Classification of minimal mass blow-up solutions for an $L^2$ critical inhomogeneous NLS

We establish the classification of minimal mass blow-up solutions of the $L^2$ critical inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^{\frac{4-2b}{N}}u = 0, \] thereby extending the celebrated result of Merle from the classic case $b=0$ to the case $0<b<\min\{2,N\}$, in any dimension $N\ge1$.


Introduction
In this paper we establish the classification of minimal mass blow-up solutions of the inhomogeneous nonlinear Schrödinger equation i∂ t u + ∆u + |x| −b |u| p−1 u = 0, u(0, ·) = u 0 ∈ H 1 (R N ), (1.1) in the case p = 1 + 4−2b N , with 0 < b < min{2, N } and any N 1, where the equation is L 2 critical, as pointed out in [7]. The case b = 0 is the classic focusing NLS equation with L 2 critical nonlinearity. The physical relevance of (1.1) with b > 0 may not appear obvious due to the singularity at x = 0. However, this model problem plays an important role as a limiting equation in the analysis of more general inhomogeneous problems of the form i∂ t u + ∆u + V (x)|u| p−1 u = 0 with V (x) ∼ |x| −b as |x| → ∞, which are ubiquitous in nonlinear optics -see [5,6,8] for more details.
We consider here strong solutions u = u(t, x) ∈ C 0 t H 1 x ([0, T ) × R N ), where T > 0 is the maximum time of existence of u. We may simply denote by u(t) ∈ H 1 (R N ) the function x → u(t, x). The solution is called global if T = +∞. If it is not the case, the blow-up alternative states that u(t) H 1 → ∞ as t ↑ T . Moreover, along the flow of (1.1), we have conservation of the L 2 norm, also known as the mass: x , and of the energy: We refer to the discussion in [7] regarding the well-posedness theory of (1.1) in H 1 (R N ). The theory is similar to the classic case b = 0: there is local wellposednessi.e. existence and uniqueness of solutions for small positive times -(and global for small initial data) in H 1 (R N ) if 1 < p < 1 + 4−2b N −2 (1 < p < ∞ if N = 1, 2); there is global well-posedness for any initial data in H 1 (R N ), provided 1 < p < 1 + 4−2b N . We are here interested in the critical case p = 1 + 4−2b N . The above invariants are related to the symmetries of (1.1) in H 1 (R N ). More precisely, if u(t, x) solves (1.1), then so do: (a) u t0 (t, x) = u(t − t 0 , x), for all t 0 ∈ R (time translation invariance); (b) u γ0 (t, x) = e iγ0 u(t, x), for all γ 0 ∈ R (phase invariance); (c) u λ0 (t, x) = λ The symmetries (a) and (b) are obvious and give rise, via Noether's theorem, to the invariance of the energy and the mass, respectively. However, it is remarkable that (1.1) indeed has the scaling symmetry (c). In the case p = 1 + 4−2b N which will be our focus here, we have (2 − b)/(p − 1) = N/2, and so x , for all λ 0 > 0. The symmetry (c) is then called the L 2 scaling, and (1.1) is said to be L 2 critical.
An important feature of (1.1) is the existence of standing wave solutions. Indeed, There exists a unique positive and radial solution of (1.3), called the ground state, which we will denote by ψ throughout the paper. We refer the reader to [7] for references about the existence and uniqueness theory for (1.3). It turns out that the ground state is a fundamental object to understand the dynamics of (1.1). Theorem 2.5 of [7] shows, for instance, that the solutions of (1.1) are global provided u 0 L 2 < ψ L 2 .
A crucial inequality for the proof of this theorem, which can be deduced from Proposition 2.2 of [7], is Indeed, since the L 2 norm and the energy are conserved, (1.4) immediately yields an a priori bound on ∇u(t) L 2 in the case u 0 L 2 < ψ L 2 , namely which implies global existence. Another interesting consequence of Proposition 2.2 of [7] is that E(ψ) = 0. Therefore, ψ lies on the submanifold of H 1 (R N ) defined by the intersection of two constraints, u L 2 = ψ L 2 and E(u) = 0. This manifold will be characterized in Proposition 2 below.
On the other hand, it follows from Theorem 3.1 of [7] that there exists a solution of (1.1) with u 0 L 2 = ψ L 2 which blows up in finite time. That is, ψ L 2 is the minimal mass for blow-up solutions of (1.1), which will henceforth be referred to as the critical mass for (1.1). Note that the proof of Theorem 3.1 of [7] relies mainly on the pseudo-conformal transformation applied to the standing wave e it ψ, and if we take into account the three invariances of (1.1) described above, we obtain a 3-parameter family (S T,λ0,γ0 ) T ∈R,λ0>0,γ0∈R of critical mass solutions of (1.1) which blow up in finite time, defined by Note that these solutions present a self-similar profile, in the sense that, for all Hence, up to a time-dependent L 2 rescaling, S T,λ0,γ0 keeps the same shape as ψ while blowing up. We refer to Section 5 for more details and comments about the pseudoconformal transformation and the construction of these critical mass solutions. We now state our main result. Then there exist λ 0 > 0 and γ 0 ∈ R such that, for all t ∈ [0, T ), where S T,λ0,γ0 is defined in (1.6).
It is worth remarking here that, since the space translation invariance of (1.1) is broken for b > 0, our conclusion is stronger than in the case b = 0, which is reflected in the absence of space translation and Galilean symmetries in (1.6). In addition, it transpires from our proof (see Step 2 in Section 6) that all of the solution mass concentrates at the origin in R N as the blow-up occurs.
Blow-up solutions of the L 2 critical NLS in the classic case b = 0 have been thoroughly investigated since the seminal works of Weinstein [14,15]. In fact, Theorem 2.5 of [7] extends a result of Weinstein [14] from the case b = 0 to the case 0 < b < min{2, N }, and Theorem 1 above extends the classification result of Merle [10] to the case 0 < b < min{2, N }. A comprehensive review of the theory of blow-up solutions for the classic focusing NLS can be found in [12], where a proof of Merle's result [12,Theorem 4.1] is presented, which is based on more recent arguments -notably a refined Cauchy-Schwarz inequality due to Banica [1].
Although more scarcely, critical mass blow-up solutions have also been investigated in the context of inhomogeneous NLS equations by several authors. For instance, was considered by Merle [11], and later by Raphaël and Szeftel [13] (in the case N = 2), where the inhomogeneity k is supposed to be smooth, positive and bounded. Merle [11] derived conditions on k for the localization of the concentration point of critical mass blow-up solutions, and for the non-existence of critical mass blow-up solutions. Raphaël and Szeftel [13] proved the existence and the classification of critical mass blow-up solutions for (1.7), provided k attains its maximum in R N .
Banica, Carles and Duyckaerts [2] studied the problem where V and g satisfy strong smoothness assumptions. Assuming that g is sufficiently flat at the origin, they proved the existence of critical mass blow-up solutions by adapting a fixed point argument developed by Bourgain and Wang [3] in the classic case of (1.1) with b = 0. It is worth noting here that problem (1.1) does not fall within the scope of [2,11,13] due to the singularity at x = 0. Moreover, our approach strongly benefits from the scaling properties of (1.1) -notably the pseudo-conformal invariance, which is not present in [2,11,13].
Our proof of Theorem 1 follows the scheme outlined in [12]. In Sections 2 and 3, respectively, we prove a variational characterization of the ground state of (1.3) and a compactness property of the flow in H 1 (R N ). In Section 4 we extend the classic virial identities to the inhomogeneous case, b > 0. In Section 5 we show that (1.1) is invariant under the pseudo-conformal transformation. Combining all these ingredients, we give the proof of Theorem 1 in Section 6.
Notation. To avoid cumbersome exponents and indices, without further notice we let p = 1 + 4−2b N throughout the paper. We also let 2 * = 2N N −2 if N 3 and 2 * = ∞ if N = 1, 2. We will often denote the Lebesgue norms · L q merely by · q , for 1 q ∞. All the integrals will be understood to be over R N , even when not specified. For x, y ∈ R N , we denote x · y their inner product, and |x| = √ x · x the Euclidean norm of x. The symbol C will denote various positive constants, the exact value of which is not essential to the analysis.

