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 Schrödinger equation $$-△u+V(x)u=K(x)|u{|}^{p-2}uin{\mathbb{R}}^N$$ where $V(x)\sim a|x{|}^{-b}$ and $K(x)\sim \mu |x{|}^{-s}$ as $|x|\to \infty$ with $0<a,\mu <+\infty$, $b<2,$ $b\ne 0$, $0<\frac{s}{b}<1$ and $p=2(N-2s/b)/(N-2).$

25pages