Three positive solutions of fourth-order problems with clamped beam boundary conditions

Abstract

We investigate the existence of three positive solutions of fourth order problems with clamped beam boundary conditions

$$\left\{\begin{array}{c}{u}^{\prime \prime \prime \prime }\left(x\right)=\lambda f\left(u\left(x\right)\right),x\in \left(-1,1\right),\hfill \\ u\left(1\right)=u\left(-1\right)=u^{\prime }\left(1\right)=u^{\prime }\left(-1\right)=0,\hfill \end{array}\right.$$

where $\lambda >0$ is a parameter, $f\in C\left[0,\infty \right)$ is nondecreasing, $f\left(0\right)=0$, $f\left(s\right)>0$ for all $s>0$. Especially, we obtain an S-shaped unbounded connected component in the symmetric positive solutions set. The proof is based on the directions of a bifurcation.