SOLUTIONS FOR THE KIRCHHOFF TYPE EQUATIONS WITH FRACTIONAL LAPLACIAN∗

Due to the singularity and nonlocality of the fractional Laplacian, the classical tools such as Sturm comparison, Wronskians, Picard–Lindelöf iteration, and shooting arguments (which are all purely local concepts) are not applicable when analyzing solutions in the setting of the nonlocal operator (−∆). Furthermore, the nonlocal term of the Kirchhoff type equations will also cause some mathematical difficulties. The present work is motivated by the method of semi-classical problems which show that the existence of solutions of the Kirchhoff type equations are equivalent to the corresponding associated fractional differential and algebraic system. In such case, the existence of the fractional Kirchhoff equation can be obtained by using the corresponding fractional elliptic equation. Therefore some qualitative properties of solutions for the associated problems can be inherited. In particular, the classical uniqueness results can be applied to this equation.


Introduction
In this note, we will consider the Kirchhoff  and its corresponding associated equation is the so-called Gagliardo (semi)norm of u (see Di Nezza, Palatucci and Valdinoci [26]), s ∈ (0, 1), N > 2s is a positive integer, λ, µ ∈ R, M : R + → R + is a continuous function whose properties will be introduced later, and the functions f and g are continuous in R.
Fiscella and Valdinoci in [10] first proposed a stationary Kirchhoff variational model as follows: 4) in bounded domains of R N . In model (1.4) the authors took into account the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string, see Appendix A in [10] for more details. Recently, the results of [10] had been extended in [3] to the non-degenerate case. Other related problems were also considered in [9,29,32,36]. When the fractional Laplacian (−∆) s is replaced by the p-Laplacian, some problems were also established by some authors, for example, see Caponi and Pucci [4], Pucci, Xiang and Zhang [27].
When s = 1, equation (1.4) is reduced to the standard Kirchhoff type equation which was first proposed by Kirchhoff in [17] to describe the transversal oscillations of a stretched string. The boundary problems then attracted several researchers mainly after the work of Lions [22], where a functional analysis approach was proposed to attack it. For more mathematical and physical background, we refer readers to [2, 3, 7, 13-15, 20-23, 26, 33-35], and the references therein.
The symbol (−∆) s with s ∈ (0, 1) is called the fractional Laplacian which can be defined by dy for x ∈ R N , where C (N, s) is a dimensional constant that depends on N and s, see Di Nezza, Palatucci and Valdinoci [26]. Due to the singularity and nonlocality of the kernel, it is evident that the theory of ordinary differential equations (ODE) itself does not provide any means to establish such results. In particular, classical tools such as Sturm comparison, Wronskians, Picard-Lindelöf iteration, and shooting arguments (which are all purely local concepts) are not applicable when analyzing radial solutions in the setting of the nonlocal operator (−∆) s , see Frank and Lenzmann [11], Frank, Lenzmann and Silvestre [12], Moroz and Van Schaftingen [25] and Di Nezza, Palatucci and Valdinoci [26]. Furthermore, the nonlocal term M will also cause some mathematical difficulties, for example, see [5-9, 13-15, 19, 20, 22, 23, 26, 27, 29, 33-36] and the references therein.
The present work is motivated by the method of semi-classical problems (see also [19], [14] and [31]). For example, we consider the semi-classical problem of the form where ε > 0 is a small parameter, typically related to the Planck constant. If u ε is a solution of (1.6) and a ∈ R N , then the function v ε (y) = u ε (a + εy) solves the related equation If v is a solution of (1.7), then the function is a solution of (1.6). See Moroz and Van Schaftingen [25] (see also Van Schaftingen and Xia [28]). In this note, we mainly prove the following facts: If Q (x) is a solution of (1.2), then is unique which implies that u (x) is also unique. If v (x) is the second solution of (1.1), in view of Theorem 2.1 (in the next section), we can obtain the corresponding second solution of (1.2), which is a contradiction. By using this fact, we can obtain some existence results from the known results of (1.2). On the other hand, our methods are also valid for the other pseudo-differential operators, for example, the operator −∆ + m 2 s/2 , see Frank, Lenzmann and Silvestre [12].

Main Results
First of all, we assume that u ∈ H s R N is a nonzero solution of (1.1) and denote For any a ∈ R N , in view of (1.7), we know that the function v c (y) = u c (a + cy) Now, we assume that Q (x) is a solution of (1.2) and denote Notice that is a solution of (1.1).
In view of Theorems 2.1 and 2.2, we have immediately the following result.
Corollary 2.1. The existence of solution for (1.1) is equivalent to the existence of solution for the pseudo-differential and algebraic system It is well known that it is very difficult to obtain the uniqueness of the solution for PDEs. However, the following result is clear.
is a unique solution of (1.1) up to translations. Example 2.1. Recently, Frank, Lenzmann and Silvestre [12] considered the nondegeneracy, regularity estimates and uniqueness results for ground state solutions of the nonlinear equation settled conjecture by Kenig et al. [16] and Weinstein [30] for any dimension N > 1, and generalized the classical uniqueness result by Amick and Toland [1] on the uniqueness of solitary waves for the Benjamin-Ono equation and the case N = 1 dimension in Frank and Lenzmann [11]. In the local case for s = 1, the uniqueness and nondegeneracy of ground states for problem (2.4) was established in a celebrated paper by Kwong [18] (see also Coffman [5] and McLeod [24]). Now, we consider the algebraic equation If 2s < N < 4s, clearly, (2.5) exists a unique positive root.
In fact, the unique family of solutions is 2 with x 0 ∈ R. Unfortunately, Corollary 2.3 is not valid because s = 1/2.

Remark 2.2.
A very special Kirchhoff function M is given by a+bmt m−1 , a, b ≥ 0, a + b > 0, m ≥ 1 and t ≥ 0, see Pucci, Xiang and Zhang [27]. The similar results can also be obtained.