Existence of ground state for fractional Kirchhoff equation with L2$L^{2}$ critical exponents

*Correspondence: ccnumathhyl@163.com 1School of Statistics and Mathematics, Zhongnan University of Economics and Law, 430073 Wuhan, P.R. China Full list of author information is available at the end of the article Abstract In this paper, we consider a class of fractional Kirchhoff equations with L2 critical exponents. By using the scaling technique and concentration-compactness principle we obtain the existence and nonexistence of ground state for fractional Kirchhoff equation with L2 critical exponent.


Introduction
In this paper, we consider the existence of ground state for the following fractional Kirchhoff equation: where a, b > 0, N > 2s > N 2 with s ∈ (0, 1), 0 ≤ γ ≤ 8s N , 2 * (s) = 2N N-2s , and V (x) is a bounded function in R N .
If s = 1, then equation (1) is related to the stationary solutions of where f (x, u) is a general nonlinear function. Equation (2) comes from free vibrations of elastic strings by taking into account the changes in length of the string produced by transverse vibrations [13]. After the pioneering works [17] and [15], equation (1) has attracted considerable attention. The existence and asymptotic behavior of nodal solutions of equation (1) were considered by Deng, Peng, and Shuai [5]. The existence and concentration behavior of positive solutions were studied in [8,9]. The uniqueness and nondegeneracy of positive solutions were obtained by Li et al. [14] and the references therein. The existence of multipeak solutions was considered in [23]. Equation (1) can be viewed as an eigenvalue problem by taking μ as an unknown Lagrange multiplier. Hence some mathematicians considered equation (1) by studying some constrained variational problems and obtained the existence of ground state of equation (1). This technique was generally used for other types of equations, for example, semilinear Schrödinger equation [11,24], Schrödinger-Poisson equation [4,12], quasilinear Schrödinger equation [29,30]; see also [1,2,18,21,22]. For s = 1, as far as we know, the first work comes from Ye [25], who considered the following minimization problem: where Using the scaling technique and concentration-compactness principle, Ye obtained the sharp existence of global constraint minimizers of problem (3). Then Zeng and Zhang [28] improved the results of [25] and obtained the sharp existence and uniqueness of global constraint minimizers of problem (3). From [25,28] we know that there is an L 2 critical exponent p * = 2 + 8 N such that problem (3) has global constraint minimizers for p < p * and no global constraint minimizers for p ≥ p * . Then, for the L 2 critical exponent, Ye [26] and Zeng and Chen [31] added a perturbation function and obtained the existence of minimizers on S c . Moreover, for the L 2 critical exponent, Ye [27] gave some mass concentration behavior. Recently, Guo, Zhang, and Zhou [7] considered the following minimization problem: where They first proved the sharp existence and nonexistence of global minimizer of problem (4) with V (x) = 0. Then, for the trapping potential V (x), they considered the existence of minimizers for problem (4). Especially, for the L 2 critical exponent, they proved that there is β p * > 0 such that problem (4) has at least one minimizer for β ≤ β p * and has no minimizers for β > β p * . Furthermore, for minimizers of problem (4) with p < p * and β > β p * , they obtained the blowup behavior of minimizers as p tends to p * . For s ∈ (0, 1), Autuori, Fiscella, and Pucci [3] obtained the existence of solutions for equation (1) with critical nonlinearity. The existence of solutions of (1) with critical exponents was also considered in [19]. The multiplicity of solutions was obtained by Pucci, Xiang, and Zhang [20] and so on. Recently, Huang and Zhang [10] considered the existence and uniqueness of minimizers for the following problem: where Using the scaling technique and some energy estimates, they obtained the existence and uniqueness of minimizers for problem (5) if p < 8s N and proved that there are no minimizers for problem (5) when p ≥ 8s N . For the existence of ground state of equation (1), we consider the following minimization problem: where and S c := u ∈ H s R N : Here H s (R N ) is the Besov space defined by It is easy to see that there are no minimizers for problem (6) if p > 8s N . Indeed, for any u ∈ S c and constant λ > 0, let u λ (x) = λ N 2 u(λx). Then Hence we can deduce that Since γ < 8s N , it is easy to see that Nγ 2 < 4s. If p > 8s N , then for λ large enough, the dominant term in (7) This means that there are no minimizers for problem (6) if p > 8s N . Therefore it seems that p = 8s N is the L 2 critical exponent for problem (6). Moreover, from (7) Hence e(c) ≤ 0 for any c > 0, and 0 < p < 2 * (s) -2. For p = 8s N , similarly to the proof of [10,28], using the Gagliardo-Nirenberg inequality (12), we have where the definition of c * is given further. If c ≤ c * , then (8) means that e(c) > 0, a contradiction to e(c) ≤ 0, which indicates that for p = 8s N and c ≤ c * , problem (6) Then we have e(c) ≤ -∞, which means that for p = 8s N and c > c * , there are no minimizers for problem (6) with V (x) = 0. In other words, for V (x) = 0, there is no minimizer for problem (6) with p = 8s N . Hence, in this paper, when the potential function V (x) satisfies some conditions, we consider the existence and nonexistence of minimizers for problem (6) with N . In addition, we consider the existence and nonexistence of ground states for equation (1) under some conditions on the function V (x). Moreover, in this paper, the energy estimate method used in [10,28] is invalid because of the existence of a potential function V (x). Hence we use the concentration-compactness principle to overcome the compactness of a minimizing sequence. Using this technique, it is natural that γ ≥ 2 is necessary by Lemma 2.6.
In this paper, we assume that Let where the function U(x) is defined in Sect. 2. We first give a nonexistence result.
Theorem 1.1 Let p = 8s N , and let V (x) satisfy (9). Then problem (6) has no minimizers if one of the following conditions holds: small enough, we have From (2) of Theorem 1.1 we know that problem (6) has minimizers if and only if the function V (x) has a negative part. Hence, in this paper, we first give a certain condition for V (x) at infinity and get the following existence result.
for some α > 0, and let a be small enough. Suppose that the function V (x) satisfies (9) and Then problem (6) has at least a minimizer.
According Theorem 1.2, we get the existence of minimizers of problem (6) for V (x) tending to 0 at infinity with some rates as |x| → ∞. Next, if we assume a general condition for V (x) at infinity, then we have the following: , and γ ∈ [2, 8s N ), and suppose that the function V (x) satisfies (9) and Then if e(c) < 0, the problem (6) has at least one minimizer.
Throughout the paper, C denotes some constant, and |u| p denotes the L p -norm of a function u.

