  Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinear
terms whose strengths are of opposite signs (the coefficients in front of the
nonlinearities), exist in various physical media (in particular, in optical and
matter-wave media). A benign competition between self-focusing cubic and
self-defocusing quintic nonlinear nonlinearities (known as cubic-quintic model)
plays an important role in creating and stabilizing the self-trapping of
D-dimensional localized structures, in the contexts of standard nonlinear
Schr\"{o}dinger equation. We incorporate an external periodic potential (linear
lattice) into this model and extend it to the space-fractional scenario that
begins to surface in very recent years---the nonlinear fractional
Schr\"{o}dinger equation (NLFSE), therefore obtaining the cubic-quintic or the
purely quintic NLFSE, and investigate the propagation and stability properties
of self-trapped modes therein. Two types of one-dimensional localized gap modes
are found, including the fundamental and dipole-mode gap solitons. Employing
the techniques based on the linear-stability analysis and direct numerical
simulations, we get the stability regions of all the localized modes; and
particularly, the anti-Vakhitov-Kolokolov criterion applies for the stable
portions of soliton families generated in the frameworks of quintic-only
nonlinearity and competing cubic-quintic nonlinear terms.
