Subcritical approximation of the Sobolev quotient and a related concentration result

Let $\Omega$ be a general, possibly non-smooth, bounded domain of $\mathbb{R}^N$, $N\geq 3$. Let $\displaystyle 2^{*}\!\!=\!{2N}\,\!/{(N-2)}$ be the critical Sobolev exponent. We study the following variational problem $$ S^{*}_{\varepsilon}=\sup\left \{ \int_{\Omega}|u|^{2^{*}\!-\varepsilon}dx: \int_{\Omega}|\nabla u|^{2}dx\leq 1, u=0 \ \text{on} \ \partial\Omega \right \}, $$ investigating its asymptotic behavior as $\varepsilon$ goes to zero, by means of $\gamp$-convergence techniques. We also show that sequences of maximizers $u_\varepsilon$ concentrate energy at one point $x_0\in \overline{\Omega}$.