  In this paper, a nonlinear Schr\"odinger equation with an attractive
(focusing) delta potential and a repulsive (defocusing) double power
nonlinearity in one spatial dimension is considered. It is shown, via explicit
construction, that both standing wave and equilibrium solutions do exist for
certain parameter regimes. In addition, it is proved that both types of wave
solutions are orbitally stable under the flow of the equation by minimizing the
charge/energy functional.
