Point interactions and singular solutions to semilinear elliptic equations

TL;DR

This paper explores the relationship between semilinear elliptic PDEs with singularities and nonlinear Schrödinger equations with point interactions, revealing new solution existence results through operator and variational methods.

Contribution

It establishes an equivalence between elliptic PDEs with singularities and Schrödinger equations with point interactions, enabling novel solution existence proofs.

Findings

Existence of infinitely many singular solutions in the focusing case.

Characterization of positive solutions via action ground states in 2D.

Existence of infinitely many singular, nodal solutions.

Abstract

We investigate the connection between semilinear elliptic PDEs with isolated singularities and stationary nonlinear Schr\"odinger equations with point interactions. In dimensions $d=2,3$, we provide a detailed equivalence result between the two problems. As a consequence, this allows us to exploit a range of operator-theoretic and variational techniques, hitherto not explicitly explored in the context of singular solutions. By leveraging this approach, in the focusing case, we provide the existence of infinitely many singular solutions by applying the Ambrosetti-Rabinowitz theory to an action functional adapted to the point interaction. When $d=2$ and relying on a suitable uniqueness result, we also characterize positive solutions in terms of action ground states, and we show the existence of infinitely many singular, nodal solutions.