Open AccessExistence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity
Author(s) -
Minggang Xia,
Xingyong Zhang,
Danyang Kang,
Cuiling Liu
Publication year - 2021
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2022037
Subject(s) - lambda , truncation (statistics) , lemma (botany) , nonlinear system , mathematics , mathematical analysis , function (biology) , beam (structure) , upper and lower bounds , term (time) , class (philosophy) , physics , mathematical physics , quantum mechanics , optics , statistics , ecology , poaceae , evolutionary biology , biology , artificial intelligence , computer science
In this paper, by using the mountain pass lemma and the skill of truncation function, we investigate the existence and concentration phenomenon of nontrivial weak solutions for a class of elastic beam differential equation with two parameters $ \lambda $ and $ \mu $ when the nonlinear term satisfies some growth conditions only near the origin. In particular, we obtain a concrete lower bound of the parameter $ \lambda $, and analyze the relationship between $ \lambda $ and $ \mu $. In the end, we investigate the concentration phenomenon of solutions when $ \mu\to 0 $, and obtain a specific lower bound of the parameter $ \lambda $ which is independent of $ \mu $.