On the orbital stability of fractional Schr\"{o}dinger equations

We show the existence of ground state and orbital stability of standing waves of fractional Schr\"{o}dinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.


Introduction
In this paper we consider the following Cauchy problem: Here n ≥ 1, 0 < s < 1, Φ : R 1+n → C and N : R n × C → C.
The equation (1.1), called fractional nonlinear Schrödinger equation, appears in many fields in science and engineering. Other domains of applications of such equations, involving the fractional powers of the Laplacian, arise in medicine (RMI and heart diseases). It is also of a great importance in astrophysics, signal processing, turbulence, and water waves, where the cases s = 1 4 and s = 3 4 are the most relevant (see [12] and references therein). We will focus our attention on the orbital stability of standing waves of this Schrödinger equation. Our results generalize those of [1] and [9]. The paper [1] seems to be the first one dealing with the orbital stability in the fractional case. The authors studied (1.1) for s = 1 2 , n = 1 with an autonomous cubic power nonlinearity. In [9] the authors extended the previous paper to general nonlinearities for n ≥ 2, but without showing the uniqueness of weak solutions. In our work, we use the concentration-compactness lemma to prove the orbital stability of standing waves, as stated by Cazenave and Lions, but without introducing a problem at infinity. See Proposition 1.1 below. We also establish the uniqueness of weak solutions to the Cauchy problem (1.1) under suitable conditions on V and f . Unlike the usual Schrödinger equation (s = 1), it is not an easy matter to show the uniqueness in the fractional setting, since we cannot utilize the standard Strichartz estimates due to a regularity loss ( [7]). Here we exploit weighted Strichartz estimates without regularity loss.
Instead, some integrability conditions on a, V are necessary to treat the weights. The details of uniqueness of weak solutions will be discussed in Section 4.2.
To present our results let us set N(x, Φ) = V (x)Φ + f (x, Φ) and describe assumptions: The functions V : R n → R and f : R n × C → C are measurable and f satisfies that , and for some ℓ with 0 < ℓ ≤ 4s n−2s and nonnegative measurable functions a, b |f (x, z)| ≤ a(x)|z| ℓ+1 , (1.2) for all x ∈ R n and for all x ∈ R n and z 1 , z 2 ∈ C. Let us also set F (x, |Φ|) = |Φ| 0 f (x, α) dα. Then we define a functional J by and also M by M(Φ) = |Φ| 2 dx. By a standing wave of (1.1) we mean a solution Φ(t, x) of the form e iωt u for some ω ∈ R, where u is a solution of the equation Some authors have studied the existence of u under suitable conditions on f . For this purpose they showed that if (u k ) is a minimizing sequence of the problem with a prescribed positive number µ, then u k → u in H s up to a subsequence, where u is a solution of (1.4) for some ω. Now by following the definition of Cazenave-Lions, we set Our first result is the existence of ground states. Proposition 1.1. Let n ≥ 1, 0 < s < 1 and 0 < ℓ < 4s n . Suppose that for n 2s < p 1 , p 2 < ∞ and f satisfies (1.2) with a ∈ L q 1 loc + L q 2 (|x| > 1) for 2n 4s−nℓ < q 1 , q 2 < ∞, and that there exist κ, R, N, δ > 0 and β, σ > 0 such that for any |z| ≤ N and |x| ≥ R and for all x, z, θ > 1. Then O µ is not empty for any µ > 0. When ℓ = 4s n , we assume that When a ∈ L ∞ , by adding some natural asymptotic conditions on V and f , one can show the existence of ground state. See [8] and [11]. But if we assume the radial symmetry of V (x), f (x, ·) in x, then using the compactness of embedding H s rad ֒→ L p , 1 2 < s < n 2 , 2 < p < 2n n−2s (see [6]), we have the following.
and f (x, ·) be radially symmetric with a ∈ L ∞ . If F satisfies (1.5) and (1.6), then O µ ∩ H s rad is not empty for any µ > 0. When ℓ = 4s n , O µ is not empty for sufficiently small µ > 0.
We say that O µ is stable if it is not empty and satisfies that for any ε > 0, there exists a Let us introduce our main result.
