Concentration phenomenon for fractional nonlinear Schr\"{o}dinger equations

We study the concentration phenomenon for solutions of the fractional nonlinear Schr\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \begin{equation}\label{e:abstract} (-\varepsilon^2\Delta)^sv+Vv-|v|^{\alpha}v=0\quad\mbox{in}\quad\mathbf R^n, \end{equation} where $n =1, 2, 3$, $\max\{\frac{1}{2}, \frac{n}{4}\}<s<1$, $1 \leq \alpha<\alpha_*(s,n)$, $V\in C^3_{b}(\mathbf{R}^n)$. Here the exponent $\alpha_*(s,n)=\frac{4s}{n-2s}$ for $0<s<\frac{n}{2}$ and $\alpha_*(s,n)=\infty$ for $s \geq\frac{n}{2}$. Then for each non-degenerate critical point $z_0$ of $V$, there is a nontrivial solution of equation (\ref{e:abstract}) concentrating to $z_0$ as $\varepsilon\to 0$.


Introduction
We mainly consider the fractional Schrödinger equation in this paper. The Schrödinger equation, often called the Schrödinger wave equation, is a fundamental equation of quantum mechanics which describes how the wave function of a physical system evolves over time. This equation is not derived from a conical set of axioms. For example, Schrödinger himself, arrived at this equation by inserting de Broglies relation into a classical wave equation. Another attempt to derive Schrödinger equation from classical physics was using Nelson's stochastic theory [59]. Hall and Reginatta [44] showed that the Schrodinger equation can be derived from the exactly uncertainty principle. It is also well known that Feynman and Hibbs used path integrals over Brownian paths to derive the standard Schrödinger equation [35].
In quantum physics, the Feynman path integral approach to quantum mechanics was the first successful attempt applying the fractality concept that was first introduced by Mandelbrot [55]. Recently, Laskin extended the fractality concept and formulated fractional quantum mechanics as a path integral over the Lévy flights paths [50,49,52,51]. Through introducing the quantum Riesz fractional derivative, he constructed the space fractional Schrödinger equation Laskin showed the hermiticity of the fractional Hamilton operator and established the parity conservation law. Energy spectra of a hydrogen like atom and of a fractional oscillator were also computed. Mathematically, (−∆) s is defined as Here P. V. is a commonly used abbreviation for 'in the principal value sense' and C(n, s) = π −(2s+n/2) Γ(n/2+s) Γ(−s) . It is well known that (−∆) s on R n with s ∈ (0, 1) is a nonlocal operator. In the remarkable work of Caffarelli and Silvestre [14], the authors express this nonlocal operator as a generalized Dirichlet-Neumann map for a certain elliptic boundary value problem with local differential operators defined on the upper half-space R n+1 x ∈ R n , t > 0}. That is, given a solution u = u(x) of (−∆) s u = f in R n , one can equivalently consider the dimensionally extended problem for u = u(x, t), which solves (1.1) Here the positive constant d s > 0 is explicitly given by The formulation (1.1) in terms of local differential operators plays a central role when deriving bounds on the number of sign changes for eigenfunctions of fractional Schrödinger operators H = (−∆) s + V . Using this idea, [37] and [38] obtain certain sharp oscillation bounds for eigenfunctions of H. In these two papers, the authors proved the uniqueness and nondegeneracy ground state solutions 0 ≤ u 0 = u 0 (|x|) ∈ H s (R n ) for the nonlinear problem which settled a conjecture by [46,68] and generalized a classical result by Amick and Toland [5] on the uniqueness of solitary waves for the Benjamin-Ono equation. If s = 1, the uniqueness and nondegeneracy for ground states of (1.2) was due to [48].
In this paper, we consider fractional Schrödinger equation where ε is the sufficiently small positive constant which is corresponding to the Plank constant, γ > 0, n ≥ 1, s ∈ (0, 1), α ∈ (0, α * (s, n)). The exponent α * (s, n) satisfies α * (s, n) = 4s n−2s , for 0 < s < n 2 , +∞, for s ≥ n 2 . (1.4) We shall seek the so-called solitary waves which is of form where v is real-valued and E is some constant in R. (1.5) solves (1.3) provided the standing wave v(x) satisfy the nonlinear eigenvalue equation For simplicity of the notation, we shall assume that γ = 1, so Equation (1.6) is reduced to Since we assume that

