Multidimensional entire solutions for an elliptic system modelling phase separation

Abstract

For the system of semilinear elliptic equations

$$\Delta V_i=V_i\sum _{j\ne i}{V}_j^2,V_i>0\text{in }{ℝ}^N,$$

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 to 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 systems and new sharp versions for monotonicity formulae of Alt–Caffarelli–Friedman type.