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

We consider solutions of the competitive elliptic system(S)-δui=-∑j≠iuiuj2in RNui>0in RNi=1,...,k. We are concerned with the classification of entire solutions, according to their growth rate. The prototype of our main results is the following: there exists a function δ=δ(k,N)∈N, increasing in k, such that if (u<inf>1</inf>, . . ., u<inf>k</inf>) is a solution of (S) andu1(x)+⋯+uk(x)≤C(1+|x|d)for every x∈RN, then d≥δ. This means that the number of components k of the solution imposes a lower bound, increasing in k, on the minimal growth of the solution itself. If N=2, the expression of δ is explicit and optimal, while in higher dimension it can be characterised in terms of an optimal partition problem. We discuss the sharpness of our results and, as a further step, for every N≥2 we can prove the 1-dimensional symmetry of the solutions of (S) 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.