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

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

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