Multi-peak solutions for singularly perturbed nonlinear Dirichlet problems involving critical growth

We consider the following singularly perturbed elliptic problem $$-{\epsilon }^2\mathrm{\Delta }u+u=f(u)\text{ in }\mathrm{\Omega },u>0\text{ in }\mathrm{\Omega },u=0\text{ on }\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Omega }$ is a domain in ${\mathbb{R}}^N(N\ge 3)$, not necessarily bounded, with boundary $\mathrm{\partial }\mathrm{\Omega }\in C^2$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a family of multi-peak solutions to the equation given above which concentrate around any prescribed finite sets of local maxima of the distance function from the boundary $\mathrm{\partial }\mathrm{\Omega }$.