Existence of Multiple Sign-Changing Solutions for a Fourth-Order Elastic Beam Equation

The aim of the present paper is to investigate the existence of multiple positive, multiple negative, and in particular, multiple sign-changing solutions depending on λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} for the following fourth-order problem: uiv+Au′′=λf(t,u)in(0,1)u(0)=u(1)=0u′′(0)=u′′(1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllll} u^{iv}+A u^{''}=\lambda f(t, u) \quad \text {in}\, (0, 1) \\ u(0)=u(1)=0 \\ u^{''}(0)=u^{''}(1)=0, \end{array} \right. \end{aligned}$$\end{document}where f:[0,1]×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:[0, 1]\times \mathbb {R}\rightarrow \mathbb {R}$$\end{document} is a function, A is a real constant and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is a positive parameter. The nonlinearity f is required to have an oscillatory behaviour.