Sharp radial Sobolev inequalities with inverse square potentials and applications to nonlinear Schrödinger equations

We prove the sharp radial Sobolev inequality with a repulsive inverse square potential. Considered all $\dot{H}^{1} $ functions, the inequality is not attained by non-trivial function. In this paper, we show that the inequality is attained under radial restriction. As its application, we put on global dynamics of solutions to nonlinear Schrödinger equation with the potential, whose energy is less than or equal to that of the optimizer to the inequality.