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We consider the following singularly perturbed nonlinear elliptic problem: where Ω is a bounded domain in $${\mathbb{R}^N (N \ge 3)}$$ with a boundary $${\partial \Omega \in C^2}$$ and the nonlinearity f is of critical growth. In this paper, we construct a solution $${u_\varepsilon}$$ of the above problem which exhibits one spike near a maximum point of the distance function from the boundary ∂Ω under a critical growth condition on f. Our result complements the study made in [] in the sense that, in that paper, only the subcritical growth was considered. [ABSTRACT FROM AUTHOR]