  In this paper we consider a class of gradient systems of type $$ -c_i \Delta
u_i + V_i(x)u_i=P_{u_i}(u),\quad u_1,..., u_k>0 \text{in}\Omega, \qquad
u_1=...=u_k=0 \text{on} \partial \Omega, $$ in a bounded domain
$\Omega\subseteq \R^N$. Under suitable assumptions on $V_i$ and $P$, we prove
the existence of ground-state solutions for this problem. Moreover, for $k=2$,
assuming that the domain $\Omega$ and the potentials $V_i$ are radially
symmetric, we prove that the ground state solutions are foliated Schwarz
symmetric with respect to antipodal points. We provide several examples for our
abstract framework.
