Multi-peak solutions for the fractional Schrödinger equation with Dirichlet datum

Let $s\in (0,1)$, $\epsilon >0$ and let $\mathrm{\Omega }$ be a bounded smooth domain. Given the problem $${\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}u+V(x)u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}uin\mathrm{\Omega },$$ with Dirichlet boundary conditions and $1<p<(n+2s)\mathrm{/}(n-2s)$, we analyze the existence of positive multi-peak solutions concentrating, as $\epsilon \to 0$, to one or several points of $\mathrm{\Omega }$. Under suitable conditions on $V$, we construct positive solutions concentrating at any prescribed set of its non degenerate critical points. Furthermore, we prove existence and non existence of clustering phenomena around local maxima and minima of $V$, respectively. The proofs rely on a Lyapunov-Schmidt reduction where three effects need to be controlled: the potential, the boundary and the interaction among peaks. The slow decay of the associated {\it ground-state} demands very precise asymptotic expansions.