  We prove that operators of the form $A=-a(x)^2\Delta^{2}$, with $|D a(x)|\leq
c a(x)^\frac{1}{2}$, generate analytic semigroups in $L^p(\mathbb{R}^N)$ for
$1<p\leq\infty$ and in $C_b(\mathbb{R}^N)$. In particular, we deduce generation
results for the operator $A :=- (1+|x|^2)^{\alpha} \Delta^{2}$,
$0\leq\alpha\leq2$. Moreover, we characterize the maximal domain of such
operators in $L^p(\mathbb{R}^N)$ for $1<p<\infty$.
