Concentration phenomena in elliptic problems with critical and supercritical growth

Abstract

This paper deals with the existence of positive solutions of problem $-\Delta u=u^{N+2\over N-2}+{\varepsilon} w(x)u^q $, with Dirichlet zero boundary condition on $\Omega$ (a bounded domain in $\mathbb R^N$), when $q\geq 1$ and $q\neq{N+2\over N-2}$. We study the existence of solutions which blow-up and concentrate at a single point of $\Omega$ whose location depends on the Robin function and on the coefficient $w$ of the perturbed term.