Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory
Volume 176 / 2003
Fundamenta Mathematicae 176 (2003), 233-249
MSC: Primary 35J50, 58E40; Secondary 58E05, 58E50.
DOI: 10.4064/fm176-3-3
Abstract
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.
