| We analyze the existence and stability of localized solutions in the one-dimensional discrete nonlinear Schrodinger (DNLS) equation with a combination of self-focusing cubic and defocusing quintic onsite nonlinearities. We provide a stability diagram for different families of soliton solutions, created numerically via parameter continuation, that suggests the (co)existence of infinitely many branches of stable localized solutions. Bifurcations which occur with the increase of the coupling constant are also studied in a numerical form through inspection of homoclinic connections, and additionally a variational approximation is developed for accurate prediction of the principal saddle-node bifurcation. Salient properties of the model, which distinguish it from the well-known cubic DNLS equation, are the existence of two different families of symmetric solitons, and, especially, stable asymmetric soliton solutions that are found in narrow regions of the parameter space. The asymmetric solutions appear from and disappear back into the symmetric ones via loops of forward and backward pitchfork bifurcations. |