Existence and Multiplicity of Constant Sign Solutions for One-Dimensional Beam Equation

Abstract

In this paper, we consider the nonlinear eigenvalue problems

$$\begin{aligned} \begin{aligned}&u''''=\lambda h(t)f(u), \quad 0<t<1, \\&u(0)=u(1)=u'(0)=u'(1)=0, \end{aligned} \end{aligned}$$

where $h\in C([0,1], (0,\infty ))$; $f\in C({\mathbb {R}},{\mathbb {R}})$ and $sf(s)>0$ for $s\ne 0$, and $f_0=f_\infty =0$, $f_0=\lim _{|s|\rightarrow 0}f(s)/s, \; f_\infty =\lim _{|s|\rightarrow \infty }f(s)/s$. We investigate the global structure of one-sign solutions by using bifurcation techniques.