We are concerned with H?lder regularity estimates for weak solutions u to nonlocal Schr?dinger equations subject to exterior Dirichlet conditions in an open set Ω ? R~N . The class of nonlocal operators considered here are defined, via Dirichlet forms, by symmetric kernels K(x, y) bounded from above and below by x - y~(-N-2s), with s ∈ (0, 1). The entries in the equations are in some Morrey spaces and the domain Ω 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?lder estimates, up to the boundary of Ω, for u and the ratio u/d~s, with d(x) = dist(x, R~N Ω). If the kernel K defines a nonlocal operator with H?lder continuous coefficients and the entries are H?lder continuous, we obtain interior C~(2s+β) regularity estimates of the weak solutions u. Our argument is based on blow-up analysis and compact Sobolev embedding.
展开▼
