For a given $s \in (0,1)$, we consider a family of positive solutions to the system of $k$ components

\begin{equation}\label{eqn var}

(-\Delta)^s u_{i,\beta} = - \beta u_{i,\beta} \sum_{j\neq i} u_{j,\beta}^2 \quad \text{in $\Omega$}

\end{equation}

or the system

\begin{equation}\label{eqn sim}

(-\Delta)^s u_{i,\beta} = - \beta u_{i,\beta} \sum_{j\neq i} u_{j,\beta} \quad \text{in $\Omega$}

\end{equation}

where $\Omega \subset \mathbb{R}^N$, with $N \geq 2$, is a smooth domain, completed by some smooth condition on the outside of $\Omega$. The previous systems arise, for instance, in the study of the phase separation in Bose-Einstein condensates under relativistic approximation and in some models of populations dynamics.

In the standard local case, that is when $s = 1$, it is known that uniformly-in-$\beta$ bounded solutions are also bounded in the Lipschitz norm, and that in singular limit $\beta \to +\infty$ the solutions converge to some limit Lipschitz profiles which share many properties. A key aspect is that, even though the proofs are different, the solutions of the two systems share many properties.

In this talk I will consider the fractional counterpart of the previous results, and show that in this case the solutions do exhibit different properties: in particular I will show that for Equation \eqref{eqn var}, the natural regularity is to be found in the H\"older space $\mathcal{C}^{0,s}$, while for Equation \eqref{eqn sim}, the regularity is still in the Lipschitz space. For both claim I will show some sub-optimal results.

This talk is based on some results obtained in collaboration with S. Terracini and G. Verzini.