A nonlinear boundary value problem associated with the static equilibrium of an elastic beam supported by sliding clamps (Q1826009)

Summary: The fourth-order boundary value problem \(d^ 4u/dx^ 4+f(x)u=e(x)\), \(0<x<\pi\); \(u'(0)=u'(\pi)=u'''(0)=u'''(\pi)=0\); where \(f(x)\leq 0\) for \(0\leq x\leq \pi\), describe the unstable static equilibrium of an elastic beam which is supported by sliding clamps at both ends. This paper concerns the nonlinear analogue of this boundary value problem, namely, \(-(d^ 4u/dx^ 4)+g(x,u)=e(x),\) \(0<x<\pi\), \(u'(0)=u'(\pi)=u'''(0)=u'''(\pi)=0\), where \(g(x,u)u\geq 0\) for a.e. x in \([0,\pi]\) and all \(u\in {\mathbb{R}}\) with \(| u|\) sufficiently large. Some resonance and nonresonance conditions on the asymptotic behavior of \(u^{-1}g(x,u)\), for \(| u|\) sufficiently large, are studied for the existence of solutions of this nonlinear boundary value problem.