A Pointwise Inequality for the Fourth Order Lane-emden Equation

We prove that the following pointwise inequality holds −∆u ≥ √ 2 (p + 1)− cn |x| a 2 u p+1 2 + 2 n− 4 |∇u|2 u in R where cn := 8 n(n−4) , for positive bounded solutions of the fourth order Hénon equation that is ∆u = |x|u in R where a ≥ 0 and p > 1. Motivated by the Moser iteration argument 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 toward constructing pointwise inequalities for partial differential equations. An interesting point is that the the coefficient 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 4 n−4 is positive.