We consider a semilinear Schrödinger equation, driven by the power degenerate second order differential operator \(\nabla \cdot (|x|^{2a} \nabla ), a\in (0,1)\) . We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli-Kohn-Nirenberg’s inequality. Depending on the parameter a and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away from the origin) and almost exponential decay. Then, and as a consequence of our variational constrcution, we completely characterize the spectral stability of the said solitons. We pose some natural conjectures, which are still open - such as the radiality of the ground states, the non-degeneracy and most importantly uniqueness.
