Normalized ground states for fractional Kirchho ﬀ equations with critical or supercritical nonlinearity

: The aim of this paper is to study the existence of ground states for a class of fractional Kirchho ﬀ type equations with critical or supercritical nonlinearity

The operator (− ) s can be seen as the infinitesimal generators of Lévy stable diffusion processes, see [1,2] for example. This operator appears in several areas such as biology, chemistry and physics (see [3][4][5][6]). Problem (1.1) is viewed as being nonlocal because of the appearance of the term b R 3 |(− ) s 2 u| 2 , which implies that Eq (1.1) is no longer a pointwise identity. This also results in lack of weak sequential continuity of the energy function associated to (1.1), so it make the study of (1.1) particularly interesting. Over the last decade, many mathematicians were particularly keen on the study of nonlinear equations involving nonlocal operators, we can look it up in [7][8][9][10][11][12][13][14] and the references therein.
It is well known that problem (1.1) arises from looking for the standing wave type solutions ϕ(x, t) = e −iλt u(x), λ ∈ R for the following time-dependent nonlinear fractional Kirchhoff equation where 0 < s < 1, i denotes the imaginary unit. The stationary case of (1.2) is the following equation When s = 1, Problem (1.3) becomes the Kirchhoff equation. In the past several years, the Kirchhoff type equations has been studied extensively by many researchers(see [15][16][17][18][19][20][21][22][23]). For all we know, the existence results to problem (1.1) have been mostly available for the case where p, q ∈ (2, 2 * s ) and λ is fixed and assigned. When a = 1, b = 0, s = 1 and µ = 0, i.e., for the Laplacian operator, Jeanjean's [24] was the first paper to prove existence of normalized solutions in purely L 2 -supercritical case. Li and Ye in [25] considered problem (1.1) with s = 1, µ = 0, N = 3, λ = −1, q ∈ (3, 6) and proved that (1.1) has at least one least energy solution by dealing with a constrained minimization problem on a manifold of H 1 (R 3 ), which is obtained by combining the Nehari manifold and the corresponding Pohozaev identity. Liu, Chen and Yang in [26] considered problem (1.1) with 2 < q < p < 2 * s and proved some existence results about the normalized solutions. However, there is few literature concerned about the normalized solutions for fractional Kirchhoff equation with critical or supercritical nonlinearity. With regard to the point, we attempt to study this kind of problem in this paper.
It is well known that the fractional order Sobolev space H := H s (R 3 ) can be defined as follows endowed with the norm and the inner product is uvdx.
According to [26], we know that Let H = H × R with the scalar product (·, ·) H = (·, ·) H + (·, ·) R and the corresponding norm and there exists a best constant S * s such that (1.4) The normalized weak solution for the problem (1.1) is obtained by looking for critical points of the following C 1 functional constrained on the L 2 -spheres in H: u c is called a ground state of (1.1) on S (c) if Since p ≥ 2 * s , the functional J µ is not well defined on H s (R 3 ) unless p = 2 * s . Moreover, we need to overcome the lack of compactness in studying critical and supercritical growth. Hence, we cannot directly use variational methods to prove the existence of normalized solutions. To overcome these difficulties, we use a new method, which came from [14,18]. The main idea of this method is to reduce the supercritical problem into a subcritical one. In comparison with previous works, this paper has several new features. Firstly, we consider the nonlinear term with supcritical growth. Secondly, we give the existence of normalized solution for the appropriate truncation problem of (1.1). Finally, the existence of a normalized ground state solution is obtained by Moser iteration method. The results in this paper extend the results in paper [4,24,26]. There have been no previous studies considering the existence of normalized ground state solutions for problem (1.1) involving supcritical growth to the best of our knowledge.
Our main result is the following: For any c > 0, there exists a µ * > 0 such that, problem (1.1) has a couple of solutions (u c , λ c ) ∈ H s (R 3 ) × R for any µ ∈ (0, µ * ]. Moreover, u c is a positive ground state, radially symmetric function and λ c < 0. Remark 1.2. When 6+8s 3 < q < 2 * s , J µ is not bounded from below on S (c), i.e., inf the minimization problem constrained on S (c) does not work. We try to look for a critical point with a minimax characterization. Although J µ has a mountain-pass geometry on S (c), the boundedness of the obtained Palais-Smale sequence is not yet clear. Motivated by [4], we try to construct an auxiliary map I µ , which on S (c)×R has the same type of geometric structure as J µ on S (c). Besides, the Palais-Smale sequence of I µ satisfies the additional condition, which is the key point to obtain the boundedness of the Palais-Smale sequence.

