J. Nonlinear Funct. Anal. 2019 (2019), Article ID 28 https://doi.org/10.23952/jnfa.2019.28 EXISTENCE RESULTS FOR A BOUNDARY VALUE PROBLEM INVOLVING A FOURTH-ORDER ELASTIC BEAM EQUATION SHAPOUR HEIDARKHANI ∗ , FARIBA GHAREHGAZLOUEI Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran Abstract. Fourth-order two-point boundary value problems are useful for material mechanics since they usually characterize the deflection of an elastic beam. In this paper, we establish the multiplicity results for a fourth-order differential equation involving Lipschitz non-linearity which models beams on elastic foundations. The approach is based on variational methods and critical point theory. Keywords. Fourth-order boundary value problem; Three solutions; Critical point; Variational methods. 2010 Mathematics Subject Classification. 34B15, 58E05, 49J40. 1. I NTRODUCTION Fourth-order boundary value problems which describe the deformations of an elastic beam in an equi- librium state whose both ends are simply supported have been extensively studied in the literature. In this paper, we study the fourth-order boundary value problem      u (iv) (x)= λ f (x, u(x)) + h(u(x)), in [0, 1], u(0)= u ′ (0)= u ′′ (1)= 0, u ′′′ (1)= μ g(u(1)), (1.1) where λ > 0, μ ≥ 0 are two parameters. f : [0, 1] × R → R is an L 1 -Carath´ eodory function, g : R → R is a nonnegative continuous function and h : R → R is a Lipschitz continuous function with the Lipschitz constant L > 0, i.e., |h(ξ 1 ) − h(ξ 2 )|≤ L|ξ 1 − ξ 2 |, ∀ξ 1 , ξ 2 ∈ R such that h(0)= 0. Problem (1.1) has the following physical description: a thin flexible elastic beam of length 1 when, along its length, f and h are added to cause deformation. Precisely, condition u(0)= u ′ (0)= 0 means ∗ Corresponding author. E-mail addresses: sh.heidarkhani@razi.ac.ir (S. Heidarkhani), f.gharehgazloo@yahoo.com (F. Gharehgazlouei). Received March 13, 2019; Accepted June 27, 2019. c 2019 Journal of Nonlinear Functional Analysis 1