Singular perturbations of systems of fractional differential equations
For a given s∈(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 Ω}
\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 Ω}
\end{equation}
where Ω⊂RN, with N≥2, is a smooth domain, completed by some smooth condition on the outside of Ω. 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-β bounded solutions are also bounded in the Lipschitz norm, and that in singular limit β→+∞ 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 (???), the natural regularity is to be found in the H\"older space C0,s, while for Equation (???), 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.