Existence of positive solutions for a critical nonlinear Schrodinger equation with vanishing or coercive potentials

Abstract: In this paper we investigate the existence of the positive solutions for the following nonlinear Schr\"odinger equation $$ -\triangle u+V(x)u=K(x)|u|^{p-2}u\ {in}\ \mathbb{R}^N $$ where $V(x)\sim a|x|^{-b}$ and $K(x)\sim \mu|x|^{-s}$ as $|x|\rightarrow\infty$ with $0<a,\mu<+\infty$, $b<2,$ $ b\neq0$, $ 0<\frac{s}{b}<1$ and $p=2(N-2s/b)/(N-2).$