Preliminaries
In this section, we give a truncation argument in order to overcome the lack of compactness in studying critical and supercritical growth. Let M > 0 be a constant. For fixed c > 0, µ > 0, M > 0, we investigate the existence of ground state for the following truncation problem To investigate (2.1), we define the the energy functional E µ : H → R by and V(c) is the Pohozaev manifold defined in lemma 2.4. Next, we give some useful preliminary lemmas to prove Theorem 2.1. Lemma 2.1. [8] If α ∈ (2, 2 * s ), there exists an optimal constant C(s, α) such that for any u ∈ H, for t ∈ (2, 2 * s ). As in [4], we introduce the useful fiber map preserving the L 2 -norm, that is, (2.5) Define the auxiliary functional I : H → R by then we can obtain that I µ is a C 1 -functional. Lemma 2.3.
[13] The map (u, τ) ∈ H → τ u ∈ H is continuous. Similar to Lemma 2.1 in [4], we can easily get the following lemma. Lemma 2.4. Let (u, λ) ∈ S (c) × R be a weak solution of Eq (2.2). Then u belongs to the set Proof. For any u ∈ S (c) and τ ∈ R, we have It is easy to see that Lemma 2.5 holds.
Lemma 2.6. Let u ∈ S (c) be arbitrary fixed, then (1) For fixed u ∈ S (c), we can easily get the conclusions (1) and (2) from the facts and β r r ≥ β q q > 4s.

Characterization of mountain pass level
As in [4], firstly, we prove that E µ (u) has the mountain pass geometry on S (c) × R in the following lemma.
By the above inequalities, we can obtain that there exists k c > 0 sufficiently small such that Lemma 3.2 holds. Next, we need to construct the minimax characterization of I µ and E µ .
. Thus we can deduce that ρ c is independent of positive numbers µ, M and ρ c ≥ γ c,µ for any µ > 0.
For the proof of (iii), by the definition of I µ , we have On the other hand, for any ϕ with satisfying ϕ ∈ T u n , by using (2.3), we have we get (iii) if we could show thatφ ∈ T v n . In fact,φ ∈ T v n follows from the following equalities

Proof of Theorem 2.1
According to Lemma 3.4 and Lemma 3.2, there exist a Palais-Smale sequence {u n } ⊂ S (c) for E µ | S (c) at level γ c,µ > 0, and it satisfies P µ (u n ) → 0 as n → ∞. By applying the Lagrange multipliers rule there exists {λ n } ⊂ R such that  (1). As P µ (u n ) → 0, we have Thus, by (4.2) we deduce that Since 6+8s 3 < q < 2 * s and (2.7), it implies that s β q q < 1 4 and β r r ≥ β q q. According to (4.3), we can deduce the boundedness of R 3 |(− ) s 2 u n | 2 , thus {u n } is bounded in H. (2). According to Lemma 2.2, we know that the embedding H s r (R 3 ) → L t (R 3 ) is compact for t ∈ (2, 2 * s ), and we can deduce that there exists u c ∈ H s r (R 3 ) such that, up to a subsequence, u n u c weakly in H, u n → u c strongly in L q (R 3 ) for q ∈ ( 6+8s 3 , 2 * s ). since {u n } ⊂ S (c) is bounded in H. By (4.1), we obtain that Using the fact that the boundedness of {u n } in H and (4.2), we can deduce that {λ n } is bounded. Hence, up to a subsequence λ n → λ c ∈ R.
(3). We claim that u c 0. We assume by contradiction that u c ≡ 0, by (4.2) we deduce that Recalling that P µ (u n ) → 0, according to (4.3), we have E µ (u n ) → 0, which is a contradiction to the assumption that E µ (u n ) → γ c,µ 0. Now, since λ n → λ c and u n → u c 0 weakly in H, together with (4.1), we know (u c , λ c ) is a couple of solutions to (2.1). By the Pohozaev identity, we obtain Combining with the (4.4) for u c , we get Since 6+8s 3 < q < 2 * s and (4.5), there exists µ 1 > 0 such that λ c < 0 for µ ∈ (0, µ 1 ]. (4). Testing (4.1) and (2.1) with u n − u c , we can obtain that Using the strong L p convergence of u n , we infer that which, being λ c < 0, implies u n → u c strongly in H. Therefore, E µ (u n ) → E µ (u c ), as n → ∞. From Lemma 2.4 and Lemma 3.2, we easily obtain that u c is a ground state of (2.1) and E µ (u c ) = m c,µ .

Proof of main result
In this section, we devote to complete the proof of Theorem 1.1. From the truncation argument in Sections 2-4, we can see that if the ground state u c of (2.1) satisfy u c ∞ ≤ M. Then u c ∈ H is a ground state of (1.1).
Lemma 5.1. Let(u c , λ c ) be a couple of solutions of problem (2.1) for µ ∈ (0, µ 1 ], then there exists a constant K c > 0 independent of µ, M > 0 such that u c ≤ K c . Proof. By Theorem 2.1 and Lemma 2.4, it is easy to see that E µ (u c ) = γ c,µ and P µ (u c ) = 0, (5.1) It follows from (5.1) and Remark 3.3 that Consequently, there exists a constant K c > 0 independent of µ, M > 0 such that u c ≤ K c .
Proof. For convenience, we replace u c with u in the following. Let L > 0 and β > 1, we first define the following functions: where u L = min{u, L}. Since Υ is an increasing function,we have Let Φ(t) = 1 2 |t| 2 and Ψ(t) = t 0 (Υ (τ)) 1 2 dτ.Then,if x > y, by Cauchy-Schwarz inequality, we have The same arguments hold for x ≤ y. Therefore, By the definition of u L , it is easy to see that |uu 2(β−1) L | ≤ L 2(β−1) u and Υ(u) ∈ H. Taking Υ(u) as a test function in Eq (2.1), and let g µ,M (x, t) = |t| q−2 t + µd M (t), we obtain dxdy.