On the singularly perturbation fractional Kirchhoff equations: Critical case

1 Introduction and main results

In this article, we are concerned with the following fractional Kirchhoff problem:

where a , b > 0 are given constants, ε is a small parameter, K ( x ) : R N → R , ( − Δ ) s is the pseudo-differential operator defined by

where ℱ denotes the Fourier transform, 2 s ∗ = 2 N N − 2 s is the standard fractional Sobolev critical exponent. Recently, when ε = 0 in (1.1), Yang and Yu [1] established uniqueness and nondegeneracy for 1 < N < 4 s . Then in this article, we will consider the high-dimensional cases, i.e., N ≥ 4 s and the associated singularly perturbation problems.

If s = 1 , equation (1.1) reduces to the well-known Kirchhoff-type problem; this problem and its variants have been studied extensively in the literature. The equation that goes under the name of Kirchhoff equation was proposed in [2] 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’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 H s ( R N ) to L 2 N N − 2 s ( R N ) :

where the fractional Sobolev space H s ( R N ) is defined by

endowed with the natural norm

From [24], we have

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

are linear combinations of the functions N − 2 s 2 u + x ⋅ ∇ u and ∂ x i u , 1 ≤ i ≤ N . 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 b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) 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 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:

where N 4 < s < 1 , 2 < p < 2 s ∗ = 2 N N − 2 s , has a unique nondegenerate positive radial solution. For the high-dimensional case, Yang [35] proved that uniqueness breaks down for dimensions N > 4 s , 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 ,

Yang and Yu [1] established uniqueness and nondegeneracy for 1 < N < 4 s . Then as a counterpart to these results, we consider the high-dimensional fractional Kirchhoff equations with critical growth. The first results of this article are collected in the following.

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 [39,40, 41,42] and references therein.

Note that if u is a (weak) solution to equation (1.1), then the following Pohozâev identity [19] holds:

Therefore, if u is a solution of equation (1.1), then

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

1. K ∈ L ∞ ( R N ) ∩ C 1 ( R N ) has finitely many critical points and set Crit ( K ) ≔ { x ∣ ∇ K ( x ) = 0 } .

2. ∀ ξ ∈ Crit ( K ) , ∃ β ∈ ( 1 , N ) , and a j ( x ) ∈ C ( R N ) with ∑ j = 1 N a j ( ξ ) ≠ 0 , such that

   K ( x ) = K ( ξ ) + ∑ j = 1 N a j ( ξ ) ∣ x j − ξ j ∣ β + o ( ∣ x − ξ ∣ β ) , x ∈ B δ ( ξ ) ,

   where δ > 0 is a small constant.

3. K satisfies

   1. ∃ ρ > 0 such that x ⋅ ∇ K ( x ) < 0 , ∀ ∣ x ∣ ≥ ρ .

   2. x ⋅ ∇ K ( x ) ∈ L 1 ( R N ) and ∫ R N ( x ⋅ ∇ K ( x ) ) d x < 0 .

4. ∑ a ξ < 0 deg loc ( ∇ K , ξ ) ≠ ( − 1 ) N ,

   where deg loc denotes the local degree and a ξ = ∑ j = 1 N a j ( ξ ) .

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 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 that may change from line to line.

2 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

which satisfies

with some constants C 2 ≥ C 1 > 0 . Let

It is not difficult to check that v ( x ) = u ( c 1 2 s x ) is a solution of equation (2.1). By the uniqueness of Q , we can deduce that

where ξ ∈ R N , λ > 0 . Furthermore,

Then, direct calculation shows that

Therefore, to find solution U ( x ) of (1.4), it suffices to find positive solutions of the above algebraic equation (2.2), and ℰ is a constant, which only depends on a , b , and Q .

Case 1: 1 < N < 4 s : In this case, we have N − 2 s 2 s < 1 , which implies that lim ℰ → + ∞ f ( ℰ ) = + ∞ . Moreover, one has f ( a ) < 0 . Consequently, there exists unique ℰ 0 > a such that f ( ℰ 0 ) = 0 , which means that (1.4) has a unique solution.

Case 2: N = 4 s : In this case, (2.2) becomes

which means that this equation has a unique positive solution

if and only if b < 1 ( − Δ ) s 2 Q 2 2 .

Case 3: N > 4 s : A simple computation implies that

which means that f ( ℰ ) has a unique maximum point

and the maximum of f ( ℰ ) is

It is easy to see that f ( ℰ 0 ) ≥ 0 implies

Since f ″ ( ℰ ) < 0 in ( 0 , + ∞ ) due to N > 4 s , we know that f ( ℰ ) is concave in ( 0 , + ∞ ) . Noting further that f ( 0 ) = − a < 0 and lim ℰ → + ∞ f ( ℰ ) = − ∞ , a sufficient and necessary condition for the solvability of equation (2.2) in ( 0 , + ∞ ) is f ( ℰ 0 ) ≥ 0 . Hence, equation (2.2) has a solution in ( 0 , + ∞ ) if and only if inequality (2.8) holds. Furthermore, we have

1. If b ( − Δ ) s 2 Q 2 2 = 2 s a 4 s − N 2 s ( N − 4 s ) N − 4 s 2 s ( N − 2 s ) N − 2 s 2 s , then equation (2.2) has exactly one positive solution ℰ 0 defined by (2.6);

2. If b ( − Δ ) s 2 Q 2 2 < 2 s a 4 s − N 2 s ( N − 4 s ) N − 4 s 2 s ( N − 2 s ) N − 2 s 2 s , then equation (2.2) has exactly two positive solutions ℰ 1 and ℰ 2 such that ℰ 1 ∈ ( 0 , ℰ 0 ) and ℰ 2 ∈ ( ℰ 0 , + ∞ ) .

3 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 ) ,

acting on L 2 ( R N ) with domain D ( L ) , where

with c = a + b ∫ R N ∣ ( − Δ ) s 2 U ∣ 2 d x and

We observe now that

Differentiating the equation

with respect of the parameters at λ = 1 , ξ = 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

Since ∂ U ∂ x i is nonradially symmetric, we have the following corollary: