Existence of stable standing waves for the nonlinear Schrödinger equation with the Hardy potential

<p style='text-indent:20px;'>In this paper, we consider the existence of stable standing waves for the nonlinear Schrödinger equation with combined power nonlinearities and the Hardy potential. In the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-critical case, we show that the set of energy minimizers is orbitally stable by using concentration compactness principle. In the <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-supercritical case, we show that all energy minimizers correspond to local minima of the associated energy functional and we prove that the set of energy minimizers is orbitally stable.</p>