2014/07/16 by Ghergu, Marius, Taliaferro, Steven D.
#35B09 #35B33 #35J61 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1407.4506
We study classical positive solutions of the biharmonic inequality -Δ2 v ≥ f(v) 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 this inequality has no C4 positive solution in any exterior domain of \mathbb 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 -Δv≥ 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 -Δ2u ≥ 0 in a punctured neighborhood of the origin in ℝn.