  We show that self-localized ground states can be created in the spin-balanced
gas of fermions with repulsion between the spin components, whose strength
grows from the center to periphery, in combination with the harmonic-oscillator
(HO) trapping potential acting in one or two transverse directions. We also
consider the ground state in the non-interacting Fermi gas under the action of
the spatially growing tightness of the one- or two-dimensional (1D or 2D) HO
confinement. These settings are considered in the framework of the
Thomas-Fermi-von Weizsacker (TF-vW) density functional. It is found that the vW
correction to the simple TF approximation (the gradient term) is nearly
negligible in all situations. The properties of the ground state under the
action of the 2D and 1D HO confinement with the tightness growing in the
transverse directions is investigated too for the Bose-Einstein condensate
(BEC) with the self-repulsive nonlinearity.
