Blow-up solutions with minimal mass for nonlinear Schrödinger equation with variable potential

Abstract: This paper studies the mass-critical variable coe cient nonlinear Schrödinger equation. We rst get the existence of the ground state by solving aminimization problem. Then we prove a compactness result by the variational characterization of the ground state solutions. In addition, we construct the blow-up solutions at the minimal mass threshold and further prove the uniqueness result on the minimal mass blow-up solutions which are pseudo-conformal transformation of the ground states.


Introduction
In this paper, we study the behavior of blowup solutions for the variable coe cient nonlinear Schrödinger equation and φ( , x) = φ , x ∈ R D , (1.2) in the case p = + − b D . Here and hereafter, D ≥ is the space dimension, < b < , φ = φ(t, x) : [ , T)×R D → C is a complex value wave function with < T ≤ ∞, i = √ − , ∆ is the Laplace operator and c ∈ ( , c * ), where c * = (D− ) is the best constant in Hardy's inequality: where S(·) is the group with in nitesimal generator i(∆ + c|x| − ).
From [2,6,15], we know that the Cauchy problem ( where · H is the usual norm on H , and H is H (R D ). Furthermore, the unique solution φ(t, x) satis es the following two conservation laws: for all t ∈ [ , T), and (1.5) Also, the L -norm of φ(t, x) is invariant under the transformation φ → φ λ (t, x) = λ D/ φ(λ t, λx) for λ > .
In other words, φ λ (t) L = φ(t) L  It is well known that (1.7) possesses a unique radial positive solution Q(x) (see [2]). Moreover, the Cauchy problem (1.6)-(1.2) has no blow-up solution in the class {φ ∈ H | φ L < Q L } (see [19]), while in the class {φ ∈ H | φ L = Q L }, there exists a unique blowup solution up to the invariances of the equation(see [10,13,14,20]). For the nonlinear Schrödinger equation with inverse-square potential, i.e. b = , c ≠ , Let φ(t, x) = e it u(x) be a standing wave solution of (1.8). Then u(x) satis es the following time-independent Schrödinger equation By solving a variational problem, Csobo and Genoud [5] proved that (1.9) has radial positive solutions to exist. Mukherjee, Nam and Nguyem [18] showed the uniqueness of radial positive solution of (1.9). In addition, Csobo and Genoud [5] proved that all H -solutions of Eq.(1.8) are global if φ L < U L and by using the pesudo-conformal transformation, they constructed the minimal mass blow-up solutions de ned as such that φ L = U L , where T ∈ R, λ > , γ ∈ R, U is the unique positive radial solution of (1.9). For the inhomogeneous nonlinear Schrödinger equation, i.e. c = , b ≠ , Let φ(t, x) = e it u(x) be a standing wave solution of (1.10). Then u(x) satis es the following time-independent Schrödinger equation By solving a variational problem, Genoud [9] proved that (1.11) has positive radial solutions to exist and the sharp condition for global existence of H -solutions was established, also, the solutions with critical mass which blow-up in nite time were proved. By Yanagida [21] and Genoud [8], the positive radial solution of (1.11) is unique. Furthermore, Combet and Genoud [4,9] constructed a -parameter family of critical mass solutions of (1.10) which blow-up in nite time, de ned by with φ L = ψ L , where T ∈ R, λ > , γ ∈ R, ψ is the unique positive radial solution of (1.11). We also refer to Chen and Tang [3], Zhang and Ahmed [23] for related contributions on the nonlinear Schrödinger equation. Motivated by the above studies, it is of interest to nd the uniqueness of the minimal mass blow-up solutions for the Cauchy problem (1.1)-(1.2). In order to solve this problem, we need a notion of a ground state. So we consider the periodic solutions of equation of the form φ(t, x) = e it u(x), then u(x) solves the nonlinear elliptic equation In section 2, we will prove the existence in H of a positive radial solution of Eq.(1.12) by using Weinstein's argument [19] and that the set of solutions of (1.12) has a "minimal" element in L called ground state. However, for c > , we are not aware of any uniqueness result of (1.12) on R D . According to [8], [18] and [21], we can assume that (1.12) has a unique positive radial solution, which we will denote by W throughout the paper. In terms of the existence of ground state, we rst show the global well-posedness condition for the Cauchy problem (1.1)-(1.2). Then, we shall exhibit blow-up solutions at the mass φ L = W L by applying the pseudo-conformal transformation to the standing waves e it W(x), which is thus the minimal mass where blowup can occur. Now we state our main result. Then there exist γ ∈ R and λ > such that for all t ∈ [ , T), where W(x) is the ground state of Eq. (1.12).
One can ask if there are other minimal mass blow-up solutions? That is, assume that φ(t, x) blows up in nite time T, and φ(t, x) L = W L for all t.
The aim of this paper is to prove this uniqueness result on the minimal mass blow-up solutions. The method is mainly based on a compactness result.
The plan of this paper is as follows. In section 2, we prove the existence of ground state by solving a minimization problem. In section 3, we prove a compactness result, crucial in the proof of Theorem 1.1. In section 4, we construct the pseudo-conformal transformation and then give the proof of Theorem 1.1.

