Nonlinear Schr\"{o}dinger equation for the twisted Laplacian in the critical case

Abstract

We prove well-posedness of solution to the nonlinear Schr\"{o}dinger equation associated to the twisted Laplacian on $\C^n$ for a general class of nonlinearities including power type with subcritical case $0\leq \alpha<\frac{2}{n-1}$, see Ratnakumar, Sohani (J. Funct. Anal. 2013). In this paper, we consider critical case $\alpha=\frac{2}{n-1}$ with $n\geq 2$. Our approach is based on truncation of the given nonlinearity $G$, which is used by Cazenave Weissler (1989). We obtain solution for the truncated problem. We obtain solution to the original problem by passing to the limit.