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.
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