Variational characterization of the ground state
We start by proving the following key proposition, which gives a variational characterization of the ground state of (1.3).
Proof. It follows from Proposition 2.2 of [7] that the ground state ψ of (1.3) is a minimizer of the Weinstein functional and that E(ψ) = 0. Therefore, for But then |v| is also a minimizer, since Furthermore, any positive minimizer is radial thanks to a result of Hajaiej [9]. Indeed, suppose v 0 is a positive minimizer that is not radial, and consider its Schwarz symmetrization v * 0 . Then Theorem 6.1 of [9] implies that Since, on the other hand, by standard properties of the Schwarz symmetrization, we get J(v * 0 ) < J(v 0 ), a contradiction. We deduce that |v| is radial. Furthermore, the Euler-Lagrange equation expressing the fact that |v| is a minimizer reads It now follows by the scaling properties of this elliptic equation, and by the uniqueness of its positive radial solution (see the discussion in [7]), that It only remains to show that w defined by w( To do this, first observe that differentiating |w| 2 ≡ 1 leads to Re(w∇w) ≡ 0, and so Now supposing |∇w| ≡ 0 on R N , we would have strict inequality in (2.2), and hence J(|v|) < J(v). This contradiction shows that, indeed, |∇w| ≡ 0 on R N . Hence, w is constant on R N , and since its modulus is 1, we deduce that there exists γ 0 ∈ R such that w ≡ e iγ0 , which completes the proof.

