Optimal Hardy–Rellich inequalities, maximum principle and related eigenvalue problem

In this paper we deal with three types of problems concerning the Hardy–Rellich's embedding for a bi-Laplacian operator. First we obtain the Hardy–Rellich inequalities in the critical dimension n=4. Then we derive a maximum principle for fourth order operators with singular terms. Then we study the existence, non-existence, simplicity and asymptotic behavior of the first eigenvalue of the Hardy–Rellich operator under various assumptions on the perturbation q.