Ground State
In this section, we prove the existence of a positive radial solution W ∈ H for (1.12). Such a solution is obtained by variational approach and called ground state. We rst state the following result.
Then one has the followings: is the positive radial solution of (1.12).

Proof. From Hardy inequality (1.3), one has that for
Let u * be the Schwarz symmetrization of u. By [11,12], one has that where u * n denotes the Schwarz symmetrization of un for any n ∈ N. Now we choose and take vn = |(u * n ) µn ,λn |. By (2.5) and (2.9), we have that Thus {vn} is bounded in H . Therefore, up to subsequence, we can suppose that {vn} has a weak limit v * ∈ H . By weak lower semi-continuity (see [17]), we obtain that It follows from the weakly sequence continuous of the functional 14) It yields that It follows that lim Thus we get the proof of (i). Now we prove (ii).
Since v * is the minimizer of (2.2), v * must satisfy the Euler-Lagrange equation By (2.14), (2.15) it follows that is the positive radial solution of (1.12). This completes the proof of (ii). Then we prove (iii) in the following. Since Thus if we assume that (1.12) has a unique positive radial solution, then for (ii) and (2.2), we get (2.3). This completes the proof.

Remark 2.1. Let ξ be the minimizer of J(u), then we de ne a set A as follows:
Moreover, W satisfying the elliptic equation (1.12). Therefore, we call W ∈ A ground state of Eq. (1.12).
In the following, we show that the ground state solutions play an important role in the global well-posedness for the Cauchy problem (1.1)-(1.2).
then the corresponding solution of the Cauchy problem (1. Proof. By the local well-posedness theory, we only need to show that R D (|∇φ| − c|x| − |φ| )dx remains bounded. From the conservation laws of mass and energy, then R D (|∇φ| − c|x| − |φ| )dx is bounded and so the solution is global. This completes the proof.

