New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms

In this paper we focus on existence and symmetry properties of solutions to the cubic Schrödinger system−Δui+λiui=∑j=1dβijuj2uiin Ω⊂RN,i=1,…d where d⩾2d⩾2, λi,βii>0λi,βii>0, βij=βji∈Rβij=βji∈R for j≠ij≠i, N=2,3N=2,3. The underlying domain Ω is either bounded or the whole space, and ui∈H01(Ω) or ui∈Hrad1(RN) respectively. We establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients, as well as in the purely cooperative case.