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

Abstract

We consider the following singularly perturbed elliptic problem \[ - {\varepsilon ^2}Δu + u = f(u){\text{ in }}Ω,{\text{ }}u > 0{\text{ in }}Ω,{\text{ }}u = 0{\text{ on }}\partial Ω, \] where $\Omega$ is a domain in ${\mathbb{R}}^N(N\ge 3)$, not necessarily bounded, with boundary $\partial \Omega \in C^2$ 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 $\partial \Omega$.

42 pages