Minimizers of constrained functionals involving a point interaction

Let E_ W R denote the expectation value of the Hamiltonian of point interaction in R³ with inverse scattering length ]0, [ and consider an energy functional I_ W R of the form I_ (u) = 12 E_ (u) + T (u), where T W R is a given nonlinear functional. We propose a set of conditions on, I_ and T under which the problem I_ (u) = \I_ (v): \|v\|₋ℂ² = ²\; \|u\|₋ℂ² = ² has a solution. As an application, we prove the existence of ground states with sufficiently small mass 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.