Nontrivial solutions of fourth-order singular boundary value problems with sign-changing nonlinear terms

In this paper, the fourth-order singular boundary value problem (BVP) $$ \aligned &u^{(4)}(t)=h(t)f(u(t)),\ \ t\in(0,1),\\ &u(0)=u(1)=u'(0)=u'(1)=0\endaligned $$ is considered under some conditions concerning the first characteristic value corresponding to the relevant linear operator, where $h$ is allowed to be singular at both $t=0$ and $t=1$. In particular, $f\colon (-\infty,\infty)\rightarrow (-\infty,\infty)$ may be a sign-changing and unbounded function from below, and it is not also necessary to exist a control function for $f$ from below. The existence results of nontrivial solutions and positive-negative solutions are given by the topological degree theory and the fixed point index theory, respectively