Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials

Abstract

The wellposedness of nonlinear Schrödinger equations (NLS) with inverse-square potentials is discussed in this article. The usual (NLS) is regarded as the potential-free case. The wellposedness of the usual (NLS) is well-known for a long time. In fact, several methods have been developed up to now. Among others, the Strichartz estimates seem to be essential in addition to the restriction on the nonlinear term caused by the Gagliardo–Nirengerg inequality. However, a parallel argument is not available when we apply such estimates to (NLS) with inverse-square potentials. Thus, we shall give only partial answer to the question in this article.

Keywords:

• nonlinear Schrödinger equation
• inverse-square potentials
• Strichartz estimates
• Hardy's inequality

AMS Subject Classifications::

• Primary 35Q55
• 35Q40
• Secondary 81Q15

Acknowledgements

The authors would like to thank the referees for reading their manuscript carefully. Especially, a lot of comments are helpful to make it as readable as possible. N. Okazawa was partially supported by Grant-in-Aid for Scientific Research (C), No.20540190. T. Yokota was partially supported by Grant-in-Aid for Young Scientists Research (B), No.20740079.