  In this paper, we study for the first time the stability of the inverse
source problem for the biharmonic operator with a compactly supported potential
in $\mathbb R^3$. Firstly, to connect the boundary data with the unknown
source, we shall consider an eigenvalue problem for the bi-Schr$\ddot{\rm
o}$dinger operator $\Delta^2 + V(x)$ on a ball which contains the support of
the potential $V$. We prove a Weyl-type law for the upper bounds of spherical
normal derivatives of both the eigenfunctions $\phi$ and their Laplacian
$\Delta\phi$ corresponding to the bi-Schr$\ddot{\rm o}$dinger operator. This
type of upper bounds was proved by Hassell and Tao for the Schr$\ddot{\rm
o}$dinger operator. Secondly, we investigate the meromorphic continuation of
the resolvent of the bi-Schr$\ddot{\rm o}$dinger operator and prove the
existence of a resonance-free region and an estimate of $L^2_{\rm comp} -
L^2_{\rm loc}$ type for the resolvent. As an application, we prove a bound of
the analytic continuation of the data from the given data to the higher
frequency data. Finally, we derive the stability estimate which consists of the
Lipschitz type data discrepancy and the high frequency tail of the source
function, where the latter decreases as the upper bound of the frequency
increases.
