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>

Variational approach for an elastic beam equation with local nonlinearities

“…There are many fourth order differential equations like problem (1.1) in material mechanics, in modelling bending equilibria of elastic beams, etc. Various interesting results on fourth-order boundary value problems have been obtained, for instance, in [1,2,9,11,13,14,[18][19][20][21][22]. Ma [14], using variational methods, has obtained the existence of at least two positive solutions for the problem…”

“…and h(0) = 0, and g : R → R is a non-positive continuous function. Xia et al [20], by using the mountain pass lemma and the truncation function, have studied the existence and concentration phenomenon of nontrivial weak solutions for problem (1.1) when the nonlinear term satisfies some growth conditions only near the origin. In particular, they have obtained a concrete lower bound of the parameter λ, analyzed the relationship between λ and µ, established the concentration phenomenon of solutions when µ tends to zero, and presented a specific lower bound of λ which is independent of µ.…”

Development of a new iterative method and its convergence analysis for nonlinear fourth‐order boundary value problems arising in beam analysis

“…The multiplicity of positive solutions and their existence for this problem have been discussed by many researchers (see previous works [8][9][10][11][12][13][14][15] and references therein). Some approximation methods have been developed to solve the considered beam equation, such as reproducing kernel Hilbert space methods 7,16 quadratic interior penalty method, 17 the iterative method by Ma and Da Silva 6 and the iterative monotone method by Alves et al 18 It is interesting to note that existing methodologies 6,16,18,19 require a large number of iterations to obtain a relatively good approximate solution to the problem.…”