A local pointwise inequality for a biharmonic equation with negative exponents

TL;DR

This paper proves a local pointwise inequality for positive solutions of a biharmonic equation with negative exponents, removing the previously assumed growth condition and establishing the inequality under more general conditions.

Contribution

It provides a positive answer to an open question by establishing a local pointwise inequality without requiring the growth condition on solutions.

Findings

Established a local pointwise inequality for solutions of biharmonic equations with negative exponents.

Removed the necessity of the growth condition in deriving the inequality.

Extended the inequality to solutions in dimensions n ≥ 3 with q > 1.

Abstract

In this paper, we are inspired by Ng\^{o}, Nguyen and Phan's [15] study of the pointwise inequality for positive $C^{4}$-solutions of biharmonic equations with negative exponent by using the growth condition of solutions. They propose an open question of whether the growth condition is necessary to obtain the pointwise inequality. We give a positive answer to this open question. We establish the following local pointwise inequality $$-\frac{\Delta u}{u}+\alpha\frac{|\nabla u|^{2}}{u^{2}}+\beta u^{-\frac{q+1}{2}}\leq\frac{C}{R^{2}}$$ for positive $C^{4}$-solutions of the biharmonic equations with negative exponent $$-\Delta^{2}u=u^{-q} \ in \ B_{R}$$ where $B_{R}$ denotes the ball centered at $x_{0}$ with radius $R$, $n\geq3$, $q>1$, and some constants $\alpha\geq0$, $\beta>0$, $C>0$.