Multiple solutions of a nonlinear elastic beam equation supported at two ends

Abstract

In this paper, several new existence theorems on pairs of solutions are obtained for the fourth order boundary value problem ${u^{(4)}}(t) = h(t,u(t))$ subject to $u(0) = u(1) = u''(0) = u''(1) = 0$. Further, if $h(t,u) = \mu f(t,u)$, we prove that there exists $ \mu ^ * > 0$ such that there is no solution to the above boundary value problem for $\mu > {\mu ^ * }$, and that there are multiple solutions of the above boundary value problem for $0 < \mu < {\mu^ * }$.