MiS Preprint Repository

We study the variational problem $$S^F_\epsilon (\Omega) =\frac{1}{\epsilon^{2^*}} \sup \left\{ \int_\Omega F(u) :\int_\Omega |\nabla u|^2 \leq \epsilon^2, \ \ u=0 \ \ on\ \ \partial \Omega \right\}$$ in possibly unbounded domains $\Omega \subset R^n$, where $n \geq 3$, $2^* =\frac{2n}{n-2}$ and F satisfies $0\leq F(t) \leq \alpha |t|^{2^*}$ and is upper semicontinuous. Extending earlier results for bounded domains we show that (almost) maximizers of $S^F_\epsilon (\Omega)$ concentrate at a harmonic center, i.e. a minimum point of the Robin function $\tau_\Omega$ (the regular part of the Green function restricted to the diagonal). Moreover we obtain the asymptotic expansion $$S^F_\epsilon (\Omega) =S^F \left( 1-\frac{n}{n-2} w^2_\infty \frac{min}{\Omega} \tau_\omega \epsilon^2 +o(\epsilon^2) \right)$$ where $S^F$ and $w_\infty$ depend only on F but not on $\Omega$ and can be computed from radial maximizers of the corresponding problem in $R^n$. The crucial point is to find a suitable definition of $\tau_\Omega (\infty)$. Interestingly the correct definition may be different from the lower semicontinuous extension of $\tau_\Omega |_{\overline{\Omega }\backslash \{ \infty \}}$ to $\infty$, at least for $n \geq 5$.