A UNIFIED APPROACH FOR ENERGY SCATTERING FOR FOCUSING NONLINEAR SCHR¨ODINGER EQUATIONS

. We consider the Cauchy problem for focusing nonlinear Schr¨odinger equation where N ≥ 1, α > 4 N and α < 4 N − 2 if N ≥ 3. We give a criterion for energy scattering for the equation that covers well-known scattering results below, at and above the mass and energy ground state threshold. The proof is based on a recent argument of Dodson-Murphy [Math. Res. Lett. 25(6):1805–1825, 2018] using the interaction Morawetz estimate.


(Communicated by Joachim Krieger)
Abstract. We consider the Cauchy problem for focusing nonlinear Schrödinger equation We give a criterion for energy scattering for the equation that covers well-known scattering results below, at and above the mass and energy ground state threshold. The proof is based on a recent argument of Dodson 1. Introduction. We consider the Cauchy problem for a class of focusing intercritical nonlinear Schrödinger equations i∂ t u + ∆u = −|u| α u, (t, x) ∈ R × R N , u(0, x) = u 0 (x), where u : R × R N → C, u 0 : R N → C, N ≥ 1 and 4 N < α < α * with It is well-known that (1) is locally well-posed in H 1 (see e.g. [3]). Moreover, local solutions enjoy the conservation of mass, energy and momentum x)| α+2 dx = E(u 0 ), (Energy) P (u(t)) = Im R N u(t, x)∇u(t, x)dx = P (u 0 ). (Momentum)
A direct computation gives This shows that the scaling (2) leaves theḢ γc -norm of initial data invariant, where Since we are interested in the mass and energy intercritical case, it is convenient to define the exponent The equation (1) admits a global non-scattering solution u(t, x) = e it Q(x), where Q is the unique positive radial solution to the elliptic equation The energy scattering for the equation (1) below the ground state threshold was first proved by Holmer-Roudenko [12] for N = 3, α = 2 and radially symmetric initial data. Duyckaerts-Holmer-Roudenko [6] later improved this result by removing the radial assumption. This result was extended to any dimension N ≥ 1 by Cazenave-Fang-Xie [9], Akahori-Nawa [1] and Guevara [11]. More precisely, we have the following result. Theorem 1.1 (Scattering below the ground state threshold [1,9,6,11]). Let N ≥ 1, where K(f ) := ∇f 2 L 2 .
Then the corresponding solution to (1) exists globally in time and scatters in H 1 in both direction, that is, there exist u ± ∈ H 1 such that The energy scattering for (1) at the ground state threshold was studied by Duyckaerts-Roudenko [7] for N = 3 and α = 2. In particular, they proved the following result. Theorem 1.2 (Scattering at the ground state threshold [7]). Let N = 3 and α = 2. Let u 0 ∈ H 1 satisfy E(u 0 )M (u 0 ) = E(Q)M (Q) (9) and K(u 0 )M (u 0 ) < K(Q)M (Q).
Then the corresponding solution to (1) exists globally in time. Moreover, it either scatters in H 1 in both directions or equals to Q − (t) up to the symmetries, where Q − (t) is a radial global solution to (1) satisfying E(Q − ) = E(Q), K(Q − (0)) = K(Q), and Q − scatters in H 1 backward in time.
Assuming V (0) < ∞, the following virial identities hold: We have the following result. Theorem 1.3 (Scattering above the ground state threshold [8]). Let u 0 ∈ H 1 satisfy and and where Then the corresponding solution to (1) exists globally and scatters in H 1 forward in time.
The proofs of Theorems 1.1, 1.2 and 1.3 are based on the concentration-compactness-rigidity argument introduced by Kenig-Merle [14]. Dodson-Murphy [5] gave an alternative simple proof for the energy scattering of (1) below the ground state threshold with N ≥ 3 and α = 4 N −1 that avoids the concentration-compact-rigidity argument. Recently, Gao-Wang [10] gave a unified proof for the energy scattering of theḢ Then T * = ∞ and u scatters in H 1 forward in time.
This result has essentially been given in [8,Theorem 3.7] using the concentrationcompactness-rigidity argument of [14] (see also Appendix). In this paper, we give an alternative simple proof in dimensions N ≥ 3 using the idea of Dodson-Murphy [5].
Suprisingly enough, Theorem 1.4 implies the energy scattering below and above the ground state threshold given in Theorems 1.1 and 1.3 respectively. More precisely, we have the following result. Lemma 1.5. Let N ≥ 1 and 4 N < α < α * . Let u 0 ∈ H 1 satisfy one of the following conditions: • (6) and (7); • (11) and (12)- (15). Then the corresponding solution to (1) satisfies (17). In particular, the solution exists globally and scatters in H 1 forward in time.
Another application of Theorem 1.4 is the following result at the ground state threshold. Lemma 1.6. Let N ≥ 1 and 4 N < α < α * . Let u 0 ∈ H 1 satisfy (9) and (10). Then the corresponding solution to (1) exists globally in time. Moreover, either it scatters in H 1 forward in time or there exists t n → ∞ and (y n ) n≥1 ⊂ R N such that for some θ ∈ R and x 0 ∈ R N as n → ∞. Remark 1. This result is weaker than that of Theorem 1.2. However, it holds for the whole range of the intercritical case and any dimensions.
The paper is organized as follows. In Section 2, we recall some useful estimates including dispersive and Strichartz estimates. In Section 3, we prove criteria for the energy scattering for the equation. Section 4 is devoted to show interaction Morawetz estimates. In Section 5, we give the proof of main results given in Theorem 1.4, Lemma 1.5 and Lemma 1.6. Finally, we gave an alternative proof of Theorem 1.4 using the concentration-compactness-rigidity argument in the Appendix.
2. Useful estimates. In this section, we recall some useful estimates needed in the sequel. Lemma 2.1 (Dispersive estimates [3]). Let r ∈ [2, ∞]. It holds that for any f ∈ L r , where (r, r ) is the Hölder's conjugate pair.
Let I ⊂ R be an interval and q, r ∈ [1, ∞]. We define the mixed norm with a usual modification when either q or r are infinity. When q = r, we use the notation L q (I × R N ) instead of L q (I, L q ).
There exists a constant C > 0 independent of I such that the following estimates hold: • (Homogeneous estimates) for any f ∈ L 2 and any Schrödinger admissible pair (q, r).
for any F ∈ L m (I, L n ) and any Schrödinger admissible pairs (q, r), (m, n).
We also have the following inhomogeneous Strichartz estimates for non Schrödinger admissible pairs.

