Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations

1 Introduction and main results

In this article, we investigate the concentrated solution to the following fractional Schrödinger equation:

under the mass constraint

where s ∈ ( 0 , 1 ) , a > 0 is an arbitrary dimensionless parameter, p ∈ ( 1 , 2 s * − 1 ) with 2 s * = 2 N N − 2 s is the fractional critical Sobolev exponent and V ( x ) satisfies

C 1 , α ( R N ) is a space that consists of those functions v that are one time continuously differentiable and whose α th -partial derivatives are Hölder continues with exponent α with 0 < α ≤ 1 .

The fractional Laplacian ( − Δ ) s , which can be regarded as the infinitesimal generator of a stable Levy process [1], is widely used in many fields such as physics, biological modeling, mathematics, and finance. On the one hand, it can be directly expressed as

where s ∈ ( 0 , 1 ) , C n , s > 0 , and P.V. represents Cauchy principal value. One could refer to [5,11] if they have more interests on it. On the other hand, by [6], this non-local operator ( − Δ ) s in R N can also be expressed in a local form via the following elliptic boundary value problem in the half-space R + N + 1 = { ( x , y ) : x ∈ R N , y > 0 } :

where u ˜ is the s -harmonic extension of u , which is defined as

with

and the constant d N , s satisfies ∫ R N p s ( x , 1 ) d x = 1 . Then, under suitable regularity, ( − Δ ) s u is the Dirichlet-to-Neumann map for this problem, and

where ω s = 2 1 − 2 s Γ ( 1 − s ) ⁄ Γ ( s ) . Moreover, it holds that

u ˜ ∈ L 2 ( y 1 − 2 s , ℬ ) for any compact set ℬ in R + N + 1 ¯ , ∇ u ˜ ∈ L 2 ( y 1 − 2 s , R + N + 1 ) and u ˜ ∈ C ∞ ( R + N + 1 ) . Without loss of generality, we may assume ω s = 1 .

For other problem with fractional Laplacian, one can refer to [2–4,6,9,10,12–16,20,21,25,27–29,31,32] and the references therein.

The aforementioned references all focus on studying the unconstrained problem. Our main emphasis is on understanding the impact of constraint conditions on the properties of the solutions. Precisely, we will discuss the existence and non-degeneracy of multiple spike solution to problem (1.1) and (1.2) with L 2 -constraint, concentrated at the isolated stable critical points of the potential V ( x ) .

With this purpose, the well-known results about the ground state of the following equation:

should be considered. Suppose N ≥ 1 , s ∈ ( 0 , 1 ) , and 1 < p < 2 s * − 1 , then the ground state solution U ∈ H s ( R N ) of (1.5) is unique, which is radial, positive, and strictly decreasing in ∣ x ∣ . Moreover, U ∈ H 2 s + 1 ( R N ) ∩ C ∞ ( R N ) and satisfies

with some constants C 2 ≥ C 1 > 0 . By Lemma C.2 of [16], ∂ U ∂ x j also has the decay estimate

for j = 1 , … , N . The linearized operator L 0 = ( − Δ ) s + 1 − p U p − 1 is non-degenerate, i.e., its kernel is given by

We first consider the following problem without constraint:

where λ > 0 is a large parameter. For any large λ > 0 , by the reduction argument, we constructed a positive k -spike solution for (1.6) of the form

with ∫ R N 1 λ ∣ ( − Δ ) s 2 h λ ∣ 2 + h λ 2 = o λ − N 2 s . Denote u λ = w λ ∫ R N w λ 2 1 2 , then we have ∫ R N u λ 2 = 1 , and there holds that

In fact, let μ = − λ , by discussing the relationship between the parameters a and μ in problem (1.1) and (1.2), we could obtain

where a * is defined as

with an assumption that N ≥ 2 through out this article.

Set

Then, one can find that a λ > 0 , and as λ → + ∞ , a λ → a 0 . Actually, we have found a k -spike solution to (1.7) with normalized L 2 -norm where a λ depends on λ . However, what we really care about is the inverse one: whether one can choose a suitable large λ = λ a > 0 , such that (1.1) and (1.2) hold with

for any fixed a > 0 close to a 0 .

