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

TL;DR

This paper investigates conditions under which positive solutions do not exist for certain nonlinear biharmonic inequalities in exterior domains, using new methods due to the failure of the maximum principle for the biharmonic operator.

Contribution

It provides lower bounds on the growth of the nonlinearity that prevent positive solutions, introducing a novel approach based on a representation formula and pointwise bounds.

Findings

No positive solutions exist under specified growth conditions.

New representation formula for biharmonic inequalities.

Method applicable where the maximum principle fails.

Abstract

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 $\mathbb{R}^n$.