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

Abstract

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 Δ²-n²(n-4⁾²/16 q(x)/|x|⁴ under various assumptions on the perturbation q.