Solutions of Schrödinger equations with inverse square potential and critical nonlinearity

In this paper, we are concerned with the following nonlinear Schrödinger equations with inverse square potential and critical Sobolev exponentequation(P)−Δu−μu|x|2+a(x)u=|u|2⁎−2u+f(x,u),u∈H1(RN), where 2⁎=2N/(N−2)2⁎=2N/(N−2) is the critical Sobolev exponent, 0⩽μ<μ¯:=(N−2)24, a(x)∈C(RN)a(x)∈C(RN). We first give a representation to the Palais–Smale sequence related to (P) and then obtain an existence result of positive solutions of (P). Our assumptions on a(x)a(x) and f(x,u)f(x,u) are weaker than the known cases even if μ=0μ=0.