Regularity estimates for nonlocal Schr\"odinger equations

We prove H"older regularity estimates up to the boundary for weak solutions $u$ to nonlocal Schr"odinger equations subject to exterior Dirichlet conditions in an open set $\mathrm{\Omega }\subset {\mathbb{R}}^N$. The class of nonlocal operators considered here are defined, via Dirichlet forms, by kernels $K(x,y)$ bounded from above and below by $\mathrm{\mid }x-y{\mathrm{\mid }}^{N+2s}$, with $s\in (0,1)$. The entries in the equations are in some Morrey spaces and the underline domain $\mathrm{\Omega }$ satisfies some mild regularity assumptions. In the particular case of the fractional Laplacian, our results are new. When $K$ defines a nonlocal operator with sufficiently regular coefficients, we obtain H"older estimates, up to the boundary of $\mathrm{\Omega }$, for $u$ and the ratio $u\mathrm{/}{d}^s$, with $d(x)=\text{dist}(x,{\mathbb{R}}^N\setminus \mathrm{\Omega })$. If the kernel $K$ defines a nonlocal operator with H"older continuous coefficients and the entries are H"older continuous, we obtain interior $C^{2s+\beta }$ regularity estimates of the weak solutions $u$. Our argument is based on blow-up analysis and compact Sobolev embedding.