Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs

Abstract

<p style='text-indent:20px;'>In this paper, we study the existence of spiky stationary solutions of the Schnakenberg model with heterogeneity on compact metric graphs. These solutions are constructed by using the Liapunov–Schmidt reduction method and taking the same strategy as that in [<xref ref-type="bibr" rid="b14">14</xref>,<xref ref-type="bibr" rid="b11">11</xref>]. First, we give the abstract theorem on the existence of multi-peak solutions for general compact metric graphs under several assumptions for the associated Green's function. In particular, we reveal that how locations of concentration points and amplitudes of spiky solutions are determined by the interaction of the heterogeneity with the geometry of the compact metric graph, represented by Green's function. Second, we apply our abstract theorem to the <inline-formula><tex-math id="M1">\begin{document}$ Y $\end{document}</tex-math></inline-formula>-shaped metric graph and the <inline-formula><tex-math id="M2">\begin{document}$ H $\end{document}</tex-math></inline-formula>-shaped metric graph in non-heterogeneity case. In particular, we show the precise effect of the geometry of those compact graphs to the locations of concentration points for these concrete graphs, respectively.