Elastic beam qquations with variable coefficients: multiple solutions under mixed nonlinearities

Abstract

This paper investigates the existence of multiple solutions for a fourth-order differential equation modelling an elastic beam,
 where the coefficients are variable, and the nonlinearities exhibit both concave and convex characteristics. Our approach is based on variational methods and critical point theorems, particularly those formulated by Ricceri, which provide a powerful framework for proving the existence of solutions in reflexive Banach spaces. By leveraging these mathematical tools, we establish that the considered problem admits at least three distinct weak solutions under specific conditions. To validate our theoretical findings,
 we present an illustrative example demonstrating how our results can be applied in practice.