  We study classical positive solutions of the biharmonic inequality
  $-\Delta^2 v \geq f(v)$
  in exterior domains in $\mathbb{R}^n$ where $f:(0,\infty)\to (0,\infty)$ is
continuous function. We give lower bounds on the growth of $f(s)$ at $s=0$
and/or $s=\infty$ such that this inequality has no $C^4$ positive solution in
any exterior domain of $\mathbb R^n$. Similar results were obtained by
Armstrong and Sirakov [ Nonexistence of positive supersolutions of elliptic
equations via the maximum principle, Comm. Partial Differential Equations 36
(2011) 2011-2047] for $-\Delta v\ge f(v)$ using a method which depends only on
properties related to the maximum principle. Since the maximum principle does
not hold for the biharmonic operator, we adopt a different approach which
relies on a new representation formula and an a priori pointwise bound for
nonnegative solutions of $-\Delta^2u \ge 0$ in a punctured neighborhood of the
origin in $\mathbb{R}^n$.
