  We address the existence and stability of localized modes in the framework of
the fractional nonlinear Schroedinger equation (FNSE) with the focusing cubic
or focusing-defocusing cubic-quintic nonlinearity and a confining
harmonic-oscillator (HO) potential. Approximate analytical solutions are
obtained in the form of Hermite-Gauss modes. The linear stability analysis and
direct simulations reveal that, under the action of the cubic self-focusing,
the single-peak ground state and dipole mode are stabilized by the HO potential
at values of the Levy index (the fractionality degree) alpha = 1 and alpha < 1,
which lead, respectively, to the critical or supercritical collapse in free
space. In addition to that, the inclusion of the quintic self-defocusing
provides stabilization of higher-order modes, with the number of local peaks up
to seven, at least.
