Bifurcation and Related Topics in Elliptic Problems

Abstract

Bifurcation theory provides a bridge between the linear world and the more complicated nonlinear world, and thus plays an important role in the study of various nonlinear problems. Nonlinear elliptic boundary value problems enjoy many nice properties that allow the use of a variety of powerful tools in nonlinear functional analysis. In the past three decades, bifurcation theory has been successfully combined with these tools to yield rather deep results for elliptic problems. Traditionally bifurcation analysis was based on local linearization techniques, but more and more global analysis is involved in modem bifurcation theory. A highlight is the global bifurcation theory of Krasnoselskii and Rabinowitz (see [Kr,Ra]), which is resulted from the use of topological degree theory, general set point theory and a linearization consideration. The final result, generally known as Rabinowitz's global bifurcation theorem (to be recalled later), has played a fundamental role in proving a great number of existence results for elliptic problems. Making use of the maximum principle, one can study elliptic problems in the framework of ordered Banach spaces. The extra order structure greatly strengthens these abstract tools. An excellent presentation of these techniques up to the late 1970s can be found in Amann's by now classical review article [Am]. In this chapter we intend to present some further results for elliptic boundary value problems, where bifurcation theory plays an important role in the proofs; we will focus on recent developments, well after [Am]. In Section 2 we discuss a new phenomenon, namely bifurcation from infinity caused by spatial 'degeneracy' in the nonlinearity and determined by 'boundary blow-up solutions'. In Section 3 we combine bifurcation argument and order structure to study a system of elliptic equations, and demonstrate that, apart from multiplicity results, these techniques can be used to discuss the stability and the profiles of the solutions. In Section 4 we present some recent fine techniques in determining the exact number of positive solutions of elliptic equations over a ball; in particular, we give the proof of a long standing conjecture on the perturbed Gelfand equation. In Section 5 we discuss the usefulness of nodal properties of solutions in global bifurcation theory. Most of the problems discussed here have an open ending; related problems and open questions can be found in the remarks at the end of the sections or subsections.