Degree theory on convex sets and applications to bifurcation

Abstract

The aim of these lectures was to give a short introduction to the use of degree the­ory ideas. In particular, the main emphasis was in the use of degree theory ideas on convex sets. It seems to the author that these ideas are a very convenient tool for a number of problems, especially problems without a variational structure. (For example, many systems do not have a variational structure). Moreover, in many applications, we are only interested in positive solutions (because of the origin of the problem). In these cases, we are looking for solutions in a cone and we naturally find we are looking at problems on a closed convex set. Working directly on a set of positive solutions also has the advantage that we automati­cally exclude solutions we are not interested in. In many of the applications, we make crucial use of the formula for the index of a non-degenerate solution for a mapping defined on a closed convex set. In the first part of the lectures, §2-3, we discuss rather briefly degree theory for mappings on Rn and for completely continuous mappings on Banach spaces. This is quite standard material and it is included here for pedagogical reasons. In §4-6, we discuss the basic degree theory of mappings defined on closed convex sets, including the basic index for­mula, and some abstract applications. In §7-9, we discuss applications to partial differential equations. In §7, we discuss applications to symmetry breaking on two-dimensional annuli. In §8, we discuss the existence of solutions with both components positive of a competing species systems with diffusion. (These are often called coexisting populations). Finally, in §9, we discuss further these mod­els when the two populations interact strongly.