Preliminary results
Since we want to consider the existence of minimizers for problem (6) with p = 8s N , we first introduce the following Gagliardo-Nirenber inequality [6]: Here the function U(x) is the unique ground state of the equation Using the Pohozaev identity and equation (13) [6,10], we can get that Lemma 2.1 Assume that V (x) ≥ 0. Then, for any c ∈ (0, c * ), we have e(c) ≥ 0.
Proof For any u ∈ S c , using the Gagliardo-Nirenberg inequality (12), we get that be small enough such that .
Then for any d ∈ [0, c), we have

The proof of theorems
Proof of Theorem 1.1 (1) From Lemma 2.3 we know that e(c) < -∞. Hence it is natural that for any c > c * , there are no minimizers for problem (6).
(3) From Lemma 2.2 we have that e(c) ≥ 0. This, together with Lemma 2.4, indicates that e(c) = 0. Similarly to the proof of (2), we can deduce that there are no minimizers for problem (6).
Proof of Theorem 1.2 Let {u n } be a minimizing sequence of e(c). From (26) we get that R N |(-) s 2 u n | 2 dx is bounded above, which, combined with R N |u n | 2 dx = c 2 , implies that {u n } is bounded in the space H s (R N ). Hence there is u ∈ H s (R N ) such that there is a subsequence of {u n }, denoted still by {u n }, such that u n u in H s (R N ). Then by the concentration-compactness principle [16] the sequence {u n } is compact. Hence the key point is excluding the case of vanishing (i.e., u = 0 in H s (R N )) and dichotomy (i.e.m u = 0 in H s (R N ) but 0 < |u| 2 < c).

This indicates that
Using (30) and (31), we can deduce that where σ (ε) → 0 as ε → 0. Let ε → 0. Then (37) contradicts to Lemma 2.6. Hence dichotomy cannot occur, and for any ε > 0, there exist R ε > 0 and {y n } ⊂ R N such that Next, we discuss this problem for two cases: {y n } is bounded and y n → ∞ as n → ∞.
(1) If {y n } is bounded from above, then (38) indicates that Since {u n } is bounded in the space H s (R N ), the Gagliardo-Nirenberg inequality gives that By Lebesgue's dominate convergence theorem we get that Similarly to the proof of (39), we obtain that From [6] we know that the norm R N |(-) s 2 u n | 2 dx satisfies weak lower semi-continuity, that is, Then the sequence {u n } has a strongly convergent subsequence, which means that u is a minimizer of e(c).
(2) If y n → ∞ as n → ∞, then from the definition of V (x) we know that V (x)|u n | γ +2 dx = 0.
This, together with (41), indicates that Using (42) and the Gagliardo-Nirenberg inequality (12), we deduce that which contradicts to Lemma 2.5. Hence y n → ∞ as n → ∞ cannot occur.
Proof of Theorem 1.3. The proof is similar to that of Theorem 1.2. We omit it.