LEAST ENERGY SOLUTIONS FOR FRACTIONAL KIRCHHOFF TYPE EQUATIONS INVOLVING CRITICAL GROWTH

. We study the following fractional Kirchhoﬀ type equation: ) , where a, b > 0 are constants, 2 ∗ s = 6 3 − 2 s with s ∈ (0 , 1) is the critical Sobolev exponent in R 3 , V is a potential function on R 3 . Under some more general assumptions on f and V , we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and

1. Introduction and main results. In this paper, we are concerned with the following fractional Kirchhoff type equation with critical growth: where a, b > 0 are constants, 2 * s = 6 3−2s with s ∈ (0, 1) is the critical Sobolev exponent in R 3 , the potential V ∈ C 1 (R 3 , R) and f ∈ C(R, R) is a subcritical perturbation. The fractional Laplacian (−∆) s is a nonlocal operator defined in the Schwartz class S (R 3 ) as (−∆) s u(x) = C(s)P.V.
where C(s) is a suitable normalized constant and P.V. means the Canchy principle value on the integral. When a = 1 and b = 0, equation (1) can be reduced to the usual fractional Schrödinger equation: The fractional Schrödinger equation was introduced by Laskin [31,32] in the context of fractional quantum mechanics, as a result of expanding the Feynman path integral, from the Brownian like to the Lévy like quantum mechanical paths. In particular, the fractional Laplacian can be understood as the infinitesimal generator of a stable Lévy diffusion processes [2]. It also has various applications in different subjects, such as, the thin obstacle problem [35,42], optimization [19], finance [12], conservation laws [4], minimal surfaces [6,8] and see [5] for further detials. The non-locality of the fractional Laplacian makes it difficult to study. To overcome this difficulty, Caffarelli and Silvestre [7] introduced the extension method that reduced this nonlocal problem into a local one in higher dimensions. This extension method has been applied successfully and a series of fruitful results have been obtained. By using the Mountain Pass and Linking Theorems, the authors in [39] proved the existence of weak solutions for the nonlocal problem involving fractional Laplacian operators, see also the works of [18,20] in the respect of variational methods. For related investigations on problem (2) involving critical growth, we refer the readers to [10,17,41] and the references therein. When s = 1, equation (1) can be reduced to the following Kirchhoff type problem which is related to the stationary analogue of the equation where f (x, u) is a given nonlinear function in R 3 × R 1 . Equation (4) was proposed by Kirchhoff in 1883 as a generalization of the classical D'Alembert's wave equations for free vibration of elastic strings, see [30]. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. In [1], the authors pointed out that the equation (4) models several physical systems, where u describes a process which depends on the average of itself. Nonlocal effect also finds its application in biological systems. For more mathematical and physical background on equation (4), we refer the readers to [3,9,14] and the references therein.
Recent studies have been focused on problem (3). In [26], He and Zou studied problem (3) in the case that f (x, t) = h(t) ∈ C 1 (R + , R + ) satisfies the variant Ambrosetti-Rabinowitz type condition ((AR) in short): i.e., for some θ > By using the Mountain Pass Theorem and the Nehari manifold, they obtained the existence and concentration behavior of ground state solutions. For the case when f (x, t) = |t| p−2 t and 3 < p ≤ 4, which does not satisfy the variant (AR) condition (5), by constructing a constrained minimization on a new manifold based on the Nehari manifold and the Pohozaev identity, Li and Ye in [33] showed that the existence of a positive ground state solution for the corresponding limiting problem of (3). Then by using a monotone method and a global compactness lemma, they proved that (3) has a positive ground state solution. After that, Guo [23] generalized the result in [33] to problem (3) with f (x, t) = h(t) ∈ C 1 (R + , R) satisfies the following general assumptions: is strictly increasing in (0, +∞).
By establishing a new Nehari-Pohozaev manifold and using a constrained minimization on the intersection of the manifold and H 1 r (R 3 ), Guo proved the limiting problem of (3) has a positive ground state solution. Combining this result and a technical condition on V : (V 1 ) V ∈ C 1 (R 3 , R) and there exists a positive constant A < a such that Guo obtained a positive ground state solution for (3).
