  We study the one-dimensional nonlinear Schr\"odinger equation with the
cubic-quintic combination of attractive and repulsive nonlinearities, and a
trapping potential represented by a delta-function. We determine all bound
states with a positive soliton profile through explicit formulas and, using
bifurcation theory, we describe their behavior with respect to the propagation
constant. This information is used to prove their stability by means of the
rigorous theory of orbital stability of Hamiltonian systems. The presence of
the trapping potential gives rise to a regime where two stable bound states
coexist, with different powers and same propagation constant.

Analysis of PDEs;Pattern Formation and Solitons;Exactly Solvable and Integrable Systems;Optics