Positive solutions of fourth order problems with clamped beam boundary conditions (Q534454)

The authors study the fourth order linear operator \(u^{(4)} + M u\) coupled with the clamped beam conditions \(u(0) = u(1) = u'(0) = u'(1) = 0\). They obtain the exact values of the real parameter \(M\) for which this operator satisfies an anti-maximum principle. When \(M < 0\), they obtain the best estimate by means of the spectral theory and, for \(M > 0\), they attain the optimal value by studying the oscillation properties of the solutions of the homogeneous equation \(u^{(4)} + M u = 0\). By using the method of lower and upper solutions, they also prove the existence of solutions of nonlinear problems coupled with these boundary conditions.