  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 $\Psi:\mathcal{U}\to\mathbb{R}$ 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).
