Abstract

In this paper, we use $\Gamma $-convergence techniques to study the following variational problem $$ S^F_{\varepsilon } := \sup \left\lbrace {\varepsilon }^{-2^*}\int _\Omega F ~dx \ :\ \int _\Omega \vert \nabla u\vert ^2~dx \le {\varepsilon }^2\, \ u=0\ {\rm on}\ \partial \Omega \right\rbrace \,, $$ where $0\le F\le \vert t\vert ^{2^*}$, with $2^*={2n \over n-2}$, and $\Omega $ is a bounded domain of ${{\mathbb{R}}^n}$, $n\ge 3$. We obtain a $\Gamma $-convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem $S^F_{\varepsilon }$. Finally, a second order expansion in $\Gamma $-convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case