Strong instability of standing waves for nonlinear Schr\"{o}dinger equations with a partial confinement

We study the instability of standing wave solutions for nonlinear Schr\"{o}dinger equations with a one-dimensional harmonic potential in dimension $N\ge 2$. We prove that if the nonlinearity is $L^2$-critical or supercritical in dimension $N-1$, then any ground states are strongly unstable by blowup.


Introduction
In this paper, we study the instability of standing wave solutions e iωt φ ω (x) for the nonlinear Schrödinger equation with a one-dimensional harmonic potential 1) where N ≥ 2, x N is the N-th component of x = (x 1 , ..., x N ) ∈ R N , ∆ is the Laplacian in x, and 1 < p < 1 + 4/(N − 2). Here, 1 + 4/(N − 2) stands for ∞ if N = 2. The Cauchy problem for (1.1) is locally well-posed in the energy space X (see [6,Theorem 9.2.6]). Here, the energy space X for (1.1) is defined by Proposition 1. Let 1 < p < 1 + 4/(N − 2). For any u 0 ∈ X there exist T max = T max (u 0 ) ∈ (0, ∞] and a unique maximal solution u ∈ C([0, T max ), X)∩ C 1 ([0, T max ), X * ) of (1.1) with initial condition u(0) = u 0 . The solution u(t) is maximal in the sense that if T max < ∞, then u(t) X → ∞ as t ր T max . Moreover, the solution u(t) satisfies the conservation laws for all t ∈ [0, T max ), where the energy E is defined by Next, we consider the stationary problem where ω ∈ R. Note that if φ(x) solves (1.3), then e iωt φ(x) is a solution of (1.1). Moreover, (1.3) can be written as is the action. The set of all ground states for (1.3) is defined by Then, we have the following result on the existence of ground states for (1.3).
Proposition 2. Let 1 < p < 1 + 4/(N − 2) and ω ∈ (−1, ∞). Then, the set G ω is not empty, and it is characterized by is the Nehari functional, and Although Proposition 2 can be proved by the standard concentration compactness argument, for the sake of completeness, we give the proof of Proposition 2 in Section 3.
Here, we remark that by Heisenberg's inequality for any ω ∈ (−1, ∞) there exist positive constants C 1 (ω) and C 2 (ω) such that for all v ∈ X. Now we state our main result in this paper.
Notice that Theorem 1 covers the physically relevant case N = 3 and p = 3 as a borderline case.
Finally, we consider the nonlinear Schrödinger equations with a partial confinement of the form The typical case is that N = 3 and d = 2. Recently, Bellazzini, Boussaïd, Jeanjean and Visciglia [2] constructed orbitally stable standing wave solutions of (1.11) for the case (see Theorem 1 and Remark 1.9 of [2]). It should be remarked that the bottom of the spectrum of −∆ + (x 2 1 + · · · + x 2 d ) is not an eigenvalue, so that unlike (1.10) with a complete confinement, the existence of stable standing wave solutions for (1.11) is highly nontrivial in the L 2 -supercritical case p > 1 + 4/N.
Although it is not clear whether the standing wave solutions constructed by [2] are ground states in the sense of (1.4) (see Definition 1.1 and Remark 1.10 of [2]), it would be safe to conclude from our Theorem 1 that the upper bound on p in (1.12) is optimal for the existence of stable standing wave solutions of (1.11).
The rest of the paper is organized as follows. In Section 2, we give the proof of Theorem 1. The proof is based on a virial type identity (2.1) associated with the scaling (2.2), the characterization of ground states (1.5) by the minimization problem on the Nehari manifold, and Lemma 1 below. We remark that the classical method by Berestycki and Cazenave [3] is not applicable to (1.1) directly. Instead, we use and modify the ideas of Zhang [21] and Le Coz [14], which give an alternative approach to the strong instability (see also [17,18,19] for recent developments).
In Section 3, we give the proof of Proposition 2. The proof is based on the standard concentration compactness argument.

Proof of Theorem 1
We define First, we derive a virial type identity.
Proof. We state formal calculations for the identity (2.1) only. These formal calculations can be justified by the classical regularization argument as in [6, Proposition 6.5.1] (see also [16]).
Let u(t, x) be a smooth solution of (1.1). Then, we have Moreover, we have Here, we consider the scaling and Thus, we have As stated above, these formal calculations can be justified by the regularization argument.
Once we have obtained Lemma 1, the rest of the proof is the same as in the classical argument of Berestycki and Cazenave [3].
is invariant under the flow of (1.1). That is, if u 0 ∈ B ω , then the solution u(t) of (1.1) with u(0) = u 0 satisfies u(t) ∈ B ω for all t ∈ [0, T max ).
Proof. This follows from the conservation laws (1.2), Lemma 1, and the continuity of the function t → P (u(t)).
Proof. Let u 0 ∈ B ω ∩ Σ and let u(t) be the solution of (1.1) with u(0) = u 0 . Then, it follows from Lemma 2 and Proposition 3 that Moreover, by the virial identity (2.1), the conservation laws (1.2) and Lemma 1, we have 1 16 Finally, we give the proof of Theorem 1.
We define Note that by (1.8), there exists a positive constant C 0 depending only on ω and p such that We also remark that by (3.1) and (1.6), we have Proof. Let v ∈ X satisfy K ω (v) = 0 and v = 0. Then, by K ω (v) = 0, the Sobolev inequality and (3.2), there exist positive constants C 1 and C 2 depending only on N, p and ω such that Since v = 0, we have J ω (v) > 0 and J ω (v) (p−1)/2 ≥ 1/C 2 . Thus, by (3.3), we have This completes the proof.
The following lemma is a variant of the classical result of Lieb [15] (see also [2,Lemma 3.4]).
Lemma 5. Assume that a sequence (u n ) n∈N is bounded in X, and satisfies lim sup n→∞ u n p+1 L p+1 > 0.
Then, there exist a sequence (y n ) n∈N in R N −1 and u ∈ X \ {0} such that (τ y n u n ) n∈N has a subsequence which converges to u weakly in X.
Then, by the definition of C 3 , we see that for any n ∈ N, there exists y n ∈ Z N −1 such that Here, we define v n = τ −y n u n . Then, we have for all n ∈ N. In particular, v n p+1 L p+1 (Q 0 ) > 0 for all n ∈ N. Moreover, by the Sobolev inequality, we have for all n ∈ N, where C 4 is a positive constant depending only on N and p.
Thus, we have Since (v n ) n∈N is bounded in X, there exist a subsequence (v n ′ ) of (v n ) and u ∈ X such that (v n ′ ) converges to u weakly in X.
We define the set of all minimizers for (1.6) by Lemma 6. The set M ω is not empty.
Proof. Let (u n ) be a sequence in X such that K ω (u n ) = 0, u n = 0 for all n ∈ N, and S ω (u n ) → d(ω). Then, by (3.2) and J ω (u n ) = S ω (u n ) → d(ω), we see that the sequence (u n ) n∈N is bounded in X.
Moreover, it follows from K ω (u n ) = 0 and Lemma 3 that Thus, by Lemma 5, there exist a sequence (y n ) in R N −1 , a subsequence of (τ y n u n ), which is denoted by (v n ), and v ∈ X \ {0} such that (v n ) converges to v weakly in X. By the weakly lower semicontinuity of J ω , we have (3.6) Moreover, by the Brezis-Lieb Lemma (see [4]), we have On the other hand, by v = 0 and (3.2), we have J ω (v) > 0. This is a contradiction. Thus, we obtain K ω (v) ≤ 0.
Finally, we give the proof of Proposition 2.