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

• Published in: AIMS mathematics Vol. 7; no. 1; pp. 579 - 605
• Main Authors: Xia, Minggang, Zhang, Xingyong, Kang, Danyang, Liu, Cuiling
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2022
• Subjects: concentration; elastic beam equation; local nonlinearity; mountain pass theorem; weak solution
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2022037

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 $.