Existence of weak solutions to some stationary Schr{\"o}dinger equations with singular nonlinearity

TL;DR

This paper establishes the existence (and sometimes uniqueness) of weak solutions for stationary Schrödinger equations with singular nonlinearities, including unbounded domains, non-local terms, varied boundary conditions, and right-hand sides beyond L^2.

Contribution

It extends previous results by addressing unbounded domains, non-local terms, diverse boundary conditions, and more general right-hand side functions for Schrödinger equations with singular nonlinearities.

Findings

Existence of weak solutions in unbounded domains.

Inclusion of non-local nonlinear terms.

Solutions with right-hand sides beyond L^2.

Abstract

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schr\''{o}dinger operator under the presence of a singular nonlinear term. Among other new facts, with respect some previous results in the literature for such type of nonlinear potential terms, we include the case in which the spatial domain is possibly unbounded (something which is connected with some previous localization results by the authors), the presence of possible non-local terms at the equation, the case of boundary conditions different to the Dirichlet ones and, finally, the proof of the existence of solutions when the right-hand side term of the equation is beyond the usual $L^2$-space.