Compactness
The main goal of this section is to prove Proposition 5 below. To do so, we first need some inhomogeneous estimates, reproduced in the following lemma for the reader's convenience. The proof can be found in [6, Appendix A] (for N = 1) and [5, Appendix A] (for N 2).
To prove Proposition 5, we also need a concentration-compactness lemma. Minor modifications to the proof of Proposition 1.7.6 in [4] yield the following result.
Then there is a subsequence (v n k ) satisfying one of the three following properties: We are now ready to prove the main result of this section.
Then there exist a subsequence of (v n ), still denoted (v n ), and γ 0 ∈ R such that Proof. The behaviour of the sequence (v n ) is constrained by the concentrationcompactness principle, as stated in Lemma 4. The proof will proceed in several steps: we will first show that property (C) holds, by ruling out (V) and (D). Then we will show that the sequence (y k ) in (C) is bounded. Using Proposition 2, this will lead to the desired conclusion.
But then, using again property (D)(iii) of Lemma 4 and inequality (1.4), we see that ∇w k 2 → 0 and ∇z k 2 → 0 as k → ∞, which in turn implies that again leading to the contradiction (3.2). Thus, to rule out (D), we need only prove claim (3.3), which we do now. Defining ξ k = v n k − w k − z k , it follows from the construction of the sequences w k and z k in the proof of [4, Proposition 1.7.6] that Since ∇ξ k 2 is bounded by property (D)(v), the Gagliardo-Nirenberg inequality then implies that ξ k q → 0 as k → ∞, ∀ q ∈ [2, 2 * ).
Hence, it follows from Lemma 3 that which proves the claim. Therefore, we conclude from Lemma 4 that there exist v ∈ H 1 (R N ) and a sequence ( Step 2: Localization. We will now show that (y k ) ⊂ R N is bounded. Suppose by contradiction that |y k | → ∞ as k → ∞ (up to a subsequence). Note that E(v n k ) can be written as with v n k (x) = v n k (x − y k ). We will show that the second term in the right-hand side of (3.6) goes to zero as k → ∞, so that which contradicts (3.1). We split the integral into two parts, as for some R > 0. First, by Hölder's inequality, where α, β 1 satisfy 1 α + 1 β = 1. Now the first factor in the right-hand side of (3.8) is finite provided β > N N −b . In fact it is possible to choose β so that β(p + 1) ∈ ( N (p+1) N −b , 2 * ) and it follows from (3.5) that On the other hand, by the Sobolev embedding theorem and the boundedness of ( v n k ) in H 1 (R N ). Hence, II can be made arbitrarily small by choosing R large enough, uniformly in k. This completes the proof of (3.7), and we conclude that the sequence (y k ) is bounded in R N .
Step 3: Conclusion. By passing to a subsequence if necessary, we can suppose that y k → y * as k → ∞, for some y * ∈ R N . Hence, Furthermore, we can also suppose that v n k ⇀ v * weakly in H 1 (R N ), and it follows from (3.1) that v * 2 = ψ 2 and ∇v * 2 ∇ψ 2 .
Remark 6. It is worth noting here that Proposition 5 can be used to prove the 'orbital stability' of the ground state ψ, in the sense of Theorem 3.7 of [12]. Moreover, the notion of stability is stronger here, since we can take x(t) ≡ 0 for the translation shift appearing in Theorem 3.7 of [12].

Virial identities
Then, for u(t) ∈ Σ, the quantity is well defined, and when u is a solution of the classic, homogeneous, NLS equation (i.e. (1.1) with b = 0), it is well known that Γ ′ and Γ ′′ have simple expressions, very useful to prove blow-up results when u 0 ∈ Σ. The following lemma shows that Γ is still a key quantity in the inhomogeneous case, b > 0. and Proof. By regularization, we may assume u smooth for the following calculation.
Using again an integration by parts and denoting ∇·v = j ∂ xj v j , we now compute To compute these last two terms, we use (1.1) and first find Similarly, we also find Since ∇(|u| p+1 ) = (p + 1)|u| p−1 Re(u∇ū) and ∇(|x| −b ) = −b|x| −b−2 x, we find by an integration by parts Similarly, since ∂ x k (|∂ xj u| 2 ) = 2 Re(∂ xj u∂ xj ∂ x kū ) for 1 j, k N , we find where we wrote δ j,k = 1 for j = k and 0 otherwise. Finally, gathering (4.4), (4.5) and (4.6), we obtain from the definition (1.2) of the energy, which concludes the proof of the lemma. which is also the key identity to classify the blow-up solutions in the homogeneous case, b = 0.

