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MiS Preprint

7/2001

Concentration of low energy extremals: Identification of concentration points

Martin Flucher, Adriana Garroni and Stefan Müller

Abstract

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