Positive solutions for classes of nonlinear reaction diffusion equations

We consider positive solutions for classes of multiparameter problems of the form -Du=lgu +mfu;W u=0;6W where Delta is the Laplacian operator, O is a bounded domain in Rn (n > 1) with smooth boundary ∂O, m > 0, l , > 0, f(0) < 0, f is sublinear, g(0) > 0, and g is superlinear. For a fixed m > 0, we will discuss existence and multiplicity results for l small and nonexistence results for l large. We prove our existence results using variational techniques and sub and super solution arguments. For the case n = 1, we develop a quadrature method and discuss the evolution of these results as m varies, using both analytical and numerical methods.; We also study positive solutions of classes of singular diffusion problems in a two patch nonhomogeneous environment where both passive and active interchanges between the two patches, and between each patch and the surrounding habitat, are allowed. We obtain existence and uniqueness results using shooting methods.