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

We consider solutions of the competitive elliptic systemequation(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)∈Nδ=δ(k,N)∈N, increasing in k  , such that if (u1,…,uk)(u1,…,uk) is a solution of (S) andu1(x)+⋯+uk(x)≤C(1+|x|d)for every x∈RN, then d≥δd≥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=2N=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≥2N≥2 we can prove the 1-dimensional symmetry of the solutions of (S) satisfying suitable assumptions, extending known results which are available for k=2k=2. The proofs rest upon a blow-down analysis and on some monotonicity formulae.