Pseudo-conformal transformation
We now establish the pseudo-conformal invariance of (1.1), which was observed by the second author in [7,Section 3]. For the reader's convenience, and also to be coherent with the notation of the present paper, we prove the following statement.
Lemma 9. Let u be a global solution of (1.1). Then, for all T ∈ R, the function u T defined by is also a solution of (1.1), defined on (−∞, T ), and has the same mass as u.

Proof. A straightforward calculation gives first
We also find It follows that x follows from the L 2 scaling invariance.
We can now construct, as announced in the introduction, a 3-parameter family of critical mass solutions of (1.1) which blow up in finite time.
Proposition 10. For all T ∈ R, λ 0 > 0 and γ 0 ∈ R, the function S T,λ0,γ0 , defined by is a critical mass solution of (1.1) defined on (−∞, T ), and which blows up with speed for some C > 0.
Proof. The proposition is a simple consequence of Lemma 9 applied to the global solution u λ0,γ0 (t, x) = e iγ0 e iλ 2 0 t λ which is nothing more than the renormalized version of the standing wave u(t, x) = e it ψ(x) under the scaling and phase symmetries.
Remark 11. Note that the blow-up solutions of the family exhibited in Proposition 10 can all be retrieved from the solution defined on (−∞, 0) and which blows up at t = 0 with speed for some C > 0. Indeed, all the solutions S T,λ0,γ0 are equal to S, up to the symmetries (a), (b) and (c) stated in the introduction. Namely, if we apply the changes u(t, x) → λ −N/2 0 u(λ −2 0 t, λ −1 0 x), u(t, x) → u(t − T, x) and finally u(t, x) → e iγ0 u(t, x) to S, we obtain S T,λ0,γ0 .
By integrating with respect to x ∈ R N , we get We can now easily prove the following refined Cauchy-Schwarz inequality for critical mass functions.
Proof. For all s ∈ R, we now have ue isθ L 2 = u L 2 = ψ L 2 , so E(ue isθ ) 0 and E(u) 0 by (1.4). The result follows from the quadratic polynomial expression (6.1) in s of E(ue isθ ), which thus must have a non-positive discriminant.
We now have all the tools to prove Theorem 1. Let u be a solution of (1.1) such that u 0 L 2 = ψ L 2 and which blows up in finite time: there exists T > 0 such that lim t↑T ∇u(t) L 2 = +∞. The core idea of the proof is to integrate the equation backwards in time, from the blow-up time, in order to show that u 0 belongs to the family of solutions (5.1). We shall proceed in four steps.
Step 1: Compactness of the flow in H 1 . Let (t n ) ⊂ R be a sequence of times such that t n ↑ T as n → +∞. Then we set u n = u(t n ), λ n = ∇u n L 2 ∇ψ L 2 , v n (x) = λ −N/2 n u n (λ −1 n x).
First note that λ n → +∞ as n → +∞, and v n L 2 = u n L 2 = u 0 L 2 = ψ L 2 from the L 2 scaling. With the change of variables y = λ −1 n x, we find Hence, we can apply Proposition 5 to (v n ), which gives γ 2 ∈ R such that, up to extracting a subsequence of (v n ), we have lim n→+∞ v n − e iγ2 ψ H 1 = 0. (6.2) Step 2: Mass concentration. We can now prove that u n concentrates all of its mass at x = 0 as n → +∞. More precisely we show, in the sense of distributions, that Indeed, for ϕ ∈ C ∞ 0 (R N ), using again the change of variables y = λ −1 n x, we find Hence, we have We conclude this step by noticing that |v n | 2 converges to |ψ| 2 strongly in L 1 (R N ) from (6.2), so the first integral converges to 0 as n → +∞. Since λ n → +∞, the second integral also converges to 0 as n → +∞ by the dominated convergence theorem, and so Step 3: u(t) ∈ Σ for all t ∈ [0, T ). Let φ ∈ C ∞ 0 (R N ) be radial and non-negative such that φ(x) = |x| 2 for |x| 1. In other words, there exists f ∈ C ∞ 0 (R, R + ) such that φ(x) = f (|x|) and f (r) = r 2 for −1 r 1.
Since u L 2 = ψ L 2 , we may apply Lemma 12 and we get, since |∇φ R | 2 Cφ R , Integrating between a fixed t ∈ [0, T ) and t n , we obtain But from the mass concentration result (6.3) in Step 2, we get Thus, letting n → +∞ in the last inequality, we obtain, for all t ∈ [0, T ) and all R > 0, Γ R (t) ≤ C(T − t) 2 .