Existence and location results for hinged beam equations with unbounded nonlinearities

Abstract

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 in [0; 1], where f and p 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.