Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent

In this paper, we study the following fractional Kirchhoff equation with critical nonlinearity \begin{document}$ \Big(a+b\int_{\mathbb{R}^3}| (-\Delta)^{\frac{s}{2}} u|^2dx\Big) (-\Delta )^su+V(x) u = K(x)|u|^{2_s^*-2}u+\lambda g(x,u), \; \text{in}\; \mathbb{R}^3, $\end{document} where \begin{document}$ a,b>0 $\end{document} , \begin{document}$ \lambda>0 $\end{document} , \begin{document}$ (-\Delta )^s $\end{document} is the fractional Laplace operator with \begin{document}$ s\in(\frac{3}{4},1) $\end{document} and \begin{document}$ 2_s^* = \frac{6}{3-2s} $\end{document} , \begin{document}$ V,K $\end{document} and \begin{document}$ g $\end{document} are asymptotically periodic in \begin{document}$ x $\end{document} . The existence of a positive ground state solution is obtained by variational method.

where a, b > 0, λ > 0, (−∆) s is the fractional Laplace operator with s ∈ ( 3 4 , 1) and 2 * s = 6 3−2s , V, K and g are asymptotically periodic in x. The existence of a positive ground state solution is obtained by variational method.

1.
Introduction. This paper deals with the existence of positive ground state solution to the following asymptotically periodic fractional Kirchhoff equation involving critical Sobolev exponent where a, b > 0, λ > 0, (−∆) s is the fractional Laplace operator defined as (see [12,Lemma 3.2]) (1.2) Here, C 3,s is a positive depending on 3 and s.
When s = 1, problem (1.1) reduces to the following classical Kirchhoff equation which is related to the stationary analogue of the Kirchhoff equation Equation (1.4) was proposed by Kirchhoff [23] as a result of extending the wellknow D'Alembert wave equation. Equation (1.3) is called to be nonlocal due to the presence of the term R 3 |∇u| 2 dx, which makes the research of such problem very difficulty. It should be pointed out that nonlocal problem (1.3) also appears in many fields such as biological systems [3]. Problem (1.3) received a great deal of attention after Lions [27] introduced a functional analysis approach, see, for example, [1,15,16,24,39] and the references therein.
Recently, Fiscella and Valdinoci [17] firstly gave a very detailed introduction of the following generalized fractional Kirchhoff equations in physical background and their applications where Ω is a bounded domain in R N , M is a Kirchhoff function(which contains the case M (t) = a + bt). The nonlocal operator L K is defined by: where K : R N \ {0} → (0, +∞) is a measurable function and satisfies following assumptions: there exists θ > 0 and s ∈ (0, 1) such that θ|x| −(N +2s) ≤ K(x) ≤ θ −1 |x| −(N +2s) , ∀x ∈ R N \ {0}.
They showed the existence of non-negative solutions for equation (1.5) under suitable conditions of M and f . We also refer to [19,20,33,35,34,36] and the references therein for more results of nonlocal operator L K (in which covers the special case (−∆) s if K(y) = |y| −(N +2s) ). The general analysis methods for elliptic PDEs involving classical Laplacian operator −∆ can not be directly applied to problem (1.1) or (1.5) since fractional operator (−∆) s is nonlocal. In [7], Cafferelli and Silvestre developed a crucial tool for transferring the nonlocal equation (1.1) into a local problem. The additional details on the extension method from [7] can be found in [6,Chapter4.2]. Zhang, Xu and Zhang [42] obtained the existence of ground states and infinitely many geometrically distinct solutions for equation (1.1) with b = 0 when V and f are asymptotically periodic and periodic in x, respectively. A similar result for the critical case f (x, u) = |u| 2 * s −2 + λg(x, u) was