Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect

Abstract

We study the ground states for the Schrödinger equation with a focusing nonlinearity and a point interaction in dimension three. We establish that ground states exist for every value of the mass; moreover, they are positive, radially symmetric, and decreasing along the radial direction and show a Coulombian singularity at the location of the point interaction. Remarkably, the existence of the ground states is independent of the attractive or repulsive character of the point interaction.