Orbital Stability of Standing Waves for the Sobolev Critical Schrödinger Equation with Inverse-Power Potential

In this paper, we study the Cauchy problem for the nonlinear Schrodinger equation with focusing inverse-power potential and the Sobolev critical nonlinearity. By considering the corresponding local minimization problem, we show that the existence of ground state solutions. Then, we prove that the solution of this equation exists globally when the initial data phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} sufficiently close to the ground states. Based on these results, we show that the set of ground states is orbitally stable.