A simple variational approach to weakly coupled competitive elliptic systems

The main purpose of this paper is to exhibit a simple variational setting for finding fully nontrivial solutions to the weakly coupled elliptic system (1.1). We show that such solutions correspond to critical points of a ${\mathcal{C}}^1$-functional $\mathrm{\Psi }:\mathcal{U}\to \mathbb{R}$ defined in an open subset $\mathcal{U}$ of the product $\mathcal{T}:=S_1\times \cdots \times S_M$ of unit spheres $S_i$ in an appropriate Sobolev space. We use our abstract setting to extend and complement some known results for the system (1.1).