by a suitable choice of E we can assume that
V − E is positive. Finally, we have the following equation where V is a positive function.
The main result of this paper is with z ε → z 0 , φ zε,ε → 0 in H 2s (R n ) and hence in C 0 b (R n ) as ε → 0, and u 0 is the unique positive radial ground state solution of (1.2).
If s = 1, (1.8) is the classical Schrödinger equation and the corresponding result of Theorem 1.1 was established by Floer and Weinstein in [36]. There are large mounts of research on this equation in the past two decades. We limit ourselves to citing a few recent papers [3,4,6,7,28,10,42,53,29], referring to their bibliography for a broader list of works, although still not exhaustive.
In the case s ∈ (0, 1), the operator (−∆) s on R n is nonlocal, while −∆ is local. As pointed out in [62], when studying the singularly perturbed equation (1.8), the standard techniques that were developed for the local laplacian do not work out-of-the-box since these techniques heavily rely on blow-up and local estimates, and need fine properties of solutions to the limiting problem. Moreover, (−∆) s may kill bumps by averaging on the whole space. Fortunately, based on the work of [37], [38] and by carefully using the cut-off function technique, we can recover the main ingredients of the Lyapunov-Schmidt reduction method in the fractional case (although the ground bound state got in [38] not decay exponentially, the speed of decay is enough for us to obtain the estimate).
The rest of this paper is organized as follows. In Section 2, we recall the notations of fractional Laplacian and some known results, especially the uniqueness and nondegenerace results of [37] and [38]. In Section 3, we prove the invertibility of the linearized operator at the ground state solution. In Section 4 and 5, we prove the main result of this paper by the Lyapunov-Schmidt reduction method.

Preliminaries
In this section, we recall some properties of the fractional order Sobolev spaces and the results of [37], [38] which are crucial in our proof of the main theorem.

Fractional order Sobolev spaces.
In this subsection, we recall some useful facts of the fractional order Sobolev spaces. For more details, please see, for example, [1], [63], [30].
Consider the Schwartz space S of rapidly decaying C ∞ functions on R n . The topology of this space is generated by the seminorms where ϕ ∈ S. Let S ′ be the set of all tempered distributions, which is the topological dual of S. As usual, for any ϕ ∈ S, we denote by the Fourier transformation of ϕ and we recall that one can extend F from S to S ′ .
When s ∈ (0, 1), the space H s (R n ) = W s,2 (R n ) is defined by (1 + |ξ| 2s )|F u(ξ)| 2 dξ < +∞ and the norm is Here the term is the so-called Gagliardo (semi) norm of u. The following identity yields the relation between the fractional operator (−∆) s and the fractional Sobolev space H s (R n ), for a suitable positive constant C depending only on s and n. When s > 1 and it is not an integer we write s = m + σ, where m is an integer and σ ∈ (0, 1). In this case the space H s (R n ) consists of those equivalence classes of functions and this is a Banach space with respect to the norm Clearly, if s = m is an integer, the space H s (R n ) coincides with the usual Sobolev space For a general domain Ω, the space H s (Ω) can be defined similarly.
On the Sobolev inequality and the compactness of embedding, one has Theorem 2.1. [1] Let Ω be a domain with smooth boundary in R n . Let s > 0, then As the usual singularly perturbed problem, when dealing with (1.8), we rescale the variable x so that the term involving V (x) appears as a small perturbation. Without loss of generality, we assume that the non-degenerate minimum point of V lies at the origin with value 1.
If s = 1, the Uniqueness and non-degeneracy of the ground state for (1.2) is due to [48].
Very recently, in the paper [38], Frank, Lenzmann and Silvestre proved uniqueness and nondegeneracy of ground state solutions for (1.2) in arbitrary dimension n ≥ 1 and any admissible exponent 0 < α < α * (s, n). This result classifies all optimizers of the fractional Gagliardo-Nirenberg-Sobolev inequality (2.4) For convenience, we summarize the results of [37] and [38] in the following theorems.  2), then there exists some x 0 ∈ R n such that u 0 (· − x 0 ) is radial, positive and strictly decreasing in |x − x 0 |. Moreover, the function u 0 belongs to H 2s+1 (R n ) ∩ C ∞ (R n ) and it satisfies
The nondegeneracy implies that 0 is an isolated spectral point of L 0 . More precisely, for all φ ∈ (kerL 0 ) ⊥ , one has for some positive constant c. By Lemma C.2 of [38], it holds that, for j = 1, · · ·, n, ∂ j u 0 := ∂ x j u 0 has the following decay estimate, It is well known that when s = 1, the ground state solution of (1.2) decays exponentially at infinity. But from Thoerem 2.3, when s ∈ (0, 1), the corresponding ground bound state solution decays like 1 |x| n+2s when |x| → ∞. Fortunately, this polynomial decay is enough for us in the estimates of our proof, see Section 3, 4 and 5.

