We study stability of solitary vortices in the two-dimensional trapped Bose-Einstein condensate (BEC) with a spatially localized region of self-attraction. Solving the respective Bogoliubov-de Gennes equations and running direct simulations of the underlying Gross-Pitaevskii equation reveals that vortices with topological charge up to S = 6 (at least) are stable above a critical value of the chemical potential (i.e., below a critical number of atoms, which sharply increases with S). The largest nonlinearity-localization radius admitting the stabilization of the higher-order vortices is estimated analytically and accurately identified in a numerical form. To the best of our knowledge, this is the first example of a setting which gives rise to stable higher-order vortices, S > 1, in a trapped self-attractive BEC. The same setting may be realized in nonlinear optics too.
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