Lemma 2.3 ([4]
). Let N ≥ 1 and I ⊂ R be an interval. Let (q, r) be a Schrödinger admissible pair with r > 2. Fix k > q 2 and define m by Then there exists C > 0, depending only on N, r and k, such that for any F ∈ L m (I, L r ).
We refer the reader to [4, Lemma 2.1] for the proof of this result.
3. Scattering criteria. Let us start with the following nonlinear estimates which follow directly from Hölder's inequality.
Lemma 3.1 (Nonlinear estimates). Let N ≥ 1, 4 N < α < α * and I ⊂ R be an interval. Denote Then the following estimates hold: then there exists a unique global solution to (1) with initial data u(T ) satisfying Here k, q and r are as in (21).
Proof. Let q, r, k and m be as in (21). Consider where I = [T, ∞) and B, L > 0 will be chosen later. We will show that the functional is a contraction on (X, d). By Remark 2, (20) and Lemma 3.1, By Strichartz estimates and Lemma 3.1, We also have There thus exists C > 0 independent of T such that for any u, v ∈ X, By choosing B = 2 e i(t−T )∆ u(T ) L k (I,L r ) and L = 2C u(T ) H 1 . Taking B sufficiently small so that CB α ≤ 1 2 , we see that Φ is a contraction on (X, d). The proof is complete. Proof. Let δ = δ(A) be as in Lemma 3.2. It follows from Lemma 3.2 that the solution satisfies Now let 0 < τ < t < ∞. By Strichartz estimates, we see that This shows that (e −it∆ u(t)) t→∞ is a Cauchy sequence in H 1 . Thus the limit exists in H 1 . By the same reasoning as above, we prove as well that for some σ > 0, where k, r are as in (21), then the solution scatters in H 1 forward in time.
Proof. By Lemma 3.3, it suffices to show that there exists T > 0 such that for some µ > 0.
To show (26), we write By Strichartz estimates, we have where γ c is as in (3) and

VAN DUONG DINH
Note that (k, l) is a Schrödinger admissible pair. By the monotone convergence theorem, there exists T 1 > 0 sufficiently large such that for any T > T 1 , Taking a = T 1 and T = t 0 with a and t 0 as in (25), we write where I : (22) and (25), we see that We next estimate F 1 . By Hölder's inequality, we have where l is as in (27), θ ∈ (0, 1) and n > r satisfy 1 r = θ l + 1 − θ n to be chosen later. Using the fact (k, l) is a Schrödinger admissible pair and On the other hand, by the dispersive estimates and Sobolev embeddings with the fact u L ∞ (R,H 1 ) ≤ A, we have for any t ≥ T , It follows that We thus obtain The above estimate holds true provided We will choose a suitable n satisfying the above conditions. By the choice of r and k, the above conditions become In the case α ≥ 1, we take 1 n = 0 or n = ∞. In the case α < 1, which together with 4 It is not hard to check that the conditions in (32) are satisfied with this choice of n.

