  We consider circular Wilson loops in a defect version of $\mathcal{N}=4$
super-Yang-Mills theory which is dual to the D3-D5 brane system with $k$ units
of flux. When the loops are parallel to the defect, we can construct both BPS
and non-BPS operators, depending on the orientation of the scalar couplings in
the R-symmetry directions. At strong 't Hooft coupling we observe, in the non
supersymmetric case, a Gross-Ooguri-like phase transition in the dual
gravitational theory: the familiar disk solution dominates, as expected, when
the operator is far from the defect while a cylindrical string worldsheet,
connecting the boundary loop with the probe D5-brane, is favourite below a
certain distance (or equivalently for large radii of the circles). In the BPS
case, instead, the cylindrical solution does not exist for any choice of the
physical parameters, suggesting that the exchange of light supergravity modes
always saturate the expectation value at strong coupling. We study the
double-scaling limit for large $k$ and large 't Hooft coupling, finding full
consistency in the non-BPS case between the string solution and the one-loop
perturbative result. Finally we discuss, in the BPS case, the failure of the
double-scaling limit and the OPE expansion of the Wilson loop, finding
consistency with the known results for the one-point functions of scalar
composite operators.
