On the Global Structure of the Set of Positive Solutions of Some Semilinear Elliptic Boundary Value Problems

In this work we analyze the structure of the set of positive solutions of a class of semilinear boundary value problems. It is shown that the global continuum of positive solutions emanating from the trivial equilibrium at the principal eigenvalue of the linearization is constituted by a regular curve if the slope of the kinetic at the trivial solution is large enough and Ω is convex. The same result holds if the support region of the species is a bounded simply connected domain of 2 close to a convex domain, in a sense to be precised later. To prove these results we have to find out the exact width of the boundary layer of a singular perturbation problem. The results about the singular perturbation problem are new and of great interest by themselves.