Recently, Tang and Chen [44] generalized and improved the results in [23] and [33] and replacing (h 4 ) and (V 1 ) by the conditions and (V 2 ) V ∈ C 1 (R 3 , R) and there exists θ ∈ [0, 1) such that respectively. It should be mentioned that the assumption (h 5 ) is weaker than (h 4 ), but the assumption (V 2 ) is stronger than (V 1 ). To obtain a ground state solution of Nehari-Pohozaev type for (3), they took the minimum on the manifoldM : However, different from [23], Tang and Chen [44] did not assume that h ∈ C 1 . Then M may not be a C 1 -manifold of H 1 (R 3 ), which means that the method used in [23] and [33] is no longer applicable. To prove that the minimum of the functional on M is a critical point, a new idea was developed in [44]. The existence results for (3) has been well developed in recent years when f (x, t) ∈ C 1 (R 3 , R), we refer the readers to [11,15,27,28,24,25,46,48] and the references therein.
As we have mentioned above, there are many papers dealt with problems (2) and (3), which have only one nonlocal term. The case that the variational problem involve double or more nonlocal terms is much more complicated and has been less studied. Very recently, Fiscella and Valdinoci in [22] proposed a stationary Kirchhoff variational model in bounded domains of R N , which took into account the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. Later on, Pucci, Xiang and Zhang in [37] investigated the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger-Kirchhoff type where (−∆) s p is the fractional p-Laplacian operator, with 0 < s < 1 < p < ∞ and ps < N . Under some assumptions on f, g, V, M , they proved problem (6) admits at least two solutions by using the Ekeland variational principle and the Mountain Pass Theorem. We refer also to [21,34,36] for related fractional Kirchhoff-type equations. The fractional Schrödinger-Poisson system has been also studied in the past several years. For example, Teng in [45] considered the following system where µ ∈ R + is a parameter and q, s, t are interdependent to each other. Motivated by the works mentioned above, the main purpose of this paper is to establish the existence of least energy solutions for problem (1) with critical growth and general subcritical perturbation. In particular, our results can be viewed as a generalization of the works in [44] to the fractional Kirchhoff type equation involving critical growth.
To establish our main results, we need the potential V ∈ C 1 (R 3 , R) satisfies the following conditions: (V 2 ) there exists a positive constant A < a such that (∇V (x), x) 3 2s < 2sAS s , where S s is given by (10) below.
On the nonlinearity f , we also need the following assumptions: (f 1 ) f ∈ C(R, R) and f (t) ≡ 0 for all t ∈ (−∞, 0); s such that f (t) ≥ µt p−1 for all t ≥ 0. The variational functional associated with equation (1) is defined by for u ∈ H s (R 3 ). We can prove that I ∈ C 1 (H s (R 3 ), R) and a critical point of I in H s (R 3 ) corresponds to a weak solution of (1). Moreover, we note that the corresponding limiting equation of (1) is: which is the Euler-Lagrange equation associated with the functional Before stating our main results, we need introduce the following Nehari-Pohozaev manifold: For any u ∈ M ∞ , we define that . Now, our first result can be stated as follows.
. Then problem (8) exists a nonnegative solutionũ such that I ∞ (ũ) = m ∞ > 0, if either 4s 3−2s < p < 2 * s for all µ > 0 or 2 < p ≤ 4s 3−2s for µ > 0 sufficiently large. Remark 1. We mention here that, to ensure the associated functional has a Mountain Pass geometry, the condition 2 * s > 4 is necessary. So, the parameter s must lie in ( 3 4 , 1). To prove Theorem 1.1, the main difficulties lie in three aspects: Firstly, the double nonlocal terms due to the presence of fractional Laplacian operator (−∆) s and integral R 3 |(−∆) s 2 u| 2 dx in equation (8), lead to some additional difficulties and make the study become interesting. Secondly, without the assumption that f ∈ C 1 (R, R), then M ∞ is not a C 1 -manifold. It seems difficult to prove that the minimum of I ∞ on M ∞ is a critical point. The arguments in [23] and [33] can not be applied in this paper and some new tricks will be developed. Thirdly, the unboundedness of the domain R 3 and the critical Sobolev exponent lead to the lack of compactness. What's more, to estimate some precise threshold value for m ∞ , we are in situation to solve a factional order algebra equation. Thus more careful analysis is needed.
