Concentration of low energy extremals: Identification of concentration points

Abstract.

We study the variational problem

$S^F_\varepsilon(\Omega) =\frac{1}{\varepsilon^{2^*}}\sup\left{\int_{\Omega}F(u):u\inD^{1,2}(\Omega), ||{\nabla u}_2 \leq\varepsilon\right},$

where $\Omega\subset{\bf R}^n, n\geq 3$, is a bounded domain, $2^*=\frac{2n}{n-2}$, F satisfies $0\leq F|t|\leq \alpha |t|^{2^*}$ and is upper semicontinuous. We show that to second order in $\varepsilon$ the value $S_\varepsilon^F(\Omega)$ 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 $\Rn$. The asymptotic expansion becomes

$S^F_\varepsilon(\Omega)= \varepsilon^{2^*} S^F \left(1 - \frac{n}{n-2}w_{\infty}^2\min_{\overline\Omega}\tau_{\Omega}\varepsilon^2+o(\varepsilon^2)\right).$

Using this we deduce that a subsequence of (almost) maximizers of $S^F_\varepsilon(\Omega)$ must concentrate at a harmonic center of $\Omega$: i.e., $\frac{| \nabla u_\varepsilon |^2}{\varepsilon^2} \stackrel{*} \rightharpoonup \delta_{x_0}$, where $x_0\in\overline\Omega$ is a minimum point of $\tau_{\Omega}$.