On the De Giorgi type conjecture for an elliptic system modeling phase separation

In this paper we study the one dimensional symmetry problem of entire solutions to the problem $$\mathrm{\Delta }u=u{v}^2,\mathrm{\Delta }v=v{u}^2,u,v>0\text{in}{\mathbb{R}}^n,$$ for all $n\ge 2$. We prove that, if a solution $(u,v)$ is a local minimizer and has linear growth at infinity, then it is one dimensional, i.e. depending only on one variable. In the proof we also obtain the global Lipschitz continuity of solutions only under the linear growth assumption.