Multiplicity of positive solutions to semi-linear elliptic problems on metric graphs (Q2062559)

Let \(\varepsilon\) be a positive suffciently small parameter and \(p>1\). The author considers positive solutions of the semi-linear elliptic equation \(-\varepsilon^2 u''+ u = u^p\) on a compact metric graph \(G\). Neumann or Dirichlet boundary conditions are given on the ends of \(G\). The author establishes the existence of the positive solution concentrated on an edge (single-peak solution).