Positive solutions of a Schrödinger equation with critical nonlinearity

Abstract

We study the nonlinear Schröodinger equation $-\Delta u+\lambda a(x)u=\mu u+u^{2^{\ast }-1},{ \ }u\in \mathbb{R}^{N},$ with critical exponent 2^*= 2 N/( N-2), N ≥ 4, where a ≥ 0, has a potential well. Using variational methods we establish existence and multiplicity of positive solutions which localize near the potential well for μ small and λ large.