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 2 − n2(n−4)2 16 q(x) |x|4 under various assumptions on the perturbation q. © 2006 Elsevier Inc. All rights reserved.