We present some results of existence for the following problem: − ∆u = a(x)g(u)+u | u | 2 − 2 , x ∈ R N (N ≥ 3), u ∈ D 1,2 ( R N ), where the function a is a sign-changing function with a singularity at the origin and g has growth up to the Sobolev critical exponent 2 ∗ = 2N/(N − 2).