Logistic Dirichlet problems with discontinuous coefficients

This paper is devoted to the study of the existence of positive solutions of semilinear Dirichlet eigenvalue problems for diffusive logistic equations with discontinuous coefficients which model population dynamics in environments with spatial heterogeneity. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of singular integral operators. Moreover, we make use of an L^p variant of an estimate for the Green operator of the Dirichlet problem introduced in the study of Feller semigroups.