*Published Paper*

**Inserted:** 7 jan 2002

**Last Updated:** 23 jul 2003

**Journal:** Ann. Scuola Norm. Sup. Pisa Cl. Sci.

**Volume:** 2 (Serie V)

**Number:** 1

**Pages:** 151-179

**Year:** 2003

**Abstract:**

In this paper, we use $\Gamma$-convergence
techniques to
study the following variational problem
$$
S^{F}_{\epsilon}(\Omega) := \sup \left\{ \epsilon^{{}-2^{}\int}_{\Omega} F(u) dx
\ :\ \int_{\Omega} \vert\nabla u\vert^{2} dx \leq \epsilon^{2\} , \
u=0\ {\rm on}\ \partial\Omega\right\}\ ,
$$
where $0\leq F(t)\leq \vert t\vert^{2^*}$, with $2^*={2n \over n-2}$, and
$\Omega$ is a bounded domain of $R^n$, $n\geq 3$.
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
$S^F_\epsilon(\Omega)$.
Finally, a second order expansion in $\Gamma$-convergence permits to identify
the concentration points of the maximizing sequences, also in some
non-homogeneous
case.

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