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

We consider positive solutions of semi-linear elliptic equations -epsilon(2)Delta u + u = u(p) on compact metric graphs, where 1 < p < infinity. For each epsilon > 0, there exists a least energy positive solution u(epsilon). We focus on the asymptotic behavior of u(epsilon) and show that u(epsilon) has exactly one local maximum point x(epsilon) and concentrates like a peak for sufficiently small epsilon. Moreover, we prove that the location of x(epsilon) is determined by the length of edges of graphs. These results are shown for the more general super-linear term f(u) instead of u(p). (C) 2020 Elsevier Inc. All rights reserved.