Based on Theorem 1.1, under the conditions (V 1 ) − (V 2 ), we can prove the existence of a least energy solution to equation (1).
. Then problem (1) exists a nonnegative nontrivial least energy solution w, if either 4s 3−2s < p < 2 * s for all µ > 0 or 2 < p ≤ 4s 3−2s for µ > 0 sufficiently large. Since we do not assume that f satisfies the variant (AR) type condition, it seems difficult to get the boundedness of any (P S) sequence even if a (P S) sequence has been obtained. In order to overcome this difficulty, we make use of the monotone method due to Jeanjean [29]. First, for λ ∈ [δ, 1], we introduce a family of C 1functionals defined as where δ ∈ (0, 1). By Proposition 1 below, for a.e. λ ∈ [δ, 1], there is a bounded . In fact, it is very difficult to prove the weak sequential continuity of I λ in H s (R 3 ) by direct calculations due to the existence of the nonlocal term , we need establish a version of global compactness lemma related to the functional J λ and its limiting functional Finally, choosing a critical point sequence {(λ n , u λn )} ⊂ [δ, 1]×H s (R 3 ) with λ n → 1 as n → ∞, we can prove that {u λn } is a bounded (P S) c1 sequence for I = I 1 . By using the global compactness lemma again, we complete the proof of Theorem 1.2. The outline of this paper is as follows. In Section 2, we devote to showing the existence of ground state solutions of Nehari-Pohozaev type for the limiting problem (8). In Section 3, we employ the monotone method developed by Jeanjean in [29] to prove Theorem 1.2.
Notations. Throughout this paper, we make use of the standard notations as follows. The Hilbert space H s (R 3 ) is defined as We endow the space H s (R 3 ) with the inner product and norm by It is well known that H s (R 3 ) is continuously embedded into L q (R 3 ) for 2 ≤ q ≤ 2 * s and compactly embedded into L q loc (R 3 ) for 1 ≤ q < 2 * s . L q (R 3 ) is the usual Lebesgue space with the standard norms From [13,16] we know that D s,2 (R 3 ) is continuously embedded into L 2 * s (R 3 ) and there exists a best constant S s > 0 such that where D s,2 (R 3 ) is defined by From assumption (V 1 ) and a > 0 is fixed, we can define an equivalent norm on the fractional Sobolev space H s (R 3 ) as 2. Ground state solutions of Nehari-Pohozaev type for (8). In this section, we will use a constrained minimization on M ∞ to get a nonnegative ground state solution of Nehari-Pohozaev type for the limiting problem (8).
Proof. Without loss of generality, we may assume that τ = 0 and set By a direct computation, we have It follows from ( Then for any t > 0,

YINBIN DENG AND WENTAO HUANG
From Lemma 2.1 and Sobolev inequality, we have where we used the fact that So, Lemma 2.2 is proved.
Proof. For any u ∈ H s (R 3 )\{0} and t > 0, we consider y(t) := I ∞ (u t ). It is easy to check that y(t) > 0 for t > 0 small and y(t) → −∞ as t → +∞, which gives that y(t) has a critical point t 0 > 0 corresponding to its maximum, i.e., y(t 0 ) = max t>0 y(t) and y (t 0 ) = 0. Thus Moreover, we claim that the critical point of y(t) is unique. Indeed, by contradiction, we suppose that there exist two points t 1 , t 2 > 0 such that G ∞ (u ti ) = 0 for i = 1, 2. Similarly to the proof of Lemma 2.2, we can deduce that This implies that t 1 = t 2 . So, t 0 > 0 is the unique critical point of y(t).
Proof. It follows from Lemma 2.3 that Thus for any u ∈ M ∞ , by (11), Lemma 2.2 and Sobolev inequality, we have As a result, we complete the proof.
Throughout this section, the norm on the H s (R 3 ) is taken as Now, we establish a splitting lemma as follows.
In the following, we give an important energy estimate for m ∞ . Proof. Let ψ ∈ C ∞ 0 (R 3 ) be a cut-off function such that ψ(x) = 1 if |x| ≤ r and ψ(x) = 0 if |x| ≥ 2r. For ε > 0, we define with κ ∈ R\{0}, τ > 0 and x 0 ∈ R 3 . From [40,45], we know that for q < 3 3−2s . In particular, since s > 3 4 , one has By Lemma 2.3 and Lemma 2.4, there exists a t ε > 0 such that Next, we claim that there exist two constants t * , t * > 0 such that t * ≤ t ε ≤ t * . Indeed, we first prove that t ε is bounded from below by a positive constant.
