A pointwise inequality for the fourth order Lane-Emden equation

Abstract

We prove that the following pointwise inequality holds \begin{equation*} -\Delta u \ge \sqrt\frac{2}{(p+1)-c_n} |x|^{\frac{a}{2}} u^{\frac{p+1}{2}} + \frac{2}{n-4} \frac{|\nabla u|^2}{u} \ \ \text{in}\ \ \mathbb{R}^n \end{equation*} where $c_n:=\frac{8}{n(n-4)}$, for positive bounded solutions of the fourth order H\'{e}non equation that is \begin{equation*} \Delta^2 u = |x|^a u^p \ \ \ \ \text {in }\ \ \mathbb{R}^n \end{equation*} for some $a\ge0$ and $p>1$. Motivated by the Moser's proof of the Harnack's inequality as well as Moser iteration type arguments in the regularity theory, we develop an iteration argument to prove the above pointwise inequality. As far as we know this is the first time that such an argument is applied towards constructing pointwise inequalities for partial differential equations. An interesting point is that the coefficient $\frac{2}{n-4}$ also appears in the fourth order $Q$-curvature and the Paneitz operator. This in particular implies that the scalar curvature of the conformal metric with conformal factor $u^\frac{4}{n-4}$ is positive.