  We consider the divergent fractional Laplace operator presented in
[Dipierro-Savin-Valdinoci, Rev. Mat. Iberoam.] and we prove three types of
results.
  Firstly, we show that any given function can be locally shadowed by a
solution of a divergent fractional Laplace equation which is also prescribed in
a neighborhood of infinity.
  Secondly, we take into account the Dirichlet problem for the divergent
fractional Laplace equation, proving the existence of a solution and
characterizing its multiplicity.
  Finally, we take into account the case of nonlinear equations, obtaining a
new approximation results.
  These results maintain their interest also in the case of functions for which
the fractional Laplacian can be defined in the usual sense.