Otherwise, we could find a sequence ε n → 0 such that t εn → 0. By the above estimations, up to a subsequence, we have (u εn ) tε n → 0 in H s (R 3 ). Therefore, which is a contradiction. On the other hand, it follows from (f 4 ) that which implies that there exists t * > 0 such that t ε ≤ t * . Thus, the claim is proved. Therefore, we conclude that Then When 4s 3−2s < p < 2 * s , we can verify that for any µ > 0, m ∞ < g(T ) for ε > 0 small. Moreover, if 2 < p ≤ 4s 3−2s and ε > 0 small, we also have m ∞ < g(T ) by choosing µ = ε −2s . Proof. Take t = 0 in Lemma 2.1, we have that We set Φ ∞ (u) : Let {u n } ⊂ M ∞ be a minimizing sequence for m ∞ such that where g is defined in Lemma 2.6. In the following, we divide the proof into three steps.
Step 1. The sequence {u n } is bounded in H s (R 3 ).
From (21)- (22), we know that Next, we only need to show the boundedness of R 3 u 2 n dx. By (f 1 ) − (f 2 ) and G ∞ (u n ) = 0, we have that for any ε > 0, there exists a C ε > 0 such that By choosing ε > 0 small enough, we obtain that {u n } is bounded in H s (R 3 ).
Step 2. There exist a sequence {y n } ⊂ R 3 and constants R, β > 0 such that Suppose by contradiction that for all R > 0, By Lemma 2.4 in [38], we have that Since G ∞ (u n ) = 0, we get From the definition of the constant S s , we have Without loss of generality, we may assume that It is easy to check that l > 0, otherwise u n → 0 as n → ∞ which contradicts m ∞ > 0. Thus, (25) implies that Since g(t) has a unique critical point T > 0 corresponding to its maximum, it follows from (26) that l ≥ T . On the other hand, from (24), one has Thus by the definition of S s and the fact that aS 3 2s which contradicts Lemma 2.6.
Proof. Suppose by contradiction that (I ∞ ) (u) = 0, there exist ρ, δ > 0 such that We first show that Otherwise, suppose that there exist ε 0 > 0 and a sequence {t n } such that u tn − u 2 ≥ ε 0 as t n → 1.
Notice that there exist two functions U 1 , U 2 ∈ C 0 (R 3 ) such that
Next, we only need to prove thatũ is nonnegative. Let us consider the functional where u + = max{u, 0} and u − = min{u, 0}. Similarly, we can obtain a nontrivial solution u of the equation Multiplying the above equation (37) by u − and integrating over R 3 , we find But we know that Thus, u − = 0 and u ≥ 0 is a solution of equation (37). Hence, we can obtain that u ≥ 0.
3. Proof of Theorem 1.2. In this section, we employ the monotone method developed by Jeanjean in [29] to prove Theorem 1.2. Since we do not assume that f satisfies (AR) condition, it seems difficult to get a bounded (P S) sequence. In order to overcome this difficulty, we introduce the following abstract result developed by Jeanjean [29].
Proposition 1. ( [29], Theorem 1.1) Let X be a Banach space equipped · , and let L ⊂ R + be an interval. We consider a family (I λ ) λ∈L of C 1 -functionals on X of the form where B(u) ≥ 0, ∀u ∈ X, and such that either A(u) → +∞ or B(u) → +∞ as u → ∞. We assume that there are two points (v 1 , v 2 ) in X, such that setting there hold, ∀λ ∈ L, Then, for almost every λ ∈ L, there is a bounded (P S) c λ sequence in X. Moreover, the map λ → c λ is continuous from the left.
To use Proposition 1, we set L = [δ, 1], where δ ∈ (0, 1), and denote a family of C 1 -functionals on H s (R 3 ) as follows Denoting as u → +∞, and B(u) = R 3 F (u)dx + 1 From Theorem 1.1, we conclude that for any λ ∈ [δ, 1], the associated limiting equation of I λ as follows: where By a standard argument (see e.g. [27,33,45]), we can verify that the functional I λ has a Mountain Pass geometry.  Arguing as in Lemma 2.2, we can establish the following lemma.
where u t is given in Lemma 2.2.
