  We study analytically and numerically the existence and orbital stability of
the peak-standing-wave solutions for the cubic-quintic nonlinear Schrodinger
equation with a point interaction determined by the delta of Dirac. We study
the cases of attractive-attractive and attractive-repulsive nonlinearities and
we recover some results in the literature. Via a perturbation method and
continuation argument we determine the Morse index of some specific
self-adjoint operators that arise in the stability study. Orbital instability
implications from a spectral instability result are established. In the case of
an attractive-attractive case and an focusing interaction we give an approach
based in the extension theory of symmetric operators for determining the Morse
index.
