Energy Scattering for Schr\"{o}dinger Equation with Exponential Nonlinearity in Two Dimensions

When the spatial dimensions $n$=2, the initial data $u_0\in H^1$ and the Hamiltonian $H(u_0)\leq 1$, we prove that the scattering operator is well-defined in the whole energy space $H^1(\mathbb{R}^2)$ for nonlinear Schr\"{o}dinger equation with exponential nonlinearity $(e^{\lambda|u|^2}-1)u$, where $0<\lambda<4\pi$.


Introduction
We consider the Cauchy problem for the following nonlinear Schrödinger equation iu t + △u = f (u), (1.1) f (u) := (e λ|u| 2 − 1)u, (1.2) in two spatial dimensions with initial data u 0 ∈ H 1 and 0 < λ < 4π. Solutions of the above problem satisfy the conservation of mass and Hamiltonian Nakamura and Ozawa [16] showed the existence and uniqueness of the scattering operator of (1.1) with (1.2). Then, Wang [19] proved the smoothness of this scattering operator. However, both of these results are based on the assumption of small initial data u 0 . In this paper, we remove this assumption and show that for arbitrary initial data u 0 ∈ H 1 (R 2 ) and H(u 0 ) ≤ 1, the scattering operator is always well-defined.
In this paper, we further study the scattering of this problem. Note that f (u) = (e λ|u| 2 − 1)u = ∞ k=1 λ k k! |u| 2k u. Nakanishi [15] proved the existence of the scattering operators in the whole energy space H 1 (R 2 ) for (1.1) with f (u) = |u| p u when p > 2. Then, Killip et al. [12] and Dodson [7] proved the existence of the scattering operators in L 2 (R 2 ) for(1.1) with f (u) = |u| 2 u. Inspired by these two works, we use the concentration compactness method, which was introduced by Kenig and Merle in [11], to prove the existence of the scattering operators for (1.1) with (1.2).
We will prove Theorem 1.2 by contradiction in Section 5. In Section 2, we give some nonlinear estimates. In Section 3, we prove the stability of solutions. In Section 4, we give a new profile decomposition for H 1 sequence which will be used to prove concentration compactness. Now, we introduce some notations: We define When q =r, we abbreviate L q t L r x as L q t,x . When q or r are infinity, or when the domain R × R 2 is replaced by I × R 2 , we make the usual modifications. Specially, we denote S(u) := u 4 For any two Banach spaces X and Y , · X∩Y := max{ · X , · Y }. C denotes positive constant. If C depends upon some parameters, such as λ, we will indicate this with C(λ). Remark 1.2. Note that 0 < λ < 4π in Theorem 1.2, we only need to prove the result for 0 < λ < 4(1 − 4ε)π, ε ∈ (0, 1/8). Hence, we always suppose that 0 < λ < 4(1 − 4ε)π in the context. Moreover, we always suppose that the initial date u 0 of (1.3) satisfies u 0 ∈ H 1 (R 2 ) and H(u 0 ) ≤ 1.

Nonlinear Estimates
In order to estimate (1.2), we need the following Trudinger inequality.
As is shown in [8] and [15] , to obtain the scattering result, it suffices to show that any finite energy solution has a finite global space-time norm. So, if Theorem 1.2 is true, we only need to prove the following theorem.
t,x (I×R 2 )) ) u X∩Y . (2.6) Using the same way as in Bourgain [3], one can split R into finitely many pairwise disjoint intervals By (2.6), As ε ∈ (0, 1/8) can be chosen small arbitrarily, by interpolation, for all admissible pairs and j = 1, 2, · · · , J. The desire result follows. (3.1) , by Strichartz estimates, (3.1) and triangle inequality, we have When A and δ = δ(σ) both are sufficiently small, standard continuity argument gives w X ≤ σ. When A is large, we only need to subdivide the time interval I and then the result follows by an iterate process.

