Strong competition versus fractional diffusion: the case of Lotka-Volterra interaction

We consider a system of differential equations with nonlinear Steklov boundary conditions, related to the fractional problem $$(-\Delta)^s u_i = f_i(x,u_i) - \beta u_i^p \sum_{j\neq i} a_{ij} u_j^p,$$ where $i = i,\dots, k$, $s\in(0,1)$, $p>0$, $a_{ij}>0$ and $\beta>0$. When $k=2$ we develop a quasi-optimal regularity theory in $C^{0,\alpha}$, uniformly w.r.t. $\beta$, for every $\alpha < \alpha_{\mathrm opt}=min(1,2s)$; moreover we show that the traces of the limiting profiles as $\beta\to+\infty$ are Lipschitz continuous and segregated. Such results are extended to the case of $k\geq3$ densities, with some restrictions on $s$, $p$ and $a_{ij}$.