Singularly perturbed nonlinear Dirichlet problems involving critical growth

Abstract

We consider the following singularly perturbed nonlinear elliptic problem: -epsilon(2)Delta u + u = f (u), u > 0 on Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain in R-N (N = 3) with a boundary partial derivative Omega is an element of C-2 and the nonlinearity f is of critical growth. In this paper, we construct a solution u(epsilon) of the above problem which exhibits one spike near a maximum point of the distance function from the boundary partial derivative Omega under a critical growth condition on f. Our result complements the study made in [9] in the sense that, in that paper, only the subcritical growth was considered.