Constant sign solutions of two-point fourth order problems

Abstract

In this paper we characterize the sign of the Green’s function related to the fourth order linear operator u(4) + M u coupled with the two point boundary conditions u(1) = u(0) = u′(0) = u′′(0) = 0. We obtain the exact values on the real parameter M for which the related Green’s function is negative in (0, 1) × (0, 1). Such property is equivalent to the fact that the operator satisfies a maximum principle in the space of functions that fulfil the homogeneous boundary conditions. When M > 0 the best estimate follows from spectral theory. When M < 0, we obtain an estimation by studying the disconjugacy properties of the solutions of the homogeneous equation u(4) + M u = 0. The optimal value is attained by studying the exact expression of the Green’s function. Such study allow us to ensure that there is no real parameter M for which the Green’s function is positive on (0, 1) × (0, 1). Moreover, we obtain maximum principles of this operator when the solutions verify suitable non-homogeneous boundary conditions. We apply the obtained results, by means of the method of lower and upper solutions, to nonlinear problems coupled with these boundary conditions. Keywords: Fourth order boundary value problem; Maximum principles; Lower and upper solutions