## Topological Methods in Nonlinear Analysis

### Conley index of isolated equilibria

Martin Kell

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 38, Number 2 (2011), 373-393.

Dates
First available in Project Euclid: 20 April 2016

https://projecteuclid.org/euclid.tmna/1461184833

Mathematical Reviews number (MathSciNet)
MR2932043

Zentralblatt MATH identifier
1272.37013

#### Citation

Kell, Martin. Conley index of isolated equilibria. Topol. Methods Nonlinear Anal. 38 (2011), no. 2, 373--393. https://projecteuclid.org/euclid.tmna/1461184833

#### References

• 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 \ref\key 2
• A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge (2002) \ref\key 3
• D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981), 840 \ref\key 4
• H. Hofer, Variational and topological methods in partially ordered Hilbert spaces , Math. Ann, 261 (1982), 493–514 \ref\key 5 ––––, A note on the topological degree at a critical point of mountainpass-type , Proc. Amer. Math. Soc., 309–315, 90 (1984) \ref\key 6
• K. P. Rybakowski, On the homotopy index for infinite-dimensional semiflows , Trans. Amer. Math. Soc., 351–382, 269 (1982) \ref\key 7 ––––, The Homotopy Index and Partial Differential Equations, Springer, Berlin, 208 (1987) \ref\key 8
• E. H. Spanier, Algebraic Topology (1966), 528