  Let $n\in\mathbb{N}, n\geq 2$. We prove that the strongly singular
differential operator \[\left(\Delta^2
+c|x|^{-4}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}, \quad c
\in \mathbb{R}, \] is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if
and only if \[c\geq \begin{cases}3(n+2)(6-n)&\mbox{for $2\leq n\leq 5$};\\[5pt]
{\displaystyle -\frac{n(n+4)(n-4)(n-8)}{16}}&\mbox{for $n\geq 6$}.\end{cases}\]
  Via separation of variables, our proof reduces to studying the essential
self-adjointness on the space $C_0^{\infty}((0,\infty))$ of fourth-order
Euler-type differential operators of the form \[
\frac{d^4}{dr^4}+c_1\left(\frac{1}{r^2}\frac{d^2}{dr^2}+\frac{d^2}{dr^2}\frac{1}{r^2}\right)+\frac{c_2}{r^4},\quad
r\in(0,\infty),\quad(c_1,c_2)\in \mathbb{R}^2,\] in $L^2((0,\infty);dr)$.
  Our methods generalize to differential operators related to higher-order
powers of the Laplacian, however, there are some nontrivial subtleties that
arise. For example, the natural expectation that for $m,n\in\mathbb{N}$, $n
\geq 2$, there exist $c_{m,n}\in\mathbb{R}$ such that
$\left(\Delta^m+c|x|^{-2m}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash
\{0\})}$ is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if and only
if $c \geq c_{m,n}$, turns out to be false. Indeed, for $n=20$, we prove that
the differential operator \[
\left((-\Delta)^5+c|x|^{-10}\right)\big|_{C_0^{\infty}(\mathbb{R}^{20}
\backslash \{0\})}, \quad c \in \mathbb{R},\] is essentially self-adjoint in
$L^2\big( \mathbb{R}^{20}; d^{20} x\big)$ if and only if $c\in [0,\beta]\cup
[\gamma,\infty)$, where $\beta\approx 1.0436\times 10^{10}$, and $\gamma\approx
1.8324\times 10^{10}$ are the two real roots of the quartic equation
\begin{align*}&3125z^4-83914629120000z^3+429438995162964368031744
z^2\\&\quad+1045471534388841527438982355353600z\\&\quad
+629847004905001626921946285352115240960000=0.\end{align*}
