Non-degeneracy of Positive Solutions for Fractional Kirchhoff Problems: High Dimensional Cases

In this paper, we establish the nondegeneracy of positive solutions to the fractional Kirchhoff problem (a+b∫RN|(-Δ)s2u|2dx)(-Δ)su+u=up,inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Big (a+b{\int _{\mathbb {R}^{N}}}|(-\Delta )^{\frac{s}{2}}u|^2\mathrm{{d}}x\Big )(-\Delta )^su+u=u^p,\quad \text {in}\ \mathbb {R}^{N}, \end{aligned}$$\end{document}where a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b>0$$\end{document}, 0<s<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s<1$$\end{document}, 1<p<N+2sN-2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\frac{N+2s}{N-2s}$$\end{document} and (-Δ)s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s$$\end{document} is the fractional Laplacian. In particular, we prove that uniqueness breaks down for dimensions N>4s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N>4s$$\end{document}, i.e., we show that there exist two non-degenerate 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, we can derive the existence of solutions to the singularly perturbation problems.


Introduction and Main Results
In this paper, we are concerned with the following fractional Kirchhoff problem where 2 * s is the standard fractional Sobolev critical exponent. Recently, Rǎdulescu and Yang [41] established uniqueness and nondegeneracy for positive solutions to (1.1) for N 4 < s < 1. Then in this paper, we will consider the high dimensional cases, i.e. N ≥ 4s. We also refer to [26,40,44,45] for critical cases and single/multi-peak solutions in this direction.
If s = 1, Eq. (1.1) reduces to the well known Kirchhoff type problem, which and their variants have been studied extensively in the literature. The equation that goes under the name of Kirchhoff equation was proposed in [28] as a model for the transverse oscillation of a stretched string in the form for t ≥ 0 and 0 < x < L, where u = u(t, x) is the lateral displacement at time t and at position x, E is the Young modulus, ρ is the mass density, h is the cross section area, L the length of the string, 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 [10], which has been generalized to arbitrary dimension N ≥ 1 by Pohožaev in [37]. We also 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. [9]). Many interesting work on Kirchhoff equations can be found in [15,27,33,43] and the references therein. We also refer to [38] 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 [20] an equation describing the behaviour 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. [1-4, 6, 8, 23-25, 34-36, 42]).
Besides, if b = 0 in (1.1), then we are led immediately to the following fractional Schrödinger equation a(− ) s u + u = u p , in R N . (1.3) This equation is related to the standing wave solutions of the time-independent fractional Schrödinger equation endowed with the natural norm From [17], we have and the fractional Gagliardo-Nirenberg-Sobolve inequality , (1.5) where S > 0 is the best constant. It follows from (1.5) that Since the fractional Laplacian (− ) s is a nonlocal operator, one can not apply directly the usual techniques dealing with the classical Laplacian operator. Therefore, some ideas are proposed recently. In [12], Caffarelli and Silvestre expressed the operator (− ) s on R N as a generalized elliptic BVP with local differential operators defined on the upper half-space R N +1 By means of Lyapunov-Schmidt reduction, concentration phenomenon of solutions was considered independently in [13,16]. For more interesting results concerning with the existence, multiplicity and concentration of solutions for the fractional Laplacian equation, we refer reader to [5,17,18]  After that, Lenzmann [31] obtained the uniqueness of ground states for the pseudorelativistic Hartree equation in 3-dimension. In [21], Frankand and Lenzmann extends the results in [7] to the case that s ∈ (0, 1) and N = 1 with completely new methods. For the high dimensional case, Fall and Valdinoci [19] established the uniqueness and nondegeneracy of ground state solutions of (1.3) when s ∈ (0, 1) is sufficiently close to 1 and p is subcritical. In their striking paper [22], Frank, Lenzmann and Silvestre solved the problem completely, and they showed that the ground state solutions of (1.3) is unique for arbitrary space dimensions N ≥ 1 and all admissible and subcritical exponents p > 0. Moreover, they also established the nondegeneracy of ground state solutions. We summarize their main results as follows.
Then the following holds.
(i) there exists a minimizer Q ∈ H s (R N ) for J (u), which can be chose a nonnegative function that solves Eq.
with some constants C 2 ≥ C 1 > 0; (iii) Q is a unique solution of (1.3) up to translation.
and T + denotes the corresponding linearized operator given by Then the following holds.
(i) Q is nondegenerate, i.e., ker T + = span{∂ x 1 Q, ∂ x 2 Q, · · · , ∂ x N Q}; (ii) the restriction of T + on L 2 rad (R N ) is one-to-one and thus it has an inverse T −1 + acting on L 2 rad (R N ); From the viewpoint of calculus of variation, the fractional Kirchhoff problem (1.1) is much more complex and difficult than the classical fractional Laplacian Eq. (1.3) as the appearance of the term b R N |(− ) s 2 u| 2 dx (− ) s u, which is of order four. So a fundamental task for the study of problem (1.1) is to make clear the effects of this non-local term. The only one uniqueness and non-degeneracy result which we know for the solution of problem (1.1) is proved in [41] for the case N 4 < s < 1, and [14,32] for the case s = 1. As in [41], let U be a ground state positive solution of (1.1) and set Then, it is easy to check thatŨ is a positive solution of (1.3) and a minimizer of J (u). Therefore, from the uniqueness result for positive solutions of problem (1.3), we know that any solution U (x) of problem (1.1) with a, b > 0 has the following form Consequently, the solvability of the problem (1.1) is simply equivalent to the solvability of the following algebraic equation in (0, +∞), This observation makes the question of uniqueness and multiplicity for solutions to problem (1.1) very simple. Therefore, our main focus of the present paper is nondegeneracy property for positive solutions of problem (1.1). The main results of this paper are collected in the following results.
Then the following statements are true: . Furthermore, problem (1.1) has exactly one solution when the equality holds, and has exactly two solutions for the other case.
Moreover, define the solution by U , then there exist some x 0 ∈ R N such that U (· − x 0 ) is radial, positive and strictly decreasing in r = |x − x 0 |. Moreover, the function U belongs to C ∞ (R N ) ∩ H 2s+1 (R N ) and it satisfies with some constants C 2 ≥ C 1 > 0; is non-degenerate if one of the following conditions holds: . By Theorem 1.2, it is now possible that we apply Lyapunov-Schmidt reduction to study the perturbed fractional Kirchhoff equation.
where V : R N → R is a bounded continuous function. We want to look for solutions of (1.8) in the Sobolev space H s (R N ) for sufficiently small ε, which named semiclassical solutions. We also call such derived solutions as concentrating solutions since they will concentrate at certain point of the potential function V . Moreover, it is expected that this approach can deal with problem (1.8) for all 1 < p < 2 * s − 1, in a unified way. To state our following results, let introduce some notations that will be used throughout the paper. For ε > 0 and y = (y 1 , Assume that V : R N → R satisfies the following conditions: The assumption (V 1 ) allows us to introduce the inner products Now we state the existence result as follows.
This paper is organized as follows. We complete the proof of Theorem 1.1 in Sect. Notation. Throughout this paper, we make use of the following notations.
• For any R > 0 and for any x ∈ R N , B R (x) denotes the ball of radius R centered at x; • · q denotes the usual norm of the space L q (R N ), 1 ≤ q ≤ ∞; • o n (1) denotes o n (1) → 0 as n → ∞; • C or C i (i = 1, 2, · · · ) are some positive constants may change from line to line.

