Nonlinear scalar field equation with point interaction

TL;DR

This paper investigates the existence and properties of solutions to a nonlinear scalar field equation with a point interaction at the origin in two and three dimensions, using variational methods and establishing key identities.

Contribution

It proves the existence of nontrivial singular solutions for a broad class of nonlinearities and establishes the Pohozaev identity with gradient estimates.

Findings

Existence of nontrivial singular solutions in 2D and 3D

Derivation of the Pohozaev identity for the problem

Qualitative analysis of solution properties

Abstract

This paper is devoted to the study of the nonlinear scalar field equation with a point interaction at the origin in dimensions two and three. By applying the mountain pass theorem and the technique of adding one dimensional space, we prove the existence of a nontrivial singular solution for a wide class of nonlinearities. We also establish the Pohozaev identity by proving a pointwise estimate of the gradient near the origin. Some qualitative properties of nontrivial solutions are also given.