Solution surfaces for semilinear elliptic equations on rotated domains
The authors consider the problem \(\Delta u+\lambda f(u)=0\) in \(\Omega_\varepsilon\), \(u=0\) on \(\partial\Omega_\varepsilon\), where \(\Omega_\varepsilon\) is a torus-like domain in \(\mathbb{R}^{n+1}\) obtained from \(\Omega\) by a translation \({1\over\varepsilon}\) and a rotation about an axis. For \(\Omega\in \mathbb{R}^2\) (with symmetry and convexity assumptions) or \(\Omega\) a ball in \(\mathbb{R}^2\) they prove the existence of a solution surface in \((\alpha,\varepsilon)\), where \(\alpha=\max u\) and \(\varepsilon\) is sufficiently small, together with some additional interesting properties.
scientific article; zbMATH DE number 1493593
scientific article