Nonexistence of Positive Supersolutions of Nonlinear Biharmonic Equations without the Maximum Principle Academic Article uri icon

abstract

  • © 2015, Taylor & Francis Group, LLC. We study classical positive solutions of the biharmonic inequality (Formula presented.) 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 C4 positive solution in any exterior domain of ℝn. Similar results were obtained by Armstrong and Sirakov 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 − Δ2 u ≥ 0 in a punctured neighborhood of the origin in ℝn.

author list (cited authors)

  • Ghergu, M., & Taliaferro, S. D.

citation count

  • 3

publication date

  • March 2015