We call a family of non-negative functions { u a } concentrating at a set of points { c 1 , c 2 , … , c k } ⊂ R N , if there exist x a , j → c j as a → a 0 for j = 1 , … , k and a non-negative functions U ∈ H s ( R N ) , which satisfies U ( x ) ≠ 0 and U ( 0 ) = max x ∈ R N U ( x ) , such that

with ∫ R N − 1 μ a ∣ ( − Δ ) s 2 h a ∣ 2 + h a 2 = o ( − μ a ) − N 2 s . We also call such an u a a k -spike solution for a close to a 0 .

The method of locating the concentration points x a , j of a solution u a satisfying (1.1) and (1.2) is standard. By use of the local Pohozaev identities, we will show that a k -spike solution of (1.1) and (1.2) must concentrate at some critical points of V ( x ) . Our first result is as follows.

The converse of Theorem 1.1 is the existence of k -spike solutions u a to (1.1) and (1.2), and we have

Moreover, if we suppose that the function V ( x ) has an isolated maximum point c 0 ∈ R N , then we can obtain a k -spike solution with k concentrated points close to the same point.

Finally, we discuss the non-degeneracy of the solution to problem (1.1) and (1.2), or the non-degeneracy of the linear operator

The study of the non-degeneracy of peak solutions is of great significance as it can aid in further understanding the properties of concentrated solutions for the equation and has numerous interesting applications, such as the construction of new concentrated solutions, etc. The significance of studying non-degeneracy is thoroughly explained in [17]. Precisely, we obtain the following result.

The construction and applications of Pohozaev identities remain vital for deducing the existence and various properties of the concentrated solutions such as the non-degeneracy of the spike solutions. However, the direct establishment of specific local Pohozaev identities proves challenging due to the inherent non-local characteristics of the fractional Laplacian. Consequently, we undertook a harmonic extension of the equation under investigation and conducted estimations for various integrals not typically encountered in conventional local Schrödinger problems, aiming to address this issue. Similar approaches can also be observed in [18] and related works.

Before concluding this section, let us make a comparison with the relevant results from previous researchers.

For the classical Bose Einstein Concentrations problem with s = 1 , p = 3 , N = 2 or 3, Luo et al. [23] recently have given a complete description of the solutions u a , which are related to the problem and concentrate at a * . That is, if u a is a solution of (1.1) and (1.2) with s = 1 , p = 3 , N = 2 or 3 and concentrates at some spikes as a → a 0 , then there holds that

where k is the number of concentration points of u a . Moreover, the existence and local uniqueness of such kind of solution are also provided in it. However, Luo et al. [23] applied a standard comparison method from the ODE theory in the proof of a non-existent result, which is of great importance for obtaining the main results, while that argument could not work in our problem. We use a sliding method corresponding to the non-local operator, which is developed by Chen et al. [8] and would be typically useful in addressing some other similar problems involving the fractional operator, to show such a result instead. Additionally, we discuss in detail the relationship between p and s in the article. In fact, we distinct the case of mass-subcritical, the mass-critical, and the mass-supercritical in our discussion, which, respectively, correspond to p − 1 < 4 s N , p − 1 = 4 s N , and p − 1 > 4 s N .

Dávila et al. [12] considered the equation

where 1 < p < 2 s * − 1 , ε is a small number, and V ( x ) is some smooth enough potential. As the small parameter ε → 0 , they built solutions of (1.11) concentrating at the stable critical points of V ( x ) (with the definition of “stable” given in [22]). However, there is not any small parameter in our present problem, which makes the concentrating behavior of the spike solutions different from that in [12]. In fact, we need to carefully analyze the relationships between different parameters and constraint conditions to initiate the appropriate limiting processes a → a 0 and set forth the framework of reduction. Moreover, we have utilized the local Pohozaev identity techniques to establish the non-degeneracy of normalized spike solutions, an aspect that has not been previously investigated in the literature.

This article is organized as follows. In Section 2, we study the necessary conditions for the existence of the concentrated solutions. Precisely, Theorem 1.1 is proved by applying the Moser iteration, a sliding method, blow-up analysis, and the local Pohozaev identities. Then, the existence of the normalized spike solutions (Theorems 1.2 and 1.3) is proved in Section 3 by the standard reduction argument. Finally, the non-degeneracy of the spike solutions is investigated in Section 4 by the use of the local Pohozaev identity techniques.

2 Necessary conditions