Asymptotic shape and location of small cores in elliptic free-boundary problems
The paper deals with the boundary value problem \[ \mu^2\Delta u+g(u)=0 \quad \text{in }\Omega,\qquad u=0 \text{ on }\partial\Omega \] with \(g(\tau) =(\tau-1)^p_+\), \(\Omega\subset\mathbb{R}^n\), \(n\geq 3\). The authors investigate the asymptotic properties of the mountain pass solution as \(\mu\to 0\) and determine the asymptotic shape and location of the cores \(A_\mu\). It is proved that the cores \(A_\mu\) of the mountain pass solutions concentrate at a harmonic center of \(\Omega\) and that they are asymptotically spherical.
scientific article; zbMATH DE number 1229589