In view of the well-posedness results in Section 4.2 below, by assuming that V, a, b are smooth and have suitable decay at infinity, we get the orbital stability for 1 2 < s < 1, 0 < ℓ ≤ 4s n and ℓ < ℓ 0 , where ℓ 0 = ∞, if n = 1, 2s−1 2s(1−s) if n = 2 and n(2s−1) (n−2s)(n−1) if n ≥ 3. The critical case ℓ = 4s n can be included when n = 1, 2, 3. Our paper is organized as follows. In Section 2 we will prove the existence of ground states by showing the compactness of the minimizing sequences of the constrained variational problem. This is a key step to show the orbital stability of standing waves. This goal is achieved in Theorem 1.3 and Theorem 1.4, which will be shown in Section 3. In the last section, we will discuss the uniqueness of solutions of the Cauchy problem for a large class of nonlinearities.
On the other hand, from the proof of Lemma 3.1 of [11] one can easily show that I µ is continuous on (0, ∞).
Let (u j ) ⊂ S µ be a minimizing sequence such that J(u j ) → I µ . From (2.1) we deduce that (u j ) is bounded in H s . To show O µ = ∅ we will use the concentration-compactness (see [14]). Let the concentration function m j be defined by Then 0 ≤ ν ≤ µ and there exists a subsequence u j (still denoted by u j ) satisfying the following properties 1 .
(2) If ν = µ, then there exists a sequence (y j ) ⊂ R n and u ∈ H s such that for any p with 2 ≤ p < s * u j (· + y j ) → u as j → ∞ in L p and given ε > 0 there exists j 0 (ε) and r(ε) such that If ν = 0, then for 0 < ℓ < 4s n we have One can verify the concentration-compactness by following the arguments in [14] or [2]. We omit the details.
Therefore ν = µ. Set u j (x) = u j (x + y j ). Then u, u j ∈ S µ and u j → u in L p for all 2 ≤ p < s * . On the other hand, (u j ) is bounded in H s . So, there is a subsequence (still denoted by u j ) converging to v weakly in H s and strongly in L p loc for any 1 ≤ p < s * . Now for any ε > 0 we can find R, j 0 > 1 such that Suppose that (y j ) is unbounded. Then up to subsequence we may assume that |y j | → ∞. Since u j → u in L 2 , u j − u(· − y j ) → 0 in the sense of distributions. But u(· − y j ) → 0 and u j → v in the sense of distributions and thus v = 0. That is, P (u j ) → 0 as j → ∞. This implies that I µ = lim j→∞ J(u j ) ≥ 0, which contradicts (2.3). So, (y j ) is bounded. Now let R * = sup j≥1 |y j |. Then for any ε > 0 we have and thus This completes the proof of Proposition 1.1.
By the same argument as above it follows that there exists a function u ∈ H s such that J(u) = I µ , provided we can show that there exists a subsequence (u j ) (denoted by u j again) P (u j ) → P (u) as j → ∞. In fact, by the compact embedding H s ֒→ L p , 2 < p < 2n n−2s , we can find a subsequence (u j ) such that u j → u in L p . So, it is clear that we deduce that for any ε > 0 there is an R > 0 such that Since V ∈ L ∞ loc and the embedding H s ֒→ L 2 loc is compact, (up to a subsequence) there exists j 0 > 0 such that for some sequence t j ∈ R and ε 0 , where Φ j (t, ·) is the solution of (1.1) corresponding to the initial data Φ j 0 . Let w j = Φ j (t j , ·). Since w ∈ S µ and J(w) = I µ , it follows from the continuity of L 2 norm and J in H s that Thus we deduce from the conservation laws that Therefore if w j has a subsequence converging to an element w ∈ H s such that w L 2 = µ and J(w) = I µ . This shows that w ∈ O µ but which contradicts (3.1). Since O µ is not empty, to show the orbital stability of O µ one has to prove that any sequence (w j ) ⊂ H s with is relatively compact in H s . Since I µ is continuous w.r.t µ ∈ (0, ∞) and ℓ ≤ 4s n , by the arguments in the proof of Proposition 1.1 we may assume that (w j ) is bounded in H s and also verify from (2.10) that by passing to a subsequence there exists w ∈ H s such that This implies w j → w in H s and thus the relative compactness.
Proof of Proposition 4.1. To show the existence of weak solutions we follow the standard regularizing argument (for instance see [2]). For this purpose we have only to verify that Here we used the Sobolev embedding H s ֒→ L r i . If n = 1 and 1 2 < s < 1, for any 1 < q 1 , q 2 ≤ ∞ we can find r i , l i ∈ [2, ∞) such that 1 If n = 1 and s = 1 2 , then for any 1 < q 1 , q 2 ≤ ∞, we can find r i , l i ∈ [2, ∞) and p i ≫ 1 such that 1 This proves Proposition 4.1.
where U(t) = e it(−∆) s . Let Ψ be another weak solution of (1.1) with the same initial data as Φ So Φ = Ψ on [−t 1 , t 2 ] for sufficiently small t 1 , t 2 . Let I = (−a, b) be the maximal interval Without loss of generality, we may assume that a < T 1 and b < T 2 . Then for a small ε > 0 we can find a < t 1 < T 1 , b < t 2 < T 2 such that Then the H s -weak solution to (1.1) is unique.
