Existence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity

Minggang Xia; Xingyong Zhang; Danyang Kang; Cuiling Liu; Minggang Xia; Xingyong Zhang; Danyang Kang; Cuiling Liu

doi:10.3934/math.2022037

AIMS Mathematics

2022, Volume 7, Issue 1: 579-605. doi: 10.3934/math.2022037

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Research article

Existence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity

Department of Systems Science and Applied Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, China

Received: 20 June 2021 Accepted: 13 September 2021 Published: 14 October 2021
MSC : 34B15, 49J35

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 $.
- weak solution,
- elastic beam equation,
- local nonlinearity,
- mountain pass theorem,
- concentration
Citation: Minggang Xia, Xingyong Zhang, Danyang Kang, Cuiling Liu. Existence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity[J]. AIMS Mathematics, 2022, 7(1): 579-605. doi: 10.3934/math.2022037

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Abstract

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 $.

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