Stabilizing vortex solitons with high values of the topological charge
S is a challenging issue in optics, studies of Bose-Einstein condensates (BECs), and other fields. To develop an approach to the solution of this problem, we consider a two-dimensional dipolar BEC under the action of an axisymmetric radially periodic lattice potential, $V(r)\ensuremath{\sim}cos(2r+\ensuremath{\delta})$, with dipole moments polarized perpendicular to the system's plane, which gives rise to isotropic repulsive dipole-dipole interactions. Two radial lattices are considered, with $\ensuremath{\delta}=0$ and $\ensuremath{\pi}$, i.e., a potential maximum or minimum at
r=0, respectively. Families of vortex gap soliton (GSs) with
S=1 and $S\ensuremath{\ge}2$, the latter ones often being unstable in other settings, are completely stable in the present system (at least up to
S=11), being trapped in different annular troughs of the radial potential. The vortex solitons with different
S may stably coexist in sufficiently far separated troughs. Fundamental GSs, with
S=0, are found too. In the case of $\ensuremath{\delta}=0$, the fundamental solitons are ring-shaped modes, with a local minimum at
r=0. At $\ensuremath{\delta}=\ensuremath{\pi}$, they place a density peak at the center.
We study the ground states of rotating atomic Bose-Einstein condensates with dipolar interactions. We present the results of numerical studies on a periodic geometry which show vortex lattice ground states of various symmetries: triangular and square vortex lattices, ``stripe crystal,'' and ``bubble crystal.'' We present the phase diagram (for systems with a large number of vortices) as a function of the ratio of dipolar to contact interactions and of the chemical potential. We discuss the experimental requirements for observing transitions between vortex lattice ground states via dipolar interactions. We finally investigate the stability of mean-field supersolid phases of a quasi-two-dimensional nonrotating Bose gas with dipolar interactions.