Remark 3.
It is easy to see that the last condition in (30) does not hold when N = 1, 2. As in [2], a space time estimate (see [2, Proposition 3.1]) is needed to overcome the logarithmic divergence of the dispersive estimate at least in 2D.

Interaction Morawetz estimates.
4.1. Variational analysis. We recall some properties of the ground state Q related to (5). The ground state Q optimizes the Gagliardo-Nirenberg inequality that is where K and H are defined in (8) and (16) respectively. Recall that Q satisfies the following Pohozaev's identities (see e.g. [3]) It follows that In particular, and We also have the following refined Gagliardo-Nirenberg inequality.

VAN DUONG DINH
Proof. The proof is essentially given in [5]. For the reader's convenience, we give some details. Using (36), we have Since H and M are invariant under the transform f → e ix·ξ f , we see that which proves the result.

4.2.
Interaction Morawetz estimate. Let η ∈ (0, 1) be a small constant and χ be a smooth decreasing radial function which satisfies For R > 0 large, we define the functions and where χ R (z) := χ(z/R) and ω N is the volume of the unit ball in R N . It is easy to see that φ and φ 1 are radial functions. We next define We collect some properties of φ, ψ and Ψ as follows. 5,16]). It holds that for j = 1, · · · , N , where P jk := δ jk − xj x k |x| 2 with δ jk the Kronecker symbol and and for all x ∈ R N .
for some small constant ρ > 0. Then there exists ν = ν(ρ) > 0 such that Proof. We first recall the Gagliardo-Nirenberg inequality Multiplying both sides of (47)  and using (36), it follows from (45) that Since N α > 4, we infer that Lemma 4.4 (Coercivity II). Let N ≥ 1 and 4 N < α < α * . Let u(t) be a H 1 solution to (1) satisfying (17). Then T * = ∞, and there exists ν = ν(Q) > 0 such that for any R > 0 and any z, ξ ∈ R N , Proof. It follows from (17), the conservation of mass and energy that sup which by the blow-up alternative implies T * = ∞. By (17), there exists ρ = ρ(Q) > 0 such that By the definition of χ and u ξ , we have for all t ∈ [0, ∞), all R > 0 and all z, ξ ∈ R N . We thus get for any R > 0 and any z, ξ ∈ R N , The estimate (48) follows directly from Lemma 4.3.
Let u ∈ C([0, ∞), H 1 ) be a solution to (1). We define the interaction Morawetz quantity for some constant A > 0. Let M R (t) be as in (51). Then it holds that Moreover, for all t ∈ [0, ∞).
Proof. By integration by parts and using the fact we have By (42), we have We also have We see that is the angular derivative centered at y, and similarly for / ∇ x u(t, y). By Cauchy-Schwarz inequality and the fact ψ − φ in non-negative, we deduce We next have − Im(u(t, y)∇u(t, y)) · Im(u(t, x)∇u(t, x)) dxdy.
5. Energy scattering. In this section, we give the proof of Theorem 1.4 and Lemma 1.5. Let us start with the proof of Lemma 1.5.
• We next consider the case u 0 ∈ H 1 satisfies (11) and (12)- (15). We will prove (17) by following the argument of [8]. It is done by several steps.

By (35), we have
Inserting it to (85), we get Consequencely, the assumption (11) is equivalent to and the assumption (13) is equivalent to Moreover, the assumption (15) is equivalent to and the assumption (14) is equivalent to

