MiS Preprint Repository

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