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

Marek Izydorek¹ ; Krzysztof P. Rybakowski²

¹Technical University Gdańsk Faculty of Technical Physics and Applied Mathematics Narutowicza 11/12 80-952 Gdańsk, Poland
²Fachbereich Mathematik Universität Rostock Universitätsplatz 1 18055 Rostock, Germany

Fundamenta Mathematicae, Tome 173 (2002) no. 1, pp. 77-100

Résumé

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}.

Détail

DOI : 10.4064/fm173-1-5

Mots-clés : recent paper presented galerkin type conley index theory certain classes infinite dimensional odes without uniqueness property cauchy problem paper apply theory strongly indefinite elliptic systems specifically study elliptic system eqalign mit delta partial quad hbox mit omega mit delta partial quad hbox mit omega quad quad hbox partial mit omega tag smooth bounded domain omega mathbb type hamiltonians class satisfying subcritical growth assumptions their first order derivatives shown angenent van der vorst cite solutions equilibria abstract ordinary differential equation dot tag defined certain hilbert space functions map continuous general differentiable nor even locally lipschitzian main result paper linearization principle which states whenever hyperbolic equilibrium conley index computed formally linearizing particular application linearization principle obtain elementary conley index based proof existence nontrivial solutions result previously established cite via morse floer homology further applications method existence multiplicity results strongly indefinite systems appear cite cite