Compactness
In this section, we will establish a compactness result which is important in the proof of Theorem 1.1. For this aim, we introduce the notion of the variational characterization of ground state solutions. Proof. It follows from Lemma 2.1 and (3.1) that v is a minimizer of J(u). Since ∇(|v|) L ≤ ∇v L , then |v| is also a minimizer. Furthermore, any positive minimizer is radial thanks to a result of Hajaiej [11]. Indeed, suppose v is a positive minimizer that is not radial, and consider its Schwarz symmetrization v * . On the other hand, ∇v * L ≤ ∇v L , v * L ≤ v L , then we get J(v * ) < J(v ), a contradiction. We deduce that |v| is radial. Furthermore, the Euler-Lagrange equation (2.17) expressing the fact that |v| is a minimizer reads It now follows by the scaling properties of this elliptic equation, there exists W ∈ A such that It only remains to show that w de ned by w(x) = v(x) |v(x)| is constant on R D . Di erentiating |w| = yields Re(w∇w) = , thus |∇v| = |∇(|v|)| + |v| |∇w| + |v|∇(|v|) · Re(w∇w), If |∇w| ≠ , then J(|v|) < J(v), which is a contradiction. Therefore, w is constant, which completes the proof. Then we review the following result, for the proofs we refer readers to [7].
In addition, we also need the following concentration-compactness Lemma which can be proved with a minor modi cation to the proof of Proposition 1.7.6 in [2]. Then there exists a subsequence {vn k } k∈N which satis es one of the following properties: (V) vn k L q → as k → ∞ for all q ∈ ( , * ).
(D) There are sequences w k , z k ∈ H and a constant α ∈ ( , ) such that: (1) dist(supp(w k ), supp(z k )) → ∞; (2) sup k∈N ( w k H + z k H ) < ∞ for all k ∈ N; (3) w k L → αM and z k L → ( − α)M as k → ∞, for some α ∈ ( , ); Here and subsequently, for abbreviation, H(φ) stands for the Hardy functional de ned by (|∇v| − c|x| − |v| )dx. We are now show the main result of this section. where W ∈ A is the ground state of (1.12).
Proof. The behavior of the sequence {vn} is constrained by the concentration-compactness Lemma. The proof will be divided into three steps: we rst show that property (C) holds in Lemma 3.2 by ruling out (V) and (D). And then prove the sequence {y k } k∈N ⊂ R D in (C) is bounded. We thus obtain the desired conclusion by using the variational characteristic of ground states.
Step 1: Since β, p + ∈ ( , * ), (V) would imply that  Therefore, again, it contradicts (3.4). Thus, to rule out (D), we just have to prove (3.5) holds. For this aim, de ning ξ k = vn k − w k − z k , it follows from the construction of the sequence w k and z k in the proof of [2] that Since ξ k L is bounded by property (D) (5), applying the inequality (2.3), one can deduce that Hence, it follows from Lemma 3.1 that which proves (3.5). Therefore, according to Lemma 3.2, there exist v ∈ H and a sequence {y Step 2: Localization. We will now show that {y k } is bounded in R D . Suppose that there exists a subsequence denoted still by {y k } such that |y k | → ∞ as k → ∞. Note that E(vn k ) can be written as (3.8) Then we will show that the second term in the right hand side of (3.8) goes to zero as k → ∞, so that which contradicts (3.2). We split the integral as for some R > . We rst observe that, by Holder's inequality, (3.11) where α, β ≥ satisfy α + β = . Then the rst term in the right hand side of (3.11) is nite provided β > D D−b . Indeed, it is possible to choose β satis es that β(p + ) ∈ ( D(p+ ) D−b , * ) and it follows from (3.7) that On the other hand, by the Sobolev embedding Theorem and the boundedness of {vn k (x − y k )} in H , the second term of (3.10) can be estimated as Hence, II can be made arbitrarily small by choosing R large enough, uniformly in k. This completes the proof of (3.9), and we conclude that the sequence {y k } ⊂ R D is bounded.
Furthermore, we can also suppose that vn k v * weakly in H . Since the Hardy functional H is weakly lowercontinuous [17], v → R D |x| −b |v| p+ dx is weakly sequentially continuous [9] and p + ∈ ( , * ), it follows So from the inequality (2.3), one has that E(v * ) ≥ , and so E(v * ) = and H(v * ) = lim k→∞ H(vn k ). (3.12) Together with v * L = W L and the variational characteristic of the ground states, it implies that v * = e iγ W for some γ ∈ R. Finally, we have vn k || H → W H = v * H , {vn k } converges to v * in H , which concludes the proof.

