Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation

We consider solutions of the competitive elliptic system $$\{\begin{array}{ll}-\mathrm{\Delta }{u}_i=-{\sum }_{j\mathrm{\ne }i}{u}_i{u}_j^2& \text{in }{\mathbb{R}}^N\\ u_i>0& \text{in }{\mathbb{R}}^N\end{array}i=1,\dots ,k\mathrm{.}$$ We are concerned with the classification of entire solutions, according with their growth rate. The prototype of our main results is the following: there exists a function $\delta =\delta (k,N)\in \mathbb{N}$, increasing in $k$, such that if $(u_1,\dots ,u_k)$ is a solution and $$u_1(x)+\cdots +u_k(x)\le C(1+\mathrm{\mid }x{\mathrm{\mid }}^d)\text{for every }x\in {\mathbb{R}}^N,$$ then $d\ge \delta$. This means that the number of components $k$ of the solution imposes an increasing in $k$ minimal growth on the solution itself. If $N=2$, the expression of $\delta$ is explicit and optimal, while in higher dimension it can be characterized in terms of an optimal partition problem. We discuss the sharpness of our results and, as a further step, for every $N\ge 2$ we can prove the $1$-dimensional symmetry of the solutions satisfying suitable assumptions, extending known results which are available for $k=2$. The proofs rest upon a blow-down analysis and on some monotonicity formulae.