Existence and location results for hinged beam equations with unbounded nonlinearities Original Research Article

This work presents some existence, non-existence and location results for the problem composed by the fourth-order fully nonlinear equation u(4)(x)+f(x,u(x),u′(x),u″(x),u‴(x))=sp(x)u(4)(x)+f(x,u(x),u′(x),u″(x),u‴(x))=sp(x) Turn MathJax on for x∈[0,1]x∈[0,1], where f:[0,1]×R4→Rf:[0,1]×R4→R and p:[0,1]→R+p:[0,1]→R+ are continuous functions and ss is a real parameter, with the Lidstone boundary conditions u(0)=u(1)=u″(0)=u″(1)=0.u(0)=u(1)=u″(0)=u″(1)=0. Turn MathJax on This problem models several phenomena, such as, the bending of an elastic beam simply supported at the endpoints. The arguments used apply a lower and upper solutions technique, a priori estimations and topological degree theory. In this paper we replace the usual bilateral Nagumo condition by some one-sided conditions, which enables us to consider unbounded nonlinearities.