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

Abstract

<p style='text-indent:20px;'>We consider positive solutions of semi-linear elliptic equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ - \epsilon^2 u'' +u = u^p $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>on compact metric graphs, where <inline-formula><tex-math id="M1">\begin{document}$ p \in (1,\infty) $\end{document}</tex-math></inline-formula> is a given constant and <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is a positive parameter. We focus on the multiplicity of positive solutions for sufficiently small <inline-formula><tex-math id="M3">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula>. For each edge of the graph, we construct a positive solution which concentrates some point on the edge if <inline-formula><tex-math id="M4">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is sufficiently small. Moreover, we give the existence result of solutions which concentrate inner vertices of the graph.</p>