Nonexistence of Positive Supersolutions of Nonlinear Biharmonic Equations without the Maximum Principle

in exterior domains in Rn where f : (0,∞) → (0,∞) is continuous function. We give lower bounds on the growth of f(s) at s = 0 and/or s = ∞ such that inequality (0.1) has no C positive solution in any exterior domain of Rn. 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 −∆v ≥ 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 −∆2u ≥ 0 in a punctured neighborhood of the origin in Rn.