Since I λ ((u 1 ) t ) → −∞ as t → +∞, there is at > 0 such that However, By the definition of t λ , one has which is a contradiction. Thus, 0 < β 0 ≤ t λ <t, ∀λ ∈ [δ, 1]. Set It follows from Lemma 3.2 and the definition of t λ that The second inequality in (40) is obtained due to Lemma 2.6.
Next, we provide the following version of global compactness lemma, which is adopted to prove that the functional I λ satisfies (P S) c λ condition for a.e λ ∈ [δ, 1]. The proof is similar to [33,34,45], so we omit detials here. 1] be fixed and {u n } be a bounded (P S) c λ sequence for I λ with c λ < g λ (T λ ). Then there exist a u ∈ H s (R 3 ) and a A ∈ R such that J λ (u) = 0, where and either (i) u n → u in H s (R 3 ), or (ii) there exist an l ∈ N and {y k n } ⊂ R 3 with |y k n | → +∞ as n → ∞ for each 1 ≤ k ≤ l, nontrivial solutions w 1 , · · · , w l of the following equation such that On the convergence of the bounded (P S) sequence {u n } for I λ , we have the following result. 1] be fixed and {u n } ⊂ H s (R 3 ) be a bounded (P S) c λ sequence for I λ with c λ < g λ (T λ ). Then there exist a subsequence of {u n }, still denoted by {u n } and a u λ ∈ H s (R 3 )\{0} such that u n → u λ in H s (R 3 ).
From (V 2 ), (21) and Sobolev inequality, we have In the following we estimate the value of J ∞ λ (w k ). For each nontrivial solution w k (k = 1, · · · , l) of problem (43), we have the following Pohozaev identity By (46), for each 1 ≤ k ≤ l, we have that 0 = (4s − 3) (J ∞ λ ) (w k ), w k + 2P ∞ λ (w k ) = 2s(a + bA 2 λ ) with G λn (u λn ) = 0 due to Pohozaev identity and I λn (u λn ) = 0. Since I λn (u λn ) = 0, from (f 1 ) − (f 2 ), we have that for any ε > 0, there exists a C ε > 0 such that By choosing ε > 0 small enough, we obtain that {u λn } is bounded in H s (R 3 ). Thus we have for any ϕ ∈ H s (R 3 ) I λn (u λn ), ϕ − I (u λn ), ϕ = (λ n − 1) where we used the fact that the map λ → c λ is continuous from the left. That is {u λn } is a bounded (P S) c1 sequence for I. By applying Lemma 3.5, we obtain a nontrivial critical point u 0 ∈ H s (R 3 ) for I and I(u 0 ) = c 1 . Finally, we end this proof by showing the existence of a least energy solution for problem (1). Let d := inf I(u) u = 0, I (u) = 0 . Then 0 ≤ d ≤ I(u 0 ) < g(T ). In fact, for any u satisfying I (u) = 0, by standard argument we see u ≥ ρ for some positive constant ρ. Similarly to the argument of (49)-(50), we infer Therefore, d ≥ 0. In the following we rule out d = 0. Suppose by contradiction that {u n } be a critical point sequence of I satisfying lim n→∞ I(u n ) = 0. From (51), we have lim n→∞ R 3 |(−∆) s 2 u n | 2 dx = 0. This conclusion combined with I (u n ), u n = 0, we can verify that lim n→∞ R 3 u 2 n dx = 0. Therefore, we obtain lim n→∞ u n = 0, which is a contradiction with u n ≥ ρ > 0 for all n ∈ N. Let {u n } ⊂ H s (R 3 ) be a sequence of nontrivial critical point of I satisfying lim n→∞ I(u n ) = d < g(T ). Similarly, we can deduce that {u n } is bounded in H s (R 3 ), i.e., {u n } is a bounded (P S) d sequence for I. Similarly to the argument in Lemma 3.5, there exists a nontrivial w ∈ H s (R 3 ) such that I(w) = d with I (w) = 0. Arguing as in the proof of Theorem 1.1, we show that the least energy solution w is nonnegative.