On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects

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

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