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 Δ 2 − n 2 ( n − 4 ) 2 16 q ( x ) | x | 4 under various assumptions on the perturbation q.
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