Reference

Pascal Bégout, and Jesus Ildefonso Diaz, Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity , TSE Working Paper, n. 13-402, April 2013.

Abstract

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schrödinger 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 L2-space

Pascal Bégout, and Jesus Ildefonso Diaz, Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity , Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, vol. 109, n. 1, March 2015, pp. 43–63.