  We study the fourth order Schr\"odinger type differential inequality
$-\Delta^2 u + \lambda V(x)u \geq a(x)u^q$ with $a,V\in
L^1_{loc}(\mathbf{R}^N)$, both nonnegative, and $\lambda>0$. We consider
nonnegative solutions without making any assumptions about their asymptotic
behavior, and choosing appropriate test functions, we find a priori estimates
for some type of subharmonic solutions. Finally, applying these a priori
estimates, we proof a Liouville type result.
