  For the system of semilinear elliptic equations \[
  \Delta V_i = V_i \sum_{j \neq i} V_j^2, \qquad V_i > 0 \qquad \text{in
$\mathbb{R}^N$} \] we devise a new method to construct entire solutions. The
method extends the existence results already available in the literature, which
are concerned with the 2-dimensional case, also in higher dimensions $N \ge 3$.
In particular, we provide an explicit relation between orthogonal symmetry
subgroups, optimal partition problems of the sphere, the existence of solutions
and their asymptotic growth. This is achieved by means of new asymptotic
estimates for competing system and new sharp versions for monotonicity formulae
of Alt-Caffarelli-Friedman type.