Classi cation
In this section, we construct nite time blow-up solutions with minimal mass W L to the Cauchy problem (1.1)-(1.2). Furthermore, we prove the uniqueness of nite time blow-up solutions at the minimal mass threshold. We rst show that the equation is invariant under the pseudo-conformal transformation, which is de ned as follows.  1)-(1.2). Then, for all T ∈ R, the function Proof. By a direct calculation we have and x T − t for the derivatives. For the nonlinear term we nd and for the potential term Then it follows that which proves the Lemma. For all T ∈ R, λ > and γ ∈ R, then the function φ T,λ ,γ ,de ned by is a minimal mass solution of the Cauchy problem (1.1)-(1.2) de ned on (−∞, T), and which blows up with speed where W ∈ A is the ground state of (1.12).
Proof. The proof may be proved by applying Lemma 4.1 to the global solution Indeed, all the solutions φ T,λ ,γ are equal to φ , , , up to the symmetries. In other words, if we apply the changes and nally φ(t, x) → e iγ φ(t, x) to φ , , , we obtain φ T,λ ,γ .
By a direct calculation we have Then the re ned Cauchy-Schwarz estimate for critical mass functions can be proved.

Lemma 4.2.
Let φ ∈ H be a function such that φ L = W L . Then for all θ ∈ C ∞ (R D ), one has Proof. For all η ∈ R, we now have φe iηθ L = φ L = W L , so E(φe iηθ ) ≥ and E(φ) ≥ by (2.3). The result follows from the quadratic polynomial expression (4.2) in η of E(φe iηθ ), which thus must have a nonpositive discriminant. This completes the proof. In the reminder of this section we prove the main result. Proof of Theorem 1.1. Let φ be a solution of the Cauchy problem (1.1)-(1.2) such that φ L = W L and which blows up in nite time. The proof then falls into several steps.
Step 1. Let {tn} n∈N ⊂ R be a sequence of times such that tn → T as n → ∞. We set We note that λn → ∞ as n → ∞, and On the other hand, by conservation of the energy, Step 2. Using the variational arguments, we prove that blowup's pro le is a Dirac function, i.e. in the sense of distributions, |φn| → W L δ .
Indeed, by scaling (4.3), we observe that for u ∈ C ∞ (R D ), Thus, we obtain (4.4) implies that |vn| converges to |W| strongly in L (R D ), so the rst integral converges to 0 as n → +∞.
Since φ L = W L and |∇ϕ R | ≤ Cϕ R , we can apply Lemma 4.2 to get By integration, we obtain that for t ∈ [ , T), Since J R (tn) → by (4.5). Thus, letting tn → T, we obtain that for all t ∈ [ , T) and all R > , Since the right-hand side of the last expression of R, we obtain that by letting R → ∞, for all t ∈ [ , T), where J(t) = R D |x| |φ| dx. From this estimate, we can extend by continuity J(t) at t = T by setting J(T) = , from which we also obtain J ′ (T) = . Moreover, since φ(t) ∈ H , xφ ∈ L (R D ) and φ is a solution of the Cauchy problem (1.1)-(1.2), we obtain J ′′ (t) = E(φ ), which nally gives, for all t ∈ [ , T), Letting t = , we nd Step 4. Determination of φ and conclusion. We nally apply identity (4.2) to φ and η = T , with θ(x) = |x| . Since ∇θ(x) = x, we obtain Note that this calculation justi es, a posteriori, the application of (4.2) with the function θ(x) = |x| ∉ C ∞ (R D ).
Hence, we have φ e i |x| T L = W L and E(φ e i |x| T ) = , and we deduce from the variational characteristic of ground state that there exist λ > and γ ∈ R such that φ = e iγ e −i |x| T λ D W(λ x).
Acknowledgements: This research was supported by the National Natural Science Foundation of China 11871138.

Con ict of interest:
Authors state no con ict of interest.