Least energy solutions to semi-linear elliptic problems on metric graphs

Abstract

We consider positive solutions of semi-linear elliptic equations −ε2Δu+u=up on compact metric graphs, where 1<p<∞. For each ε>0, there exists a least energy positive solution uε. We focus on the asymptotic behavior of uε and show that uε has exactly one local maximum point xε and concentrates like a peak for sufficiently small ε. Moreover, we prove that the location of xε is determined by the length of edges of graphs. These results are shown for the more general super-linear term f(u) instead of up.