Proof of Theorem 1.1
In this section, we analyze the existence of solutions for the following fractional Kirchhoff problem As mentioned in the introduction, we know that any solution to (2.1) has the following form where Q being the unique positive radial solution to the following problem Let Q be the uniquely positive solution of (2.2) and also a minimizer of J (u). Consider the equation Therefore, to find solution U (x) of (2.1), it suffices to find positive solutions of the above algebraic Eq. (2.3).
which means that this equation has a unique positive solution 2s , (2.6) which means that f (E) has a unique maximum point and the maximum of f (E) is , then Eq. (2.3) has exactly two positive solutions E 1 and E 2 such that E 1 ∈ (0, E 0 ) and E 2 ∈ (E 0 , +∞).
Up to now, we have proved Theorem 1.1. Next, we analyze the asymptotic behavior of solution obtained above as b → 0. In the case 1 < N ≤ 4s, if we denote by E 0 the unique positive solution to Eq.
, Eq. (2.1) has exactly two solutions E 1 and E 2 such that (2.10) Correspondingly, problem (2.1) has exactly two solutions From (2.10), we can see that lim b→0 bE 2 ≥ lim b→0 bE 0 = +∞. Hence, By a similar analysis, we have lim b→0 bE 1 = 0, and the following conclusion is true.

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 for any u ∈ H s (R N ) in the weak sense. The linearized operator L + of L at U is defined as It is easy to see that for any ϕ ∈ H s (R N ), We also denote by Ker(L) the kernel space of a linear operator L, that is In the sequel, we always use U (x) to denote a positive solution to the equation L(u) = 0 in H s (R N ). We divide the proof of Theorem (1.2) into the following series of lemmas.

Lemma 3.1 Ker(T
For any fixed i ∈ {1, 2, . . . , N }, taking partial derivative with respect to x i on both sides of the above Eq. (3.1), we obtain This implies that T + ∂U ∂ x i = 0 for any fixed i ∈ {1, 2, . . . , N }. Therefore, Since ∂U ∂ x i is non-radially symmetric, we have the following corollary: We have Therefore, Since, for any fixed i, up to a translation, the function is a positive solution to the equation L(u) = 0, we know that U (x) has the following form with Q(x) ∈ H s (R N ) being the unique positive solution to the Eq. (1.3). Therefore, This completes the proof. .

