Universal gradient estimates of Δu + a(x)up(ln⁡(u+c))q = 0 on complete Riemannian manifolds

Abstract

In this paper, we study the elliptic non-linear equation $\Delta u+a\left(x\right)u^p{(\ln (u+c\left)\right)}^q=0$ on a complete Riemannian manifold with Ricci curvature bounded from below. By applying Nash-Moser iteration, we establish universal gradient estimates for positive solutions to the equation, where $c\ge 1$ and $a\left(x\right)$ is allowed to change sign. As an application, we obtain Liouville theorems when the manifold has non-negative Ricci curvature and $a\left(x\right)$ is constant.