Conley index in Hilbert spaces and a problem of Angenent and van der Vorst

Volume 173 / 2002

Marek Izydorek, Krzysztof P. Rybakowski Fundamenta Mathematicae 173 (2002), 77-100 MSC: Primary 37B30, 47J35; Secondary 34G20, 58E05. DOI: 10.4064/fm173-1-5


In a recent paper [9] we presented a Galerkin-type Conley index theory for certain classes of infinite-dimensional ODEs without the uniqueness property of the Cauchy problem. In this paper we show how to apply this theory to strongly indefinite elliptic systems. More specifically, we study the elliptic system $$\eqalign{ -{\mit\Delta} u={}&\partial_vH(u,v,x)\quad\ \hbox{in ${\mit\Omega}$,}\cr -{\mit\Delta} v={}&\partial_uH(u,v,x)\quad\ \hbox{in ${\mit\Omega}$,}\cr u={}&0,\quad v=0\quad\ \hbox{in $\partial{\mit\Omega}$,}\cr}\tag{$A1$} $$ on a smooth bounded domain $\Omega$ in $\mathbb R^N$ for “$-$”-type Hamiltonians $H$ of class $C^2$ satisfying subcritical growth assumptions on their first order derivatives. As shown by Angenent and van der Vorst in \cite{AV}, the solutions of $(A1)$ are equilibria of an abstract ordinary differential equation $$ \dot z=f(z)\tag{$A2$} $$ defined on a certain Hilbert space $E$ of functions $z=(u,v)$. The map $f: E\to E$ is continuous, but, in general, not differentiable nor even locally Lipschitzian. The main result of this paper is a Linearization Principle which states that whenever $z_0$ is a hyperbolic equilibrium of $(A2)$ then the Conley index of $\{z_0\}$ can be computed by formally linearizing $(A2)$ at $z_0$. As a particular application of the Linearization Principle we obtain an elementary, Conley index based proof of the existence of nontrivial solutions of $(A1)$, a result previously established in \cite{AV} via Morse–Floer homology. Further applications of our method to existence and multiplicity results for strongly indefinite systems appear in \cite{CR} and \cite{IR2}.


  • Marek IzydorekTechnical University Gdańsk
    Faculty of Technical Physics
    and Applied Mathematics
    Narutowicza 11/12
    80-952 Gdańsk, Poland
  • Krzysztof P. RybakowskiFachbereich Mathematik
    Universität Rostock
    Universitätsplatz 1
    18055 Rostock, Germany

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