Linear Profile Decomposition
In this section, we will give the linear profile decomposition for Schrödinger equation in H 1 (R 2 ). First, we give some definitions and lemmas.
(Enlarged group, [18]) For any phase θ ∈ R/2πZ, position x 0 ∈ R 2 , frequency ξ 0 ∈ R 2 , scaling parameter λ > 0, and time t 0 , we define the unitary transformation g θ,ξ0,x0,λ,t0 : or in other words Let G ′ be the collection of such transformations. We also let G ′ act on global space-time function u : R × R 2 → C by defining or equivalently Lemma 4.1. (Linear profiles for L 2 sequence, [13]) Let u n be a bounded sequence in L 2 x (R 2 ). Then (after passing to a subsequence if necessary) there exists a family φ (j) , j = 1, 2, · · · of functions in L 2 x (R 2 ) and group elements g n ∈ G ′ for j, n = 1, 2, · · · such that we have the decomposition is such that its linear evolution has asymptotically vanishing scattering size: Moreover, for any j = j ′ , Furthermore, for any l ≥ 1 we have the mass decoupling property for any j ≤ l, we have Remark 4.1. If the orthogonal condition (4.3) holds, then (see [13]) , then (see [2], [13]), for any 0 < θ < 1 , then (see [4], Lemma 5.5) x (R 2 )(see [13]), after passing to a subsequence in n, rearrangement, translation, and refining φ (j) accordingly, we may assume that the parameters satisfy the following: Our main result in this section is the following lemma: . Then up to a subsequence, for any J ≥ 1, there exists a sequence φ α in H 1 (R 2 ) and a sequence of group elements g nα = g θnα,ξnα,xnα,λnα,tnα ∈ G ′ such that Here, for each α, λ nα and ξ nα must satisfy λ nα ≡ 1 and ξ nα ≡ 0, orλ nα → ∞; (4.9) Moreover, for any α = α ′ , one has the same orthogonal conditions as (4.3). For any J ≥ 1, one has the following decoupling properties Then, we have By Lemma 4.1, after passing to a subsequence if necessary, we can obtain with the stated properties 1)-4) and (4.1)-(4.5). Denote Step 1. We prove that k and for each fixed N , By (4.2) and lim N →∞ lim sup n→∞ R N L 2 (R 2 ) = 0, (4.18) holds obviously. For (4.15), we prove it by induction. For every k, suppose that k . (4.20) Using (4.5), (g By direct calculation, ).
Specially, as C ∞ c is dense in L 2 , we can also suppose F φ α ∈ C ∞ c and hence φ α ∈ H 1 (R 2 ).

The Proof of Theorem 1.2
Let u be a solution of (1.3), by Strichartz estimate and (2.6), .

(5.1)
When u 0 L 2 (R 2 ) ≪ 1, by standard continuity argument, we have To prove Theorem 1.2, one only needs to prove that the critical mass m 0 is infinite. We will prove that by contradiction.
Proposition 5.1. Suppose that the critical mass m 0 is finite. Let u n : R×R 2 → C for n = 1, 2, · · · be a sequence of solutions and t n ∈ R be a sequence of times such that lim sup n→∞ M (u n ) = m 0 and Then there exists a sequence of x n = x n (t n ) ∈ R 2 such that u n (t n , x + x n ) has a subsequence which converges strongly in L 2 x (R 2 ).
Proof. We can take t n = 0 for all n by translating u n in time. Thus, where Λ 1 and Λ ∞ were defined by (4.31). Suppose that where t nα ∈ R and h nα ∈ G. By (4.11), We define the nonlinear profile v α : R × R 2 → C as follows: ⋄ if t nα → +∞, we define v α to be the global solution of (1.3) which scatters to e it△ φ α when t → +∞.
⋄ if t nα → −∞, we define v α to be the global solution of (1.3) which scatters to e it△ φ α when t → −∞.
⋄ if t nα → +∞, we define v α to be the global solution of iu t + △u = |u| 2 u which scatters to e it△ φ α when t → +∞.
⋄ if t nα → −∞, we define v α to be the global solution of iu t + △u = |u| 2 u which scatters to e it△ φ α when t → −∞.
If we defineũ for n, J = 1, 2, · · · , then we have the following two lemmas: By the definition ofũ n , we havẽ and Thus, by triangle inequality, it suffices to show that Using (4.13) and the same estimates as in (3.3), we have , and e it△ R(n, J) x (R×R 2 ) = 0. By (5.15), (5.8) was obtained.
Since there is only one profile now, we havẽ u n = T hn v + e it△ R n .
By Lemma 3.1 (with 0 as the approximate solution and u n (0) as the initial data), we have lim n→∞ S ≥0 (u n ) = 0 which contradicts one of the estimates in (5.4). When λ n ≡ 1, ξ n ≡ 0 and t n → −∞, the argument is similar and we can obtain a contradiction by using the other half of (5.4). Now, the only case left is λ n ≡ 1, ξ n ≡ 0 and t n ≡ 0, . In this case, we have M (u n (0) − h n φ) = M (R n ) → 0 as n → ∞.
Thus (h n ) −1 u n (0) = e iθn u n (0, x + x n ) converges to φ in L 2 x (R 2 ). After passing to a subsequence if necessary and refining φ, the desired result follows.
Let {u n } be the sequence given in Proposition 5.1 and suppose u n (0, x + x n ) converges to u 0 strongly in L 2 x (R 2 ), then M (u 0 ) ≤ m 0 . Let u be the global solution with initial data u(0) = u 0 , by Lemma 3.1, we must have S ≥0 (u) = S ≤0 (u) = +∞.
By the definition of m 0 , M (u 0 ) ≥ m 0 and hence M (u 0 ) = m 0 .
Since u is locally in L 4 t,x , for ∀t n ∈ R, we have S ≥tn (u) = S ≤tn (u) = +∞.
Using Proposition 5.1 for {u(t n )}, we have u(t n , x + x(t n )) converges in L 2 x (R 2 ). By Ascoli-Arzela Theorem, that is Once we proved Proposition 5.3, we can say that m 0 = ∞ and thus Theorem 1.2 is true. In order to prove Proposition 5.3, we need the following two lemmas.