Multi-layer local minimum solutions of the bistable equation in noncylindrical domains (Q1377540)

The semilinear elliptic problem \[ -\Delta u + f(u) = 0\quad \text{in}\quad \Omega,\;\frac{\partial u}{\partial n} \quad\text{on} \partial \Omega \] is studied in an unbounded tube-shaped domain \(\Omega\) in \(\mathbb{R}^d\). Multi-layer local minimum solutions are constructed by minimizing the corresponding functional \(E\) near some approximate solutions. Here \[ E(u) = \int_{\Omega} \big[\frac12 |\nabla u|^2 + W(u)\big]dx \] with \(W(u) = \int_{-1}^u f(w)dw\). The basic assumption is that \(W\) is a \(C^2\) function which has exactly two global minima at \(-1\) and \(1\), \(W(-1)=W(1)=0, \;W''(-1)>0,\;W''(1)>0\). A certain characterization of Palais-Smale sequences and the fact that the functional takes higher values away from the approximate solutions play an essential role.