Conley index of isolated equilibria

Abstract

In this paper we study stable isolated invariant sets and show that the zeroth singular homology of the Conley index characterizes stability completely. Furthermore, we investigate isolated mountain pass points of gradient-like semiflows introduced by Hofer in [Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), 493-514] and show that the first singular homology characterizes them completely.

The result of the last section shows that for reaction-diffusion equations \begin{align*} u_{t}-\Delta u& = f(u),\\ u_{|\partial\Omega}& = 0, \end{align*} the Conley index of isolated mountain pass points is equal to $\Sigma^{1}$ - the pointed $1$-sphere. Finally we generalize the result of [T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z. 233 (2000), 655-677, Proposition 3.3] about mountain pass points to Alexander-Spanier cohomology.