Existence and symmetry results for competing variational systems

Abstract

In this paper we consider a class of gradient systems of type -ciΔ ui+Vi(x)ui = Pui(u), u1, ... ,uk > 0 in Ω, u1 = ... = uk = 0 on ∂Ω, in a bounded domain Ω ⊆ ℝN. 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. © 2012 Springer Basel AG.