Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory

Marek Izydorek¹ ; Krzysztof P. Rybakowski²

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

Fundamenta Mathematicae, Tome 176 (2003) no. 3, pp. 233-249

Résumé

Let ${\mit\Omega}$ be a bounded domain in $\mathbb R^N$ with smooth boundary. Consider the following elliptic system: $$\eqalign{ -{\mit\Delta} u=\partial_vH(u,v,x)\quad\ \hbox{in ${\mit\Omega}$,}\cr -{\mit\Delta} v=\partial_uH(u,v,x)\quad\ \hbox{in ${\mit\Omega}$,}\cr u=0,\quad v=0\quad\ \hbox{in $\partial{\mit\Omega}$.}\cr} \tag{ES} $$ We assume that $H$ is an even “$-$”-type Hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions. We show that if $(0,0)$ is a hyperbolic solution of~(ES), then (ES) has at least $2|\mu|$ nontrivial solutions, where $\mu=\mu(0,0)$ is the renormalized Morse index of $(0,0)$. This proves a conjecture by Angenent and van der Vorst.

Détail

DOI : 10.4064/fm176-3-3

Keywords: mit omega bounded domain mathbb smooth boundary consider following elliptic system eqalign mit delta partial quad hbox mit omega mit delta partial quad hbox mit omega quad quad hbox partial mit omega tag assume even type hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions hyperbolic solution has least nontrivial solutions where renormalized morse index proves conjecture angenent van der vorst