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

Abstract

<abstract><p>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 $.</p></abstract>