Global Structure of Positive Solutions of Fourth-Order Problems with Clamped Beam Boundary Conditions

In this paper, we investigate the global structure of positive solutions of
$$ \begin{cases} u''''(x)=\lambda h(x)f(u(x)), & 0<x<1, \\ u(0)=u(1)=u'(0)=u'(1)=0,& \end{cases} $$
where $\lambda > 0$ is a parameter, $h\in C[0,1]$, $f\in C[0,\infty)$ and $f(s)>0$ for $s>0$. We show that the problem has three positive solutions suggesting suitable conditions on the nonlinearity. Furthermore, we also establish the existence of infinitely many positive solutions. The proof is based on the bifurcation method.