In this talk, we present some results about ground states at fixed mass of a L2-subcritical Nonlinear Schrödinger Equation in presence of a point interaction and, possibly, of an attractive Coulomb potential. First, we present the problem in dimension two and three. We prove that ground states exist for every value of the mass and, up to gauge invariance, they are positive, radially symmetric, decreasing along the radial direction and present a logarithmic singularity where the interaction is placed. In order to obtain qualitative features of the ground states, we refine a classical result on rearrangements and move to equivalent variational formulations of the problem.
Then, we present some future developments, among them the possibility to study nonlinear models on hybrids, i.e. particular multi-dimensional manifolds introduced in the eighties by Exner and Šeba in the linear setting. These results have been obtained in collaboration with R. Adami, R. Carlone, M. Gallone and L. Tentarelli.