  For an arbitrary nonempty, open set $\Omega \subset \mathbb{R}^n$, $n \in
\mathbb{N}$, of finite (Euclidean) volume, we consider the minimally defined
higher-order Laplacian $(- \Delta)^m\big|_{C_0^{\infty}(\Omega)}$, $m \in
\mathbb{N}$, and its Krein--von Neumann extension $A_{K,\Omega,m}$ in
$L^2(\Omega)$. With $N(\lambda,A_{K,\Omega,m})$, $\lambda > 0$, denoting the
eigenvalue counting function corresponding to the strictly positive eigenvalues
of $A_{K,\Omega,m}$, we derive the bound $$ N(\lambda,A_{K,\Omega,m}) \leq (2
\pi)^{-n} v_n |\Omega| \{1 + [2m/(2m+n)]\}^{n/(2m)} \lambda^{n/(2m)}, \quad
\lambda > 0, $$ where $v_n := \pi^{n/2}/\Gamma((n+2)/2)$ denotes the
(Euclidean) volume of the unit ball in $\mathbb{R}^n$.
  The proof relies on variational considerations and exploits the fundamental
link between the Krein--von Neumann extension and an underlying (abstract)
buckling problem.