established in [25]. For further details about problem (1.1) with b = 0, we refer to [5,11,14] and the references therein. In [4], Autuori, Fiscella and Pucci studied problem (1.1) involving a generalized Kirchhoff function, and they obtained the the existence and the asymptotic behavior of nonnegative solutions. Pucci, Xiang and Zhang [31] investigated a nonhomogeneous fractional p-Laplacian equation of Schrödinger-Kirchhoff type. We also make reference to [8] and the referencrs therein for more results of fractional Kirchhoff type involving generalized Kirchhoff function, in which contains the so-called degenerate case (when b = 0). Zhang et al. [43] used the s-harmonic extension technique in [7] and established the Pohožaev identity of (1.1), they obtained some results on the existence of ground state solutions for s ∈ [ 3 4 , 1) and proved the non-existence result for s ∈ (0, 3 4 ]. Here we refer to [2,9,10,26,29,30,32] and the references therein for more results of the fractional Kirchhoff type equations involving variational methods. We also refer to [13] for more results about nonlinear and nonlocal problems in the whole space. Inspired by the works formulated above, in the present paper, we obtain the existence of ground states to asymptotically periodic problem (1.1). However, according to our current knowledge, it seems that this problem was not studied in literatures before.
There are some difficulties in our problem. The first one is that the appearance of the Kirchhoff term. On the one hand, one does not know in general that for any . Thus, one cannot verify directly that I ε satisfies the (P S) ccondition. On the other hand, as said above, the fractional Laplacian operator (−∆) s is nonlocal. The combination of the Kirchhoff term and fractional Laplacian operator (−∆) s make the exact value of c (see (2.2)) very complicated, since the solvability of a fractional order algebra equation is still unknown (see, e.g., [28]). However, for (1.1) with ε = 1, this difficulty can be overcome (see e.g., [21,22]). In order to overcome this difficulty, we will use an idea from [28]. The second one is that we can not use the Nehari manifold in a standard way since g does not belongs to C 1 -class. To overcome the nondifferentiability of the Nehari manifold, we shall make use of the argument developed by Szulkin and Weth in [37]. Those difficulties stop us from using the standard variational methods. This paper is organized as follows. In section 2 we collect some necessary preliminary Lemmas which will be used later. In section 3, we prove that the existence of positive ground state solutions for asymptotically periodic problem (1.1).
2. Preliminary results. In this section, we firstly define the homogeneous fractional Sobolev space D s,2 (R 3 ) as follows , endowed with the natural norm From [12, Proposition 3.4 and Proposition 3.6], we know that It is well known that the embedding , and there exists a best constants S s > 0 such that which is called the fractional Sobolev critical exponent. Moreover, the embedding We consider the Hilbert space H s (R 3 ) endowed with one of the following norms: From the hypotheses (V ), the norms || · || and || · || p are equivalent to the standard and the norm The energy functional associated with (1.1), I : H s (R 3 ) → R is defined as Obviously, I is well-defined in u ∈ H s (R 3 ) and I ∈ C 1 (H s (R 3 ), R). Moreover, . Therefore, a critical point of I is a weak solution of (1.1). Let us define by N the Nehari manifold [41] associated to I, given by The least energy on N is defined by We firstly recall the following vanishing Lemma [14].

