On stability and instability of standing waves for the nonlinear Schrödinger equation with an inverse-square potential

Abstract

We consider the focusing nonlinear Schrödinger equation with inverse square poten-Using the profile decomposition obtained recently by the first author [1], we show that in the L 2 -subcritical case, i.e. 0 < α < 4 d , the sets of ground state standing waves are orbitally stable. In the L 2 -critical case, i.e. α = 4 d , we show that ground state standing waves are strongly unstable by blow-up.

Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential

“…We should point out that, among the methods used in the study of orbital stability of standing waves, the profile decomposition method plays an important role in recent studies, see [4], [13] and [38]. In [4], the authors considered a Schrödinger equation with inverse-square potential, i.e. (1.1) with N ≥ 3, α = 2 and γ < (N − 2) 2 /4.…”

Existence, non-existence and blow-up behaviour of minimizers for the mass-critical fractional non-linear Schrödinger equations with periodic potentials

“…Note that theorem 1.4 does not give any information on the blow-up rate of (−Δ) s/2 u a L 2 . Under an additional assumption on the external potential, we gave a detailed description of the blow-up behaviour of minimizers for I(a).…”

“…using the profile decomposition, Bensouilah-Dinh-Zhu [2] proved the existence of minimizers for I(a) for any a > 0 in the mass-subcritical case. The existence, non-existence and blow-up behaviour of minimizers for J(b) has been extended to ring-shaped potentials in [19], multi-well potentials [20], ellipseshaped potentials [18] and rotating trap potentials [21].…”

Existence and orbital stability of standing waves to a nonlinear Schrödinger equation with inverse square potential on the half-line

“…There has been considerable interest recently in the study of the Schrödinger equation with inverse-square potential in three and higher dimensions. Classification of the so-called minimal mass blow-up solutions, global well-posedness, and stability of standing wave solutions were studied in [1,6,8,22]. In the papers by Bensouilah et al [1], and by Trachanas and Zographopoulos [22] the authors establish orbital stability of ground state solutions in the Hardy subcritical (c < (N − 2) 2 /4) and Hardy critical (c = (N − 2) 2 /4) case respectively for dimensions higher that three.…”

“…Classification of the so-called minimal mass blow-up solutions, global well-posedness, and stability of standing wave solutions were studied in [1,6,8,22]. In the papers by Bensouilah et al [1], and by Trachanas and Zographopoulos [22] the authors establish orbital stability of ground state solutions in the Hardy subcritical (c < (N − 2) 2 /4) and Hardy critical (c = (N − 2) 2 /4) case respectively for dimensions higher that three. In both cases, orbital stability is proved by showing the precompactness of minimizing sequences of the energy functional on an L 2 constraint.…”