This paper is concerned with nonlinear elliptic problems of the type (P_ε): -Δu = u^(N+2)/(N-2) + εu, u>0 in Ω; u=0 on ∂Ω, where Ω is a smooth and bounded domain in R^N, N≥4, and ε>0. We show that if the u_ε are solutions to (P_ε) which concentrate at a point x_0 as ε goes to 0, x_0 cannot be on the boundary of Ω and is a critical point of the regular part of the Green's function. Conversely, we show that for N≥5 and any non-degenerate critical point x_0 of the regular part of the Green's function, there exist solutions of (P_ε) concentrating at x_0 as ε goes to 0.