A local pointwise inequality for a biharmonic equation with negative exponents

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 $$-\frac{\mathrm{\Delta }u}{u}+\alpha \frac{\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2}{u^2}+\beta u^{-\frac{q+1}{2}}\le \frac{C}{R^2}$$ for positive $C^4$-solutions of the biharmonic equations with negative exponent $$-{\mathrm{\Delta }}^{2}u=u^{-q}in{B}_R$$ where $B_R$ denotes the ball centered at $x_0$ with radius $R$, $n\ge 3$, $q>1$, and some constants $\alpha \ge 0$, $\beta >0$, $C>0$.