Positive solutions of nonlinear elliptic boundary value problems

This dissertation focuses on the study of positive steady states to classes of nonlinear reaction diffusion (elliptic) systems on bounded domains as well as on exterior domains with Dirichlet boundary conditions. In particular, we study such systems in the challenging case when the reaction terms are negative at the origin, referred in the literature as semipositone problems. For the last 30 years, study of elliptic partial differential equations with semipositone structure has flourished not only for the semilinear case but also for quasilinear case. Here we establish several results that directly contribute to and enhance the literature of semipositone problems. In particular, we discuss existence, non-existence and multiplicity results for classes of superlinear as well as sublinear systems. We establish our results via the method of sub-super solutions, degree theory arguments, a priori bounds and energy analysis