Singular solutions of nonlinear Schrِdinger problem in bounded piecewise Dini-smooth Jordan domain

We prove the existence of positive singular solutions of the nonlinear equation Δ u − μ u = f ( . , u ) on Ω ∖ { 0 } , where Ω is a bounded simply connected piecewise Dini-smooth Jordan domain in R 2 , μ and f are, respectively, a signed measure on Ω and a Borel measurable function, required to satisfy suitable assumptions related to a new functional class J.