We study minimal mass blow-up solutions of the focusing L 2 critical nonlinear Schrödinger equation with inverse-square potential, i∂tu + ∆u + c |x| 2 u + |u| 4 N u = 0, with N 3 and 0 < c < (N−2) 2 4. We first prove a sharp global well-posedness result: all H 1 solutions with a mass (i.e. L 2 norm) strictly below that of the ground states are global. Note that, unlike the equation in free space, we do not know if the ground state is unique in the presence of the inverse-square potential. Nevertheless, all ground states have the same, minimal, mass. We then construct and classify finite time blow-up solutions at the minimal mass threshold. Up to the symmetries of the equation, every such solution is a pseudo-conformal transformation of a ground state solution.2010 Mathematics Subject Classification. 35Q55 ; 35B44 ; 35C06.

In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point argument. Unlike the unperturbed case, a noteworthy difficulty here arises from the possible non-unitarity of the semigroup generating the corresponding linear evolution. We then show that the equation is Hamiltonian and we establish several stability/instability results for its standing waves. Our analysis relies on a detailed study of the spectral properties of the linearization of the equation, and on the well-known 'slope condition' for orbital stability.( 1.15) This group action leaves (1.12) invariant. The corresponding notion of orbital stability is the following.Definition 1.2. For a fixed ω 0 ∈ R, the standing wave e iω0t Φ ω0 is orbitally stable if the following holds: for any > 0 there is a δ > 0 such that, if U (t) is a solution of (3.3), then we haveOtherwise, Φ ω0 is said to be orbitally unstable.In addition to orbital stability, we will also prove some linear instability results. Writing a solution U of (1.12) in the form U (t) = e iω0t (Φ ω0 + V (t)), we have that, at first order, V satisfies the linearized equationwhere L ω is defined in (1.20).

In our work, we establish the existence of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We apply a profile decomposition argument to overcome the difficulty arising from the non-compactness of the setting. We obtain convergent minimizing sequences by comparing the problem to the problem at “infinity” (i.e., the equation without inverse square potential). Finally, we establish orbital stability/instability of the standing wave solution for mass subcritical and supercritical nonlinearities respectively.

We investigate the properties of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We first establish the existence of standing waves. Then, by a variational characterization of the ground states, we establish the orbital stability of standing waves for mass sub-critical nonlinearity. Owing to the non-compactness and the absence of translational invariance of the problem, we apply a profile decomposition argument. We obtain convergent minimizing sequences by comparing the problem to the problem at "infinity" (i.e., the equation without inverse square potential). Finally, we establish orbital instability by a blow-up argument for mass super-critical nonlinearity.

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