  We predict the existence of stable fundamental and vortical bright solitons
in dipolar Bose-Einstein condensates (BECs) with repulsive dipole-dipole
interactions (DDI). The condensate is trapped in the 2D plane with the help of
the repulsive contact interactions whose local strength grows $\sim r^{4}$ from
the center to periphery, while dipoles are oriented perpendicular to the
self-trapping plane. The confinement in the perpendicular direction is provided
by the usual harmonic-oscillator potential. The objective is to extend the
recently induced concept of the self-trapping of bright solitons and solitary
vortices in the pseudopotential, which is induced by the repulsive local
nonlinearity with the strength growing from the center to periphery, to the
case when the trapping mechanism competes with the long-range repulsive DDI.
Another objective is to extend the analysis for elliptic vortices and solitons
in an anisotropic nonlinear pseudopotential. Using the variational
approximation (VA) and numerical simulations, we construct families of
self-trapped modes with vorticities $\ell =0$ (fundamental solitons), $\ell
=1$, and $\ell =2$. The fundamental solitons and vortices with $\ell =1$ exist
up to respective critical values of the eccentricity of the anisotropic
pseudopotential, being stable in the entire existence regions. The vortices
with $\ell =2$ are stable solely in the isotropic model.
