  The thesis studies linear and semilinear Dirichlet problems driven by
different fractional Laplacians. The boundary data can be smooth functions or
also Radon measures. The goal is to classify the solutions which have a
singularity on the boundary of the prescribed domain. We first remark the
existence of a large class of harmonic functions with a boundary blow-up and we
characterize them in terms of a new notion of degenerate boundary trace. Via
some integration by parts formula, we then provide a weak theory of
Stampacchia's sort to extend the linear theory to a setting including these
functions: we study the classical questions of existence, uniqueness,
continuous dependence on the data, regularity and asymptotic behaviour at the
boundary. Afterwards we develop the theory of semilinear problems, by adapting
and generalizing some sub- and supersolution methods. This allows us to build
the fractional counterpart of large solutions in the elliptic PDE theory of
nonlinear equations, giving sufficient conditions for the existence. The thesis
is concluded with the definition and the study of a notion of nonlocal
directional curvatures.