GUANGZE GU, XIANHUA TANG AND YOUPEI ZHANG
We need the following results to overcome the non-differentiability of N .
If the assumptions (V ), (K) and (g 1 ) − (g 4 ) hold, then the following statements hold: Moreover, which is contradiction. So h (t) = 0 for all t > t u . Moreover, it is easy to see that tu ∈ N if and only if t = t u by g (t) = t −1 I (tu), tu . This completes the proof of Lemma 2.2(i).
(ii) Since u ∈ S 1 and t u u ∈ N , we obtain this implies that t u ≥ τ > 0. If there is {w n } ⊂ W with t n := t wn → +∞, then there exists a w ∈ W such that w n → w in H s (R 3 ) since W is compact. Direct computations yield On the other hand, by v n := t n w n ∈ N and condition (g 3 ), we deduce that which yields a contradiction. Thus, the conclusion (ii) holds.
Hence, we can choose some ρ ∈ (0, 1] such that I(u) ≥ α > 0 with ||u|| = ρ. Proof. If this result does not hold, then there exist a sequence {u n } ⊂ N and a d > 0 such that u n → +∞ and I(u n ) ≤ d. Let v n = un un , going to a subsequence if necessary, we may assume that It is easy to see that ξ(z) is continuous function on R 3 . There exists a R > 0 such that It follows from the continuity of ξ and the compactness ofB R+1 (0) that there exists y n ∈B R+1 (0) such that ξ(y n ) = sup We claim that lim sup n→+∞ B1(yn) |v n | 2 * s dx > 0. Otherwise, lim n→+∞ B1(yn) |v n | 2 * s dx = 0.
By Lemma 3.4 in [40] we have and v n → 0 in L 2 * s (R 3 ). In view of the interpolation inequality, we get that as n → +∞, which is a contradiction. Thus, for large n, which yields a contradiction.

3−2s 4s
sT ) for ε small enough. Thus we completed the proof of Lemma 2.7.
3. Proof of Theorem 1.1. In this section we first study the following periodic problem associated to (1.1) The weak solutions of (3.1) are the critical points of the C 1 −functional I p : H s (R 3 ) → R given by

GUANGZE GU, XIANHUA TANG AND YOUPEI ZHANG
Moreover, for any u, v ∈ H s (R 3 ), we get The Nehari manifold associated to I p is defined by We define the number c p by c p := inf u∈Np I p (u), which is the ground energy corresponding to (3.1).
The following result is important for proofing Theorem 1.1, one also consult [42] or [25].
We may suppose Φ (w n ) → 0 by using the Ekeland variational principle. Then, by Lemma 2.6 we obtain that I (u n ) → 0 and I(u n ) = Φ(w n ) → c, where u n = m(w n ) ∈ N . Moreover, by Lemma 2.6, u n is bounded in H s (R 3 ). Then, up to a subsequence, Next, we verify that I (u) = 0. If u = 0, we get I (u) = 0. If u = 0, we claim that (3.5) Indeed, we may suppose that By the fact that I (u n ) → 0, u is a solution of the following equation Then GUANGZE GU, XIANHUA TANG AND YOUPEI ZHANG which yields that h 1 (t) > 0 for t > 0 small. Thus, there exists a t 0 ∈ (0, 1) such that h 1 (t 0 ) = I (t 0 u), t 0 u = 0, that is, t 0 u ∈ N . By (2.3) we have this contradiction implies that (3.5) holds true and hence I (u) = 0. We next continue our arguments by distinguishing the following two cases: u = 0 and u = 0 Case 1 u = 0. In this case, u ∈ N and I(u) ≥ c. By (2.3) we have which implies that I(u) ≤ c. Therefore, I(u) = c. Case 2 u = 0. That is {u n } is vanishing or non-vanishing. If {u n } is vanishing, i.g., g(x, u n )u n dx = 0 and lim Thus, By (3.9) we have (3.10) Denote R 3 |u n | 2 * s dx = L 2 * s + o n (1). By (3.10) we obtain that which implied L >T . Therefore, g(x, u n )u n − G(x, u n ) dx which contradicts with Lemma 2.7. Therefore, {u n } is non-vanishing. Then there exist x n ∈ R 3 and δ 0 > 0 such that B1(xn) |u n (x)| 2 dx ≥ δ 0 . (3.11) Without loss of generality, we assume that x n ∈ Z N with |x n | → ∞. Denotẽ u n byũ n (·) := u n (· + x n ), up to a subsequence,ũ n ũ in H s (R 3 ),ũ n → u in L q loc (R 3 ) for q ∈ [2, 2 * s ) andũ n (x) →ũ(x) a.e. in R 3 . By using (3.11) we haveũ = 0.