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

We consider positive solutions of semi-linear elliptic equations\begin{document}$ - \epsilon^2 u'' +u = u^p $\end{document}on compact metric graphs, where \begin{document}$ p \in (1,\infty) $\end{document} is a given constant and \begin{document}$ \epsilon $\end{document} is a positive parameter. We focus on the multiplicity of positive solutions for sufficiently small \begin{document}$ \epsilon $\end{document} . For each edge of the graph, we construct a positive solution which concentrates some point on the edge if \begin{document}$ \epsilon $\end{document} is sufficiently small. Moreover, we give the existence result of solutions which concentrate inner vertices of the graph.