Existence and symmetry results for competing variational systems

In this paper we consider a class of gradient systems of type $-c_i Δu_i + V_i(x)u_i=P_{u_i}(u),\quad u_1,..., u_k>0 \text{in}Ω, \qquad u_1=...=u_k=0 \text{on} \partial Ω,$ in a bounded domain $Ω\subseteq \R^N$. 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 $Ω$ 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.

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