Linearized operator at the ground state solution
In this section, we study the linearized operator at the ground state solution u 0 .
3.1. Linearized operator. Denote by · 0 and · 2s the norm in L 2 = L 2 (R n ) and H 2s = H 2s (R n ) respectively. We have the following lemma.
In fact, let and η is a smooth cut-off function such that Then {ηψ i } be a bounded sequence of function in H 2s (Ω 1 ). Since H 2s (Ω 1 ) is a Hilbert space, there exists a function f ∈ H 2s (Ω 1 ) such that By the compactness of embedding i 0 2s : H 2s (Ω 1 ) → L 2 (Ω 1 ) (Theorem 2.2), we have that ηψ i → f in L 2 (Ω 1 ). Then . However, by the argument in Claim 1, ψ i weakly converges to 0 in L 2 (Ω). Therefore, (3.16) So we have Claim 2.
(3.17) From (3.11), (3.13), (3.9) and (3.17), we obtain that (3.21) By (3.19), Estimating the first term on the right hand side of (3.22), we obtain where ρ > 0 and B ρ is the ball centered at 0 with radius ρ in R n . By the Hölder inequality, Since (V i − 1)∂ j u 0 → 0 uniformly in B ρ for some fixed ρ > 0 and ψ i 0 is bounded, we have Again by the Hölder inequality and V ∈ C 3 b , Since ∂ j u 0 ∈ L 2 (R n ) for 1 ≤ j ≤ n and ψ i is bounded in L 2 (R n ), we have The second term (3.22) goes to 0 since ψ i converges weakly to 0. Therefore, we have Hence, by (3.18) and (3.28), we have On the other hand, for any ϕ ∈ H 2s (R n ), So for all 1 ≤ i < ∞, the spectrum of G i is contained in [1, ∞). Therefore, G i is invertible, and the operator norm of G −1 i from L 2 (R n ) to H 2s (R n ) is not greater then 1. Thus, we obtain By (3.29), this is impossible. This completes the proof.

Nonlinear problem
In this section, we shall prove that for each sufficiently small z and ε, there is an element φ z,ε in K ⊥ z,ε such that π ⊥ z,ε S ε (u z,ε + φ z,ε ) = 0. From now on, we assume s > max{ n 4 , 1 2 }. By the expansion (3.3), we have For simplicity, denote π ⊥ z,ε S ε (u z,ε ) by S ⊥ z,ε , and π ⊥ z,ε N z,ε by N ⊥ z,ε . From Lemma 3.2, we know that L z,ε is invertible. Then Equation (4.1) is equivalent to a fixed point of the map M z,ε on H 2s (R n ) given by 3) We will prove that M z,ε is a contraction on a suitable neighborhood of 0.
Lemma 4.1. There exist positive constants C, δ independent of z and ε, such that for all φ 1 , φ 2 ∈ H 2s with φ 1 2s ≤ δ, φ 2 2s ≤ δ, it holds that and Proof. Since by assumption s > n 4 , it follows from Theorem 2.1 that H 2s (R n ) ֒→ L 4 (R n ) (4.6) and (4.7) By (3.4) and the imbedding above, we have, for φ 2s ≤ 1, where θ 1 is a positive function with value not greater than 1. For the second inequality, we compute 2s . Here θ 2 and θ 3 are functions with similar property as θ 1 .
In the following, for any function f , we denote the maximum of f on the closed ball B r of radius r at z by f r (z).

The reduced problem and proof of the main theorem
In this section, we shall prove the main result of this paper.
This completes the proof.

The proof of the main result.
Proof of Theorem 1.1. By assumption 0 is a non-degenerate critical point of V , so the image set of S := ∂B 1 by v 0 is diffeomorphic to S. By Lemma 5.1, for sufficiently small ε, v ε (S) is also diffeomorphic to S. Then there is a point z 0 ∈ B 1 such that v ε (z 0 ) = 0. In fact, if not, then for all z ∈ B 1 , v ε (z) = 0. Letṽ ε (z) = vε(z) |vε(z)| . Since v ε (S) is diffeomorphic to S, we haveṽ ε (S) = S andṽ ε (B 1 ) = S. By the Brouwer fixed point theorem, it is impossible.