Positive solutions of fourth order problems with clamped beam boundary conditions Original Research Article

In this paper we make an exhaustive study of the fourth order linear operator View the MathML sourceu(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 View the MathML sourceu(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.