What question did this study set out to answer?

To establish the sharp radial Sobolev inequality involving a repulsive inverse square potential and analyze its implications for nonlinear Schrödinger equations.

April 23, 2026Open Access

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

Puntos clave

To establish the sharp radial Sobolev inequality involving a repulsive inverse square potential and analyze its implications for nonlinear Schrödinger equations.
Proving the sharp radial Sobolev inequality for all $dot{H}^{1}$ functions.
Analyzing the inequality's attainment under radial restrictions.
Investigating the global dynamics of solutions to nonlinear Schrödinger equations with specific potential.
The sharp radial Sobolev inequality is attained under radial restrictions.
Non-trivial functions do not achieve the inequality generally, highlighting its uniqueness.
The solutions demonstrate specific global dynamics for energies at or below the optimizer's energy.

Resumen

We prove the sharp radial Sobolev inequality with a repulsive inverse square potential. Considered all Ḣ^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.

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