  We study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann
extension of the perturbed Laplacian $-\Delta+V$ defined on
$C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a
bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth
domains which contains all convex domains, along with all domains of class
$C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish
the Weyl asymptotic formula \[ #\{j\in\mathbb{N} |
\lambda_{K,\Omega,j}\leq\lambda\} = (2\pi)^{-n} v_n |\Omega|
\lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) {as} \lambda\to\infty, \] where
$v_n=\pi^{n/2}/ \Gamma((n/2)+1)$ denotes the volume of the unit ball in
$\mathbb{R}^n$, and $\lambda_{K,\Omega,j}$, $j\in\mathbb{N}$, are the non-zero
eigenvalues of $H_{K,\Omega}$, listed in increasing order according to their
multiplicities. We prove this formula by showing that the perturbed Krein
Laplacian (i.e., the Krein--von Neumann extension of $-\Delta+V$ defined on
$C^\infty_0(\Omega)$) is spectrally equivalent to the buckling of a clamped
plate problem, and using an abstract result of Kozlov from the mid 1980's. Our
work builds on that of Grubb in the early 1980's, who has considered similar
issues for elliptic operators in smooth domains, and shows that the question
posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl
asymptotic formula continues to have an affirmative answer in this nonsmooth
setting.
