On the singularly perturbation fractional Kirchho ﬀ equations: Critical case

: This article deals with the following fractional Kirchho ﬀ problem with critical exponent where > a b , 0 are given constants, ε is a small parameter, s 4 . We ﬁ rst prove the nondegeneracy of positive solutions when = ε 0 . In particular, we prove that uniqueness breaks down for dimensions > N s 4 , i.e., we show that there exist two nondegenerate positive solutions which seem to be completely di ﬀ erent from the result of the fractional Schrödinger equation or the low - dimen sional fractional Kirchho ﬀ equation. Using the ﬁ nite - dimensional reduction method and perturbed argu -ments, we also obtain the existence of positive solutions to the singular perturbation problems for ε small.

, 0 are given constants, ε is a small parameter, = * − 2 s N N s 2 2 with < < s 0 1 and ≥ N s 4 . We first prove the nondegeneracy of positive solutions when = ε 0. In particular, we prove that uniqueness breaks down for dimensions > N s 4 , i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε small.

Introduction and main results
In this article, we are concerned with the following fractional Kirchhoff problem: , 0 are given constants, ε is a small parameter, ( ) → K x : N , ( ) −Δ s is the pseudo-differential operator defined by for ≥ t 0 and < < x L 0 , where ( ) = u u t x , is the lateral displacement at time t and at position x, is the Young's modulus, ρ is the mass density, h is the cross-sectional area, L is the length of the string, and p 0 is the initial stress tension. Problem (1.2) and its variants have been studied extensively in the literature. Bernstein obtains the global stability result in [3], which has been generalized to arbitrary dimension ≥ N 1 by Pohozaev in [4]. We point out that such problems may describe a process of some biological systems dependent on the average of itself, such as the density of population (see, e.g., [5]). Many interesting work on Kirchhoff equations can be found in [6][7][8][9] and references therein. We also refer to [10] for a recent survey of the results connected to this model.
On the other hand, the interest in generalizing the model introduced by Kirchhoff to the fractional case does not arise only for mathematical purposes. In fact, following the ideas of [11] and the concept of fractional perimeter, Fiscella and Valdinoci proposed in [12] an equation describing the behavior of a string constrained at the extrema in which appears the fractional length of the rope. Recently, problem similar to (1.1) has been extensively investigated by many authors using different techniques and producing several relevant results (see, e.g., [13][14][15][16][17][18][19][20][21][22][23]).
Besides, if = b ε , 0 in (1.1), then we are led immediately to the following fractional Schrödinger equation: which is also of practical interest and importance. For instance, it arises as the Euler-Lagrange equation of the functional The classification of the solutions would provide the best constant in the inequality of the critical Sobolev imbedding from ( ) where the fractional Sobolev space ( ) H s N is defined by And we also define the homogeneous fractional Sobolev space Since the fractional Laplacian ( ) −Δ s is a nonlocal operator, one cannot apply directly the usual techniques dealing with the classical Laplacian operator. By using the moving plane method of integral form, Chen et al. [25] proved that every positive regular solution of (1.3) is radially symmetric and monotone about some point, and therefore assumes the form Let us observe that which actually reflects the invariance of the equation under the above scaling and translations. Then Dávila et al. [26] proved that the solution above is nondegenerate in the sense that all bounded solutions to the equation We also refer to [27][28][29][30][31][32][33] for recent works for the nonlocal problems.
From the viewpoint of calculus of variation, the fractional Kirchhoff problem (1.1) is much more complex and difficult than the classical fractional Laplacian equation (1.3) as the appearance of the term , which is of order four. So a fundamental task for the study of problem (1.1) is to make clear the effects of this nonlocal term. Recently, Rǎdulescu and Yang [34] established uniqueness and nondegeneracy for positive solutions to Kirchhoff equations with subcritical growth. More precisely, they proved that the following fractional Kirchhoff equation: , has a unique nondegenerate positive radial solution. For the highdimensional case, Yang [35] proved that uniqueness breaks down for dimensions > N s 4 , i.e., there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. As one application, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, they also derive the existence of solutions to the singularly perturbation problems [36,37]. For the critical problem (1.1) with = ε 0, Furthermore, problem (1.4) has exactly one solution when the equality holds and has exactly two solutions for the other case.
Moreover, define the solution by U , which is of the form , if one of the following conditions holds: , where + is defined as By Theorem 1.2, it is now possible that we apply finite-dimensional reduction to study the perturbed fractional Kirchhoff equation (1.1). Our problem is motivated by an interesting work [38], in which the following local problem was studied Equation (1.6) can be derived from the following scalar curvature equation: where Δ g 0 and S 0 denote the Laplace-Beltrami operator and the scalar curvature of the N -dimensional Riemann manifold ( ) M g , 0 , respectively. Equation (1.7) and its variants have been studied extensively by the mathematicians, and the reader can check [39-42] and references therein.
Note that if u is a (weak) solution to equation (1.1), then the following Pohozâev identity [19] holds: Obviously, equation (1.1) does not have any solution if ⋅∇ < x K 0 or ⋅∇ > x K 0. Thus, in order to ensure the existence of solutions of equation (1.1), it is natural to suppose that K has critical points. More precisely, we assume that has finitely many critical points and set Now we state the existence result as follows.
This article is organized as follows. We complete the proof of Theorem 1.1 in Section 2 and prove Theorem 1.2 in Section 3. In Section 4, we present some basic results and explain the strategy of the proof of Theorems 1.3 and 1.4. Notation. Throughout this article, we make use of the following notations.
• For any > R 0 and for any ∈ are some positive constants that may change from line to line.

Proof of Theorem 1.1
In this section we prove Theorem 1.1. Our methods depend on the following result for the well-known fractional critical problem By using the moving plan method of integral form, Chen et al. [25] proved that every positive regular solution of (2.1) is radially symmetric, monotone about some point, and therefore assumes the form is a solution of equation (2.1). By the uniqueness of Q, we can deduce that Then, direct calculation shows that , it suffices to find positive solutions of the above algebraic equation (2.2), and is a constant, which only depends on a b , , and Q.
which means that this equation has a unique positive solution if and only if (2.6) and the maximum of ( ) f is It is easy to see that . Noting further that , a sufficient and necessary condition for the solvability of equation if and only if inequality (2.8) holds. Furthermore, we have , then equation (2.2) has exactly one positive solution 0 defined by (2.6);

Nondegeneracy results
In this section, we prove the nondegeneracy results of Theorem 1.2. For positive constants a b , , we define the differential operator L as It is easy to see that for any We observe now that Differentiating the equation s with respect of the parameters at = = λ ξ 1, 0, we see that the functions annihilate the linearized operator around Q; namely, they satisfy the equation With no loss of generality, we assume that ( ) (| |) = U x U x is the unique positive radial energy solution to equation (2.1). Then we have Ker  span  2  2 , , , , .
is nonradially symmetric, we have the following corollary: Proof. From the definition of L 2 , and U is the solution of the equation Proof. Noting that H s N is a positive solution to the equation ( ) = L u 0, we know that ( ) U x has the following form: H s N being the unique positive solution to equation (1.3). Therefore, Proof. Direct computation shows that ( ) is indeed a radial solution to equa- is the unique radial solution to equation = + φ 0 in D rad up to a constant, where D rad contains all the radial functions in Recall that U is a ground state solution of (1.2). It follows from above that c is a constant independent of U under the assumptions of Theorem 1. that v satisfies On the singular perturbation fractional Kirchhoff equations  1105 By applying [43, Theorem 3.3], we conclude that (see, e.g., [44]). We then conclude from (3.4) to (3.6) that . We can describe the action of + L more precisely. For each l, the action of + L on the radial factor in l is given by Here ( ) −Δ l s is given by spectral calculus and the known formula Applying arguments similar to that used in [43] and [45], one can verify that each + L l , enjoys a Perron-Frobenius property, that is, if l is an eigenvalue, then E is simple and the corresponding eigenfunction can be chosen strictly positive. Moreover, we have > for ≥ l 2 in the sense of quadratic forms (see, e.g., [45]).
The proof is completed. □

Singularly perturbation problem
In this section, we are concerned with the singularly perturbation fractional Kirchhoff equation It is known that every solution to (4.1) is a critical point of the energy functional ( ) → I D : , given by , which is the fractional Sobolev space equipped with the inner product and norm given by , respectively. It is standard to verify that ( ( )) ∈ I C D . , it requires more careful estimates in the procedure, which is more complicated than the case of the fractional Schrödinger equation.

The abstract perturbation method
In this subsection, we state the abstract results we will use in the rest of the article. They are reported below for the reader's convenience. Let E be a Hilbert space and let ( ) be given. Consider the perturbed functional Suppose that I 0 satisfies (1) I 0 has a finite dimensional manifold of critical points Z; let is a Fredholm operator with index zero; Hereafter, we denote by Γ the functional | G Z . (i) z¯is nondegenerated; (ii) z¯is a proper local minimum or maximum; (iii) z¯is isolated and the local topological degree of ′ Γ at z¯, Then for | | ε small enough, the functional I ε has a critical point u ε such that → u zε as → ε 0.
To apply Theorem 4.1, we set Then Z is an + N 1 dimensional manifold of critical points for the function I 0 corresponding to (1.3). In order to apply the abstract setting we will check the assumptions on I 0 introduced above. The nondegeneracy condition comes from Theorem 1.2, so we only need to prove the following result. Then ( ) is a Fredholm operator with zero index.
Proof. Recalling that a.e. on . Thus, we can obtain On one hand, since On the other hand, by Vitali convergence theorem, we have Therefore, we have ∥ ∥ ′ − ′ → φ φ 0 n as → ∞ n . Hence, ′ is a compact operator. □ Up to now, we have proved that I 0 satisfies assumptions (1)-(3). As described above, one has which is nothing but a Poincaré-Melnikov-type function. Our next goal is to show that, locally near any ∈ z Z, there exists a manifold Z ε , diffeomorphic to Z which is a natural constraint for I ε . By this we mean that ∈ u Z ε and | ′ I ε Zε implies ( ) In this way, the search of critical points of I ε on E is reduced to the search of critical points of | ′ I ε Zε . In the sequel, we define 1,2, , , .
ε ε , 0 0 , the following properties hold: It is easy to prove that Z ε is a natural constraint for I ε , that is to say the critical point of I ε on Z ε is also a critical point of is a critical point of I ε on Z ε , then ( ) ′ I u ε is orthogonal to T Z u ε , where T Z u ε is the tangent space to Z ε at u. On the other hand, from Lemma 4.2(ii) we can know that ( ) provided ε is small enough. Thus, ( ) . Moreover, we have that will turn out to be a critical point of I ε . From the analysis above, then we turn to solve the finite-dimensional functional ( ) λ ξ Γ , .

Behavior of ( ) λ ξ Γ ,
We begin by proving some general properties of ( ) λ ξ Γ , . First of all, it is convenient to extend ( ) λ ξ Γ , by continuity to = λ 0 for all fixed ∈ ξ N by setting As a consequence, we can further extend Γ by symmetry to + N 1 as a C 1 function. We will use the same symbol Γ for such a function. Moreover, from (4.2), we can find that In the sequel, we are going to introduce some lemmas which is crucial in studying the finite dimensional functional ( ) λ ξ Γ , .  From ( )( ) K ii 3 , we can deduce that there exists R 0 such that for each ≥ R R 0 , min , min .
x B x B By a direct calculation, we can prove that where is an isolated critical point of Γ and the following properties hold: Thus, by (4.8) and (4.9), we can prove this conclusion.