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We investigate the ground states for the focusing, subcritical nonlinearSchr\"odinger equation with a point defect in dimension two, defined as theminimizers of the energy functional at fixed mass. We prove that ground statesexist for every positive mass and show a logarithmic singularity at the defect.Moreover, up to a multiplication by a constant phase, they are positive,radially symmetric, and decreasing along the radial direction. In order toovercome the obstacles arising from the uncommon structure of the energy space,that complicates the application of standard rearrangement theory, we move tothe study of the minimizers of the action functional on the Nehari manifold andthen establish a connection with the original problem. An ad hoc result onrearrangements is given to prove qualitative features of the ground states.

Ground states for the planar NLSE with a point defect as minimizers of the constrained energy