MiS Preprint Repository

We study the variational problem $$S_{ϵ}^F(\mathrm{\Omega })=\frac{1}{{ϵ}^{2^{\ast }}}sup\{{\int }_{\mathrm{\Omega }}F(u):u\in D^{1,2}(\mathrm{\Omega }),||\mathrm{\nabla }u|{|}_2\le ϵ\}$$ where $\mathrm{\Omega }\subset R^n$, $n\ge 3$, is a bounded domain, $2^{\ast }=\frac{2n}{n-2}$ and F satisfies $0\le F(t)\le a{t}^{2^{\ast }}$ and is upper semicontinuous. We show that to second order in $ϵ$ the value $S_{ϵ}^F(\mathrm{\Omega })$ only depends on two ingredients. The geometry of $\mathrm{\Omega }$ enters through the Robin function ${\tau }_{\mathrm{\Omega }}$ (the regular part of the Green's function) and F enters through a quantity $w_{\mathrm{\infty }}$ which is computed from (radial) maximizers of the problem in $R^n$. The asymptotic expansion becomes $$\begin{gathered} S_{ϵ}^F(\mathrm{\Omega })={ϵ}^{2^{\ast }}{S}^F(1-\frac{n}{n-2}{\omega }_{\mathrm{\infty }}^2\begin{array}{}min\\ \bar{\mathrm{\Omega }}\end{array}{\tau }_{\omega }{ϵ}^2=o({ϵ}^2) \\ \bar{\mathrm{\Omega }}) \end{gathered}$$ Using this we deduce that a subsequence of (almost) maximizers of $S_{ϵ}^F(\mathrm{\Omega })$ must concentrate at a harmonic center of $\mathrm{\Omega }$, i.e., $\frac{|\mathrm{\nabla }{u}_{ϵ}{|}^2}{{ϵ}^2}\rightharpoonup {\delta }_{x_0}$, where $x_0\in \bar{\mathrm{\Omega }}$ is a minimum point of ${\tau }_{\mathrm{\Omega }}$.