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

Masataka Shibata

doi:10.3934/cpaa.2021147

Article Contents

2021, Volume 20, Issue 12: 4107-4126. Doi: 10.3934/cpaa.2021147

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

Masataka Shibata

Department of Mathematics, Meijo University, 1-501 Shiogamaguchi, Tempaku-ku, Nagoya 468-8502, Japan

Received: January 31, 2021

Revised: June 30, 2021

Early access: September 2021

Published: December 2021

The author was supported by JSPS KAKENHI Grant Numbers 18K03356, 18K03362

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Abstract

We consider positive solutions of semi-linear elliptic equations

$ - \epsilon^2 u'' +u = u^p $

on compact metric graphs, where $ p \in (1,\infty) $ is a given constant and $ \epsilon $ is a positive parameter. We focus on the multiplicity of positive solutions for sufficiently small $ \epsilon $. For each edge of the graph, we construct a positive solution which concentrates some point on the edge if $ \epsilon $ is sufficiently small. Moreover, we give the existence result of solutions which concentrate inner vertices of the graph.

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Mathematics Subject Classification: Primary: 34B45; Secondary: 35J20.

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