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For an arbitrary nonempty, open set $\Omega \subset \mathbb{R}^n$, $n \in\mathbb{N}$, of finite (Euclidean) volume, we consider the minimally definedhigher-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 theeigenvalue counting function corresponding to the strictly positive eigenvaluesof $A_{K,\Omega,m}$, 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 fundamentallink between the Krein--von Neumann extension and an underlying (abstract)buckling problem.

A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets