Positive solutions of fourth order problems with clamped beam boundary conditions

In this paper we make an exhaustive study of the fourth order linear operator u(4)+Mu coupled with the clamped beam conditions u(0)=u(1)=u′(0)=u′(1)=0u(0)=u(1)=u′(0)=u′(1)=0. We obtain the exact values on the real parameter MM for which this operator satisfies an anti-maximum principle. Such a property is equivalent to the fact that the related Green’s function is nonnegative in [0,1]×[0,1][0,1]×[0,1]. When M<0M<0 we obtain the best estimate by means of the spectral theory and for M>0M>0 we attain the optimal value by studying the oscillation properties of the solutions of the homogeneous equation u(4)+Mu=0.By using the method of lower and upper solutions we deduce the existence of solutions for nonlinear problems coupled with this boundary conditions.