Abstract
In this lecture, I want to discuss how Morse inequalities apply in more general situations where mappings are not bounded below, where there are degenerate critical points, and where we have some informa·tion of hmv solutions of the corresponding differential equations join critical· points. We then discuss an application to elliptic partial differential equation and some counterexamples which show that our estimates for the number of solutions of the elliptic problem are, in a sense, best possible. The proof of most of the results given depend upon the homotopy index of Conley [3]. However, nearly all the results can be understood without: knowing the homotopy index.
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