.2), we have Then, for 0 ≤ µ < µ * , we define the functional and Consider the minimization problem and The following result shows that for 0 ≤ µ < µ * , S s,a , S s,b are achieved.Proof.Here we only give the proof of process for S s,a being achieved.With minor changes, we can also get that S s,b is achieved by a pair of radially symmetric nonnegative functions.Let {(u n , v n )} n be a minimizing sequence of S s,a , that is as n → ∞ and Inequality in (2.5) enables us to find C > 0 such that and the Sobolev embedding Ḣs (R So we may find λ n > 0 and Similarly, we can get that where ṽn (x) = λ N−2s 2 n v n (λ n x).By simple computation, we can get I(u n , v n ) = I( ũn (x), ṽn (x)), and then {( ũn (x), ṽn (x))} n is also a minimizing sequence of S s,a .We can also show that {( ũn (x), ṽn (x))} n is bounded in H. Hence, we may assume We claim that { x n λ n } n is uniformly bounded in n.Indeed, for any 0 < κ < 2s, we observe, by the H ölder inequality, that .
By the rearrangement inequality, see [17,Theorem 3.4], we have Therefore, Now, suppose on the contrary, that Thus by the H ölder inequality, it follows that which contradicts (3.9).Hence, { x n λ n } n is uniformly bounded, and there exists R > 0 such that The compact Sobolev embedding Ḣs (R N ) → L 2 loc (R N ) implies that there exists ũ satisfing which means ũ ≡ 0. Similarly we can get ṽ ≡ 0. By a Brézis-Lieb-type lemma, see [19, Lemma 2.4], we obtain and ≥ S s,a ≥ S s,a = S s,a .
For any α, β > 1 with α + β = 2 * s,b , and 0 < µ < µ * , we define the following best Hardy-Sobolev-type constant: We may prove, as in [32] with minor changes, that Λ where τ min ≥ 0 is a minimal point of f (τ) and therefore a root of the equation Proof.We mimic the proof of Theorem 1.1 in [10].By the definition of f (τ) defined in (3.11), it follows that lim Thus min τ≥0 f (τ) must be achieved at τ min ≥ 0. Furthermore, direct calculation shows that there exists a positive constant C such that From the fact that f (τ min ) = 0, we deduce that τ min is a root of the following equation Suppose that {w n } n ⊂ Ḣs (R N ) is a minimizing sequence for Λ s,b .Let τ 1 , τ 2 > 0 to be chosen later.Taking u n = τ 1 w n and v n = τ 2 w n in (3.8), we have .
Choose τ 1 and τ 2 in (3.14) such that τ 2 τ 1 = τ min .Passing to the limit as n → ∞ we have On the other hand, let {(u n , v n )} n be a minimizing sequence of S s,b and define z n = τ n v n , where From the Young inequality it follows that Thus by (3.16) we have Consequently, As n → ∞, we have From (3.15) and (3.18) it follows that Thus, the proof is complete.

Proof of Theorem 1.2
In this section, we investigate the existence of solutions for problem (1.1).We first give some technical lemmas so that we can use the mountain pass lemma to seek critical points of problem (1.1).The Nehari manifold related to I is given by Then a minimizer of the minimization problem is a solution of problem (1.1).In order to establish the existence of solutions for problem (1.1), we set where  The proof of Lemma 4.1 is standard.The further details with minor changes can be derived as in the proof of [29,Theorem 4.2].
However, the following lemma reveals that c s < c * .
Proof.By Lemma 3.1, there exist minimizers (u and Obviously, max t≥0 We may verify that the function f a (t) attains its maximum at , and the function f b (t) attains its maximum at Thus it yields that  Indeed, if there exist t 1 > 0, t 2 > 0 such that and which yields a contradiction and hence the assertion follows.
Next we show that the functional I satisfies the geometrical conditions in the mountain pass lemma without the (PS) condition in [2].(i) There exist positive numbers ρ and α 0 such that I(u, v)| (u,v) H =ρ ≥ α 0 for all u, v ∈ Ḣs (R N ).
(ii) There exists e ∈ H with e > ρ such that I(e) < 0.
The following result implies the non-vanishing of (PS) c sequences. and which implies that {(u n , v n )} n is uniformly bounded in H. Now we give the proof by contradiction.Suppose s,a .

Lemma 3 . 1 .
If 0 ≤ µ < µ * , then S s,a and S s,b are achieved respectively by a pair of radially symmetric and nonnegative functions.