Existence and location results for hinged beam equations with unbounded nonlinearities

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) for x ∈ [0, 1], where f : [0, 1] × R4 → R and p : [0, 1] → R are continuous functions and s is a real parameter, with the Lidstone boundary conditions u(0) = u(1) = u(0) = u(1) = 0. 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. © 2009 Elsevier Ltd. All rights reserved.