Newtonian potential and positive solutions of Poisson equations

Abstract

We study the properties of the well-known Newtonian potential operator including its domain, codomain, compactness, relation with the Poisson’s equation and the spectral radii of the weight Newtonian potential operators. These new properties are then applied to study the existence of positive classical solutions of nonlinear Poisson’s equations in bounded open sets, which are required neither to be connected nor smooth on their boundaries. The connectedness and smoothness are common requirements for studying the existence of classical or weak solutions of linear or nonlinear elliptic boundary value problems in the literature. Some applications to population models are provided.