Existence and symmetry results for competing variational systems

In this paper we consider a class of gradient systems of type $$begin{array}{ll} -c_{i} Delta u_{i} + V_{i}(x)u_{i} = P_{u_i}(u),qquad u_{1}, ldots, u_{k} >; 0; text{in}; Omega, quad u_{1} = cdots = u_{k} = 0 text{ on } partial Omega, end{array}$$ in a bounded domain ${Omega subseteq mathbb{R}^N}$ . 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 Ω 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.