*Accepted Paper*

**Inserted:** 29 jul 2015

**Last Updated:** 7 dec 2016

**Journal:** Annales de l'Institut Henri PoincarĂ© (C) Nonlinear Analysis

**Year:** 2016

**Doi:** 10.1016/j.anihpc.2016.04.001

**Links:**
link preliminary version

**Abstract:**

We consider a family of positive solutions to the system of $k$ components \[ -\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^2 \qquad \text{in $\Omega$}, \] where $\Omega \subset \R^N$ with $N \ge 2$. It is known that uniform bounds in $L^\infty$ of $\{\mathbf{u}_{\beta}\}$ imply convergence of the densities to a segregated configuration, as the competition parameter $\beta$ diverges to $+\infty$. In this paper we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of $\mf{u}_\beta$ in terms of entire solutions to the limit system \[ \Delta U_i = U_i \sum_{j\neq i} a_{ij} U_j^2. \] Moreover, we develop a uniform-in-$\beta$ regularity theory for the interfaces.