VAN DUONG DINH
In fact, to see (90), we use (14) to have where the last equality comes from (86).
By the continuity argument, we have for some t 0 > 0 sufficiently small. By taking t 0 smaller if necessary, we can assume that In fact, if z (0) > 2 g(λ 0 ), then we have (94) by the continuity argument. Otherwise, if z (0) = 2 g(λ 0 ), then using the fact and (90), we have z (0) > 0. This shows (94) by taking t 0 > 0 sufficiently small. Now, let 0 > 0 be a small constant so that We will prove by contradiction that Assume that it does not hold and let By (96), t 1 > t 0 . By continuity, and By (84), As consequence, g(V (t)) > g(λ 0 ) for all t ∈ [t 0 , t 1 ], thus V (t) = λ 0 and by continuity, We prove that there exists a universal constant D > 0 such that Indeed, by the Taylor expansion of g near λ 0 with the fact g (λ 0 ) = 0, there exists a > 0 such that If V (t) ≥ λ 0 + 1, then (101) holds by taking D large. If λ 0 < V (t) ≤ λ 0 + 1, then by (100) and (102), we get . However, by (95) and (101), we have provided 0 is taken small enough, thus, contradicting (98) and (99). This proves (97). Note that we have also proved (101) for all t ∈ [t 0 , T * ). This together with (93) imply (91) with δ 0 = min δ 1 , √ 0 D . Collecting the above cases, we finish the proof. Before giving the proof of Lemma 1.6, let us recall the following compactness of optimizing sequence for the Gagliardo-Nirenberg inequality due to [15]. Lemma 5.1 ([15]). Let N ≥ 1 and 4 N < α < α * . Let (f n ) n≥1 is a sequence of H 1 functions satisfying Then there exists a subsequence still denoted by (f n ) n≥1 and a sequence (y n ) ⊂ R N such that for some θ ∈ R and x 0 ∈ R N as n → ∞.
Proof. For the reader's convenience, we give some details. Since (f n ) n≥1 is a bounded sequence in H 1 satisfying M (f n ) = M (Q) for all n ≥ 1, we use the concentration-compactness lemma of Lions [15] to have: there exists a subsequence still denoted by (f n ) n≥1 satisfying one of the following three possibilities: • Vanishing: where 2 * := 2N N −2 if N ≥ 3 and 2 * = ∞ if N = 1, 2. • Compactness: There exists a sequence (y n ) n≥1 ⊂ R N such that for all ε > 0, there exists R(ε) > 0 such that for all n ≥ 1, It was shown in [15] that: if the vanishing occurs, then f n → 0 strongly in L r for any 2 < r < 2 * ; and if the compactness occurs, then up to a subsequence, u n (·+y n ) → f strongly in L r for any 2 ≤ r < 2 * for some f ∈ H 1 . We see that the vanishing cannot occur since If the dichotomy occurs, then, by the Gagliardo-Nirenberg inequality, we have Similarly, By (103) and the fact N α 4 > 1, we see that which is a contradiction. Thus, the compactness must occurs, then there exist a subsequence still denoted by (f n ) n≥1 , a function f ∈ H 1 and a sequence (y n ) n≥1 ⊂ R N such that f n (·+y n ) → f strongly in L r for any 2 ≤ r < 2 * and weakly in H 1 . We have and and On the other hand, by the sharp Gagliardo-Nirenberg inequality, we have hence K(f ) ≥ K(Q), so K(f ) = lim n→∞ K(f n (·+y n )) = K(Q). Thus, f n (·+y n ) → f strongly in H 1 and f is an optimizer for the sharp Gagliardo-Nirenberg inequality. By the characterization of ground state (see e.g. [15]) and taking into account that M (f ) = M (Q), we have f (x) = e iθ Q(x − x 0 ) for some θ ∈ R and x 0 ∈ R N . The proof is complete.
Proof of Lemma 1.6. Let u 0 ∈ H 1 satisfies (9) and (10). We first note that (9) Thus (10) becomes K(u 0 ) < K(Q). We will show that for all t ∈ [0, T * ), where T * is the maximal forward time of existence. In fact, assume by contradiction that there exists t 0 ∈ [0, T * ) such that K(u(t 0 )) = K(Q). It follows from (34) that This shows that u(t 0 ) is an optimizer for Gagliardo-Nirenberg inequality (33). By the characterization of ground state, we have u(t 0 ) = Q up to symmetries. By the uniqueness of solution to (1), we have u(t) = e it Q which contradicts (10). We thus prove (105), and, by the blow-up alternative, T * = ∞. We are now consider two cases.
We see that t n → ∞. In fact, if it is not true, then up to a subsequence t n → t 0 as n → ∞. By continuity, u(t n ) → u(t 0 ) strongly in H 1 . It follows that u(t 0 ) is an optimizer for the sharp Gagliardo-Nirenberg inequality, and we have a contradiction. We now apply Lemma 5.1 to f n = u(t n ) to get (up to a subsequence), there exists (y n ) n≥1 ⊂ R N such that for some θ ∈ R and x 0 ∈ R N as n → ∞. The proof is complete.
where k, r are as in (21). Moreover, for fixed J, we have the asymptotic Pythagorean expansions Proof.
where k, r are as in (21)   The proof of Proposition 3 is done by several steps.