Conley index in Hilbert spaces and a problem of Angenent and van der Vorst
Tom 173 / 2002
Streszczenie
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}.
