Minimizers of mass-constrained functionals involving a nonattractive point interaction

TL;DR

This paper establishes conditions for the existence of minimizers in mass-constrained functionals with nonattractive point interactions in three dimensions, using compactness arguments to rule out vanishing and dichotomy, with applications to nonlinear equations.

Contribution

It introduces a novel approach combining strategies to prevent vanishing and dichotomy, ensuring minimizer existence for specific nonlinear problems with point interactions.

Findings

Existence of minimizers for small mass in nonlinear problems

Conditions that guarantee compactness of minimizing sequences

Application to Kirchhoff-type and Schrödinger-Poisson systems

Abstract

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.