Proof of Proposition 4.3. For the uniqueness we will use the following weighted Strichartz estimate (see for instance Lemma 6.2 of [3] and Lemma 2 of [4]).
Lemma 4.4. Let n ≥ 2 and 2 ≤ q < 4s. Then we have and C is independent of t 1 , t 2 .
In [3] it was shown that The inequality (4.2) can be derived by Sobolev embedding on the unit sphere. Here D σ = √ 1 − ∆ σ , ∆ σ is the Laplace-Beltrami operator on the unit sphere.
We can proceed with the almost same way as the proof of uniqueness for high-d case. The only difference is the range of ℓ. For the proof we need 1 m 2 , 1 m 2 ≥ 0, for which we must have , ℓ < n(2s − 1) 2s(n − 2s) , n ≥ 2, respectively. In 2-d, the former is bigger than the latter and vice versa in high-d.
By exactly the same way, we have the following. 4.2. Well-posedness. By using the argument of [2] one can show that the uniqueness implies actually well-posedness and conservation laws: • Φ depends continuously on ϕ in H s , • M(Φ(t)) = M(ϕ) and J(Φ(t)) = J(ϕ) ∀ t ∈ (−T min , T max ).
We leave the details to the readers. In this section we remark on the global well-posedness. (1) If f (x, |τ |) ≤ 0 for all x and τ , and Φ is the unique solution to (1.1), then . The continuity argument implies the global well-posedness that T min = T max = ∞.
(4) If ℓ = 4s n , then since n−2s . In [3] the authors considered the well-posedness for Hartree type nonlinearity by using various Strichartz estimates. Indeed, they utilized weighted or angularly regular Strichartz estimate to control the Hartree type nonlinearity.
However, if the power type nonlinearity f is involved, then the situation is quite different.
It is not easy to handle angular regularity for which we need a high regularity of f . To avoid this we assume the radial symmetry of f and initial data.
Under the fractional and power type setting, an alternative Besov norm is useful, which is stated as follows: for 0 < s < 1, 1 ≤ r < ∞ The following is the local well-posedness result.
Proof of Proposition 4.7. For simplicity we only consider the well-posednss on [0, T ]. Let (X ρ T , d X ) be a metric space with metric d X defined by L q T B denotes L q t ([0, T ]; B) for some positive T and Banach space B. Since H s and B s r 0 are reflexive Banach space, one can readily show that X ρ T is complete. We define a mapping N on X ρ T by We use the standard contraction mapping argument. For any Φ ∈ X ρ T we have from (4.5) and (4.6) with q 0 , r 0 that From the fractional Leibniz rule, we have (4.10) On the other hand, since 1 r ′ 0 = ℓ r + 1 r 0 for 1 r = n−2s n(ℓ+2) which equals 1 r 0 − s n , from the condition (1.2) and Sobolev embedding B s r 0 ֒→ L r it follows that .
So, we get as above f (· + y, Φ(· + y)) − f (·, Φ(·)) L r ′ Now let us turn to the nonlinear estimate (4.9). We take Hölder's inequality in t-variable From the condition of ℓ we have 1/q 1 ≥ 0, and 1/q 1 = 0 when ℓ = 4s n−2s . Thus we get If ℓ < 4s n−2s , then since U(·)ϕ L q 0 T B s r 0 ϕ H s , T can be chosen to be dependent only on C and ϕ H s to guarantee N (Φ) ∈ X ρ T . If ℓ = 4s n−2s , then we first choose T, ρ such that C(T ρ + T 1 q 1 ρ ℓ+1 ) ≤ ρ/2 and then choose smaller T such that U(·)ϕ L q 0 T B s r 0 ≤ ρ 2 , which means N (Φ) ∈ X ρ T . Now we show that N is a Lipschitz map for sufficiently small T . Let Φ, Ψ ∈ X ρ T . Then from the same estimates as above we have Thus for smaller T and ρ the mapping N is a contraction and there is a fixed point Φ of N satisfying (4.1). The uniqueness and time continuity follows easily from the equation (4.1) and Strichartz estimate. We omit the details.
Remark 1. The mass and energy conservations are straightforward from the uniqueness.
One can also show the conservation laws by the argument for Strichartz solutions of [15].