New conditions and multiple sign changing solutions for a fourth-order elastic beam equations

Abstract

The aim of the present paper is to investigate the existence of multiple positive, multiple negative, and in particular, of multiple sign changing solutions depending on $\lambda$ for the following fourth-order problem
$$\left\{\begin{array}{lllll}u^{iv}+B u=\lambda f(t, u) \quad \textrm{in}\: (0, 1)\medskip\\u(0)=u(1)=0\medskip\\u^{''}(0)=u^{''}(1)=0,\end{array} \right.$$ where $f:[0, 1]\times\R\to\R$ is a function, $B$ is a real constant and $\lambda$ is a positive parameter. The nonlinearity $f$ is required to have an oscillatory behaviour.