Minimizers of constrained functionals involving a point interaction

Let E_α W → R denote the expectation value of the Hamiltonian of point interaction in R³ with inverse scattering length α ∈ ]0, ∞[ and consider an energy functional I_α W → R of the form I_α (u) = 1/2 E_α (u) + T (u), where T W → R is a given nonlinear functional. We propose a set of conditions on ρ, I_α and T under which the problem I_α (u) = inf _α (v) : \|v\|L²² = ρ²\; \|u\|L²² = ρ² has a solution. As an application, we prove the existence of ground states with sufficiently small mass ρ for the following nonlinear problems with a point interaction: (i) a Kirchhoff-type equation, (ii) the Schr\"odinger--Poisson system and (iii) the Schr\"odinger--Bopp--Podolsky system.