Minimizers of mass-constrained functionals involving a nonattractive point interaction

We establish conditions to ensure the existence of minimizer for a class of mass-constrained functionals involving a nonattractive point interaction in three dimensions. The existence of minimizers follows from the compactness of minimizing sequences which holds when we can simultaneously rule out the possibilities of vanishing and dichotomy. The proposed method is derived from the strategy used to avoid vanishing in Adami, Boni, Carlone & Tentarelli (Calc. Var. 61, 195 (2022)) and the strategy used to avoid dichotomy in Bellazzini & Siciliano (J. Funct. Anal. 261, 9 (2011)). As applications, we prove the existence of ground states with sufficiently small mass for the following nonlinear problems with a point interaction: a Kirchhoff-type equation and the Schr\"odinger-Poisson system.