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

作者：Ghergu Marius; Taliaferro Steven D^*

来源：Communications in Partial Differential Equations, 2015, 40(6): 1029-1069.

DOI：10.1080/03605302.2015.1008364

• Nonexistence
• Supersolution
• Critical exponent
• Biharmonic equation
• 35A01
• 35B09
• 35J91
• 35R45
• 35B33

We study classical positive solutions of the biharmonic inequality
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in exterior domains in Double-struck capital R- n where f: (0, infinity) -> (0, infinity) is continuous function. We give lower bounds on the growth of f(s) at s = 0 and/or s = infinity such that inequality (0.1) has no C (4) positive solution in any exterior domain of Double-struck capital R- n . Similar results were obtained by Armstrong and Sirakov for - Delta 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 - Delta(2) u >= 0 in a punctured neighborhood of the origin in Double-struck capital R- n .

• 出版日期2015-6-3