Topological methods for finding nontrivial solutions of elliptic boundary value problems

Abstract

ONE WAY of demonstrating the existence of a solution of a nonlinear equation is to consider the behavior of the operator associated with the equation on the boundary of a large ball in a function space. Then using some topological tools like homotopy or degree theory, one deduces the existence of a solution within the ball. However, in many instances a simple solution is already known; for example the zero function might be a solution. At this point, the topological tool may not give us any more information than we already know. But all is not lost for if we consider a little ball around the solution and study the behavior of the operator on its boundary, we may compare it with the behavior on the boundary of the large ball and using our topological tools conclude that there exists another solution in the annulus between. It is the purpose of this paper to study such problems. Nirenberg [l] has considered the existence question on a ball with a large radius for the equation x K(x) = 0 where K is compact and Cronin [2] has extended some of his results to an annulus. In Section 1 we extend Nirenberg’s result by considering the problem in an annulus instead of the ball. We obtain conditions for the existence of solutions and then apply the results to Fredholm operators of index i > 0. We then relate Cronin’s results [2] to ours. Section 2 deals with applications to elliptic partial differential equations. Here we use degree theory to obtain the existence of nontrivial solutions. The main references of this chapter are the works of Nirenberg [l, 31. In determining the degree of specific vector fields, we draw from the texts of Cronin [4] and Krasnosel’skii [5]. We also present some simple examples to illustrate our results. 1 wish to express my appreciation to my thesis advisor, Professor Louis Nirenberg, for his assistance and encouragement.