Then
Ker Recall that U is a ground state solution of (1.1). It follows from above that c is a constant independent of U under the assumptions of Theorem 1.1. Hence, U solves (1.3) with c = a + b (− ) s 2 U 2 2 . We then can rewrite (3.4) as By applying Proposition 1.2, we conclude that where ψ = x · ∇U . Multiplying (3.6) by (− ) s U and integrating over R N , we see that Note that and (see e.g. [39]). We then conclude from (3.7)-(3.9) that It follows from Lemma 3.3 that This implies that On the other hand, for any ϕ(x) ∈ Ker (L + ), we have To prove ϕ ∈ span ∂U ∂ x 1 , ∂U ∂ x 2 , · · · , ∂U ∂ x N , it follows from Corollary 3.1 that there exists a unique radial function ψ 1 (r ) ∈ L 2 rad (R N ) such that ψ 1 (r ). Then, from (3.10) and (3.11), we have T + (W ) = 0. Therefore, it follows from Lemma 3.1 that there are some real numbers a i such that This implies that any solution ψ(x) to the Eq. (3.10) has the following form Since ϕ(x) is a solution to (3.10), we conclude that for some real numbers a i . Noting that ϕ(x) and ∂U ∂ x i are in Ker (L + ), we can conclude from (3.12) that L + (ψ 1 (r )) = 0. That is ψ 1 (r ) ∈ Ker (L + ) . Hence, it follows from Lemma 3.4 that ψ 1 (r ) ≡ 0. Now, from (3.12), we have for some real numbers a i . This implies that ϕ ∈ span ∂U ∂ x 1 , ∂U ∂ x 2 , · · · , ∂U ∂ x N . From the arbitrariness of ϕ, we see that Ker is non-degenerate. This completes the proof of Theorem 1.2.
singularly perturbed problems. Here, we take the following problem as an example: where V : R N → R satisfies the following conditions: That is, V is of α-th order Hölder continuity around x 0 .
It is known that every solution to Eq. (4.1) is a critical point of the energy functional I ε : H ε → R, given by for u ∈ H ε . It is standard to verify that I ε ∈ C 2 (H ε ) . So we are left to find a critical point of I ε . Since the procedure of Lyapunov-Schmidt reduction is the same as in [41], we just state some Lemmas and explain the strategy of the proof. Readers interested in the full proof shall refer to [41].

Finite Dimensional Reduction
We will restrict our argument to the existence of a critical point of I ε that concentrates, as ε small enough. For δ, η > 0, fixing y ∈ B δ (x 0 ), we define where we denote E ε,y by We are looking for a critical point of the form For this we introduce a new functional J ε : M ε,η → R defined by J ε (y, ϕ) = I ε U ε,y + ϕ , ϕ ∈ E ε,y .
In fact, we divide the proof of Theorem 1.3 and 1.4 into two steps: Step 1 for each ε, δ sufficiently small and for each y ∈ B δ (x 0 ), we will find a critical point ϕ ε,y for J ε (y, ·) (the function y → ϕ ε,y also belongs to the class C 1 (H ε ) ); Step 2 for each ε, δ sufficiently small, we will find a critical point y ε for the function That is, we will find a critical point y ε in the interior of B δ (x 0 ).
It is standard to verify that y ε , ϕ ε,y ε is a critical point of J ε for ε sufficiently small by the chain rule. This gives a solution u ε = U ε,y ε + ϕ ε,y ε to Eq. (4.1) for ε sufficiently small in virtue of the following lemma.

Lemma 4.2
Assume that V satisfies (V 1 ) and (V 2 ). Then, there exists a constant C > 0, independent of ε, such that for any y ∈ B 1 (0), there holds for ϕ ∈ H ε . Here α denotes the order of the Hölder continuity of V in B r 0 (0).

Lemma 4.3
There exists a constant C > 0, independent of ε and b, such that for i ∈ {0, 1, 2}, there hold
We analyze the asymptotic behavior of j ε with respect to ε first. By Lemmas 4.2, 4.3, 4.4 and 4.6, we have j ε (y) = I ε U ε,y + O l ε ϕ ε + ϕ ε (4.5) Now consider the minimizing problem Assume that j ε is achieved by some y ε in B δ (x 0 ). We will prove that y ε is an interior point of B δ (x 0 ).
To prove the claim, we apply a comparison argument. Let e ∈ R N with |e| = 1 and η > 1. We will choose η later. Let z = η e ∈ B δ (0) for a sufficiently large η > 1. By the above asymptotics formula, we have Applying the Hölder continuity of V , we derive that where η > 1 is chosen to be sufficiently large accordingly. Note that we also used the fact that κ α/2. Thus, by using j (y ) ≤ j (z ) we deduce That is, If y ∈ ∂ B δ (0), then by the assumption (V 2 ), we have for some constant 0 < c 0 1 since V is continuous at x = 0 and δ is sufficiently small. Thus, by noting that B > 0 from Lemma 4.4 and sending → 0, we infer from (4.6) that c 0 ≤ 0.
We reach a contradiction. This proves the claim. Thus y is a critical point of j in B δ (x 0 ). Then Theorems 1.3 and 1.4 now follows from the claim and Lemma 4.1.