  We consider optimization problems in the fractional order Sobolev spaces
$H^s(\Omega)$, $s\in (0,1)$, with sparsity promoting objective functionals
containing $L^p$-pseudonorms, $p\in (0,1)$. Existence of solutions is proven.
By means of a smoothing scheme, we obtain first-order optimality conditions. An
algorithm based on this smoothing scheme is developed. Weak limit points of
iterates are shown to satisfy a stationarity system